diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhsmq" "b/data_all_eng_slimpj/shuffled/split2/finalzzhsmq"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhsmq"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nFirst notions of \\textit{artificial neural networks (ANNs)} -- in the context of supervised learning -- date back to the early 1940s~\\cite{McCulloch1943}. Although considerable research has been conducted in the succeeding decades, people had to wait for the emergence of powerful computers, and the accompanying (significant) drop in costs for storing and processing data, until ANNs turned into highly competitive machine learning algorithms~\\cite{GoodfellowBengio2016,hastie01statisticallearning}. Nowadays, ANNs represent the state of the art in a variety of supervised learning tasks, such as image, sound or object recognition \\cite{HeZRS16,SimonyanZ14,AbdelHamidMJDPY14}.\n\n\nHowever, while the strength of ANNs undoubtedly results from their huge amount of parameters, their enormous complexity also makes them rather incomprehensible. \nAlthough numerous works have introduced methods for interpreting the decisions made by ANNs \\cite{nguyen2016synthesizing,erhan2009visualizing,zhou2016learning}, these methods lack explanations for malicious and intentional behaviour against ANNs.\n\nOne possibility for fooling ANNs is the addition of carefully crafted noise, so-called \\textit{adversarial perturbations} (cf. Figure~\\ref{fig:adv_example}), to the model's input \\cite{SzegedyZSBEGF13,goodfellow2014,GuR14,TabacofV15,FawziMF16,CarliniW16,Carlini017,KurakinGB17a}. It should be noted that these perturbations do not need to be crafted individually per network; instead, ANNs that are trained for a similar task often can be fooled by the same adversaries \\cite{SzegedyZSBEGF13,goodfellow2014,PapernotMGJCS16}.\n\n\\newpage\nWhile a plethora of works have shown the advantages of (deep learning) neural networks, their black-box characteristic makes them very vulnerable. Hence, in order to enable a more frequent and, even more importantly, more secure integration of neural networks in our daily lives, we need to ensure their robustness against adversarial attacks. \n\nThe issue of a network's vulnerability and how to defend it against adversarial attacks will be investigated in this work. As such, our contributions can be summarized as follows:\n\\begin{compactenum}\n    \\item We present a \\textbf{novel method to craft transferable adversaries}, which in turn helps to improve the understanding of a network's vulnerability against adversarial attacks.\n    \\item We introduce a novel approach to regularize decision boundaries and, thus, \\textbf{enhance the resilience of neural networks against adversarial attacks}. Compared to previous attempts, our approach does not require additional backward-passes, which can decrease training speed significantly (cf. Table \\ref{tab:training_time}).\n\\end{compactenum}\n\\begin{figure}[t!]\n\\centering\n    \\begin{overpic}[width=0.95\\linewidth, trim = 2mm 0mm 0mm 0mm, clip]{figures\/example_adver.png}\n        \\put (31.5,11) {\\large{$+~\\varepsilon~\\cdot$}}\n        \\put (70,11) {\\large{$=$}}\n    \\end{overpic}\n   \n    \\caption{Example for an adversary (from left to right): the original image (classified as \\textit{great panda} with a confidence of 99.99\\%), adversarial noise with $\\varepsilon=3$ and the resulting adversarial image (classified as \\textit{saxophone} with a confidence of 83.8\\%).}\n  \\label{fig:adv_example}\n\\end{figure}\n\nThe remainder of this manuscript is structured as follows. In Section~\\ref{sec:related} we set our work into context. We then introduce a method for crafting transferable adversaries in Section~\\ref{sec:transfer}, and list possible defense strategies against adversarial attacks in Section~\\ref{sec:counteractions}. Thereafter, we provide our experimental setup in Section~\\ref{sec:experiments} and discuss our findings in Section~\\ref{sec:results}. At last, Section~\\ref{sec:conclusion} concludes our work.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\nSeveral methods to craft adversarial examples have been published in recent years \\cite{goodfellow2014,MoosaviDezfooli15,Carlini017,madry2018towards}. In principle, one can categorize these methods into \\textit{single-} and \\textit{multi-step} attacks: the former only require a single gradient while the latter calculate several gradients. Kurakin et al.~\\cite{KurakinGB16a} found in their work that multi-step adversaries are less transferable than single-step ones and they concluded that models are indirectly robust against multi-step adversaries. Yet, in line with the findings of \\cite{dong2018boosting}, we observed that the transfer of adversaries between models is rather easy when using a so-called \\emph{ensemble attack}, i.e., a multi-step attack, which is executed on an ensemble of models simultaneously \\cite{dong2018boosting}.\n\nAs generating adversarial perturbations is rather simple, we will at first address the characteristics of these perturbations along with their accompanying risks of black-box attacks (see Section~\\ref{sec:charact}). Note that in contrast to white-box methods, black-box approaches have no access to the model's gradient. Thereafter, in Section~\\ref{sec:resil}, we briefly outline selected strategies to enhance resilience.\n\n\\subsection{Characteristics of Adversaries}\n\\label{sec:charact}\nSzegedy et al.~\\cite{SzegedyZSBEGF13} crafted artificial noise, which causes ANNs to misclassify images whose original versions they classified correctly. As a result, it could be demonstrated that already slight changes, which are hardly recognizable to the human eye, are sufficient to fool the network. As further demonstrated by \\cite{SzegedyZSBEGF13,goodfellow2014}, and also in this work, adversarial perturbations are not model-specific, but instead generalize well over networks with different hyper-parameters. In addition, even networks trained on disjoint data sets, yet fulfilling a similar classification task, are likely vulnerable to the same adversarial image \\cite{goodfellow2014,PapernotMGJCS16}. Thus, two models with different architectures, hyper-parameters, and trained on different data sets, often misclassify the same images. This property can be used to execute black-box attacks as demonstrated by \\cite{PapernotMGJCS16}. \n\nSeveral works have been conducted to examine the effects of adversaries. The neighborhoods of adversarial examples and their original counterparts were investigated by \\cite{TabacofV15}. The authors found that while already small and random noise applied to adversaries often caused it to be shifted back to their original class, the classification of the original images did not change (even when rather large random noise was applied). Therefore, the authors concluded that adversarial examples likely inhabit small sub-spaces within the decision space.\n\nNext, \\cite{tramer2017space} investigated these subspaces by measuring the number of orthogonal attack directions and found that the number of these directions is rather large. \nThey concluded that the more orthogonal attack directions exist at a given point, the larger these adversarial subspaces are and the more likely adversaries transfer between models (since larger subspaces are more likely to intersect between different models).\nThis may be the explanation on \\textit{why} adversarial perturbations are often universal across different model designs and often universal across datasets.\n\n\\subsection{Enhancing Resilience through Adversarial Training}\n\\label{sec:counteractions}\n\\begin{figure}[t!]\n\\flushright\n    \\begin{overpic}[width=0.92\\columnwidth]{figures\/example_alpha.png}\n        \\put (-8,11) {{$(1-\\alpha)~\\cdot$}}\n        \\put (29.5,11) {{$+~\\alpha~\\cdot$}}\n        \\put (70,11) {{$=$}}\n    \\end{overpic}\n   \n\t\\caption{Idea of our proposed method: linear combination of the original image (\\textit{great panda}) with a randomly selected image (\\textit{saxophone}) using weight $\\alpha \\in (0, 0.5)$ (here: $\\alpha = 0.3$). The resulting training image (right) inherits the class of the original image. This allows us to generate, on the one hand, more training examples and, on the other one, training examples which lie in close proximity to the decision boundaries.}\n  \\label{fig:alpha_example}\n\\end{figure}\n\\label{sec:resil}\n\nDefense strategies against adversarial attacks can mainly be distinguished into two categories: approaches that either change (i) a model's topology, or (ii) the way a model is trained. Gu and Rigazio~\\cite{GuR14} stated that a model's robustness \\enquote{is more related to intrinsic deficiencies in the training procedure and objective function than to [its] topology}.\n\n\n\nUnder the concept of \\textit{Adversarial Training}, several works have introduced regularizers which decrease a model's vulnerability to adversarial perturbations. The first one was introduced by Goodfellow et al.~\\cite{goodfellow2014}. Here, the objective function of the \\textit{Fast Gradient Sign Method} (FGSM) \\cite{goodfellow2014} is added to the classification loss.\nFor every training image, a single-step adversary is calculated and the network's sensitivity to this adversary is minimized. Thus, \\textit{Adversarial Training} acts as a regularizer to improve resilience \\cite{goodfellow2014,KurakinGB16a}.\n\n\nThe training method of \\cite{goodfellow2014} only uses a single back-propagation step to calculate adversarial perturbations. In contrast to the aforementioned approach, other works introduced a min-max optimization method \\cite{Huang2015LearningWA,shaham2015}. It prepares the network for the worst-case scenario under a limited perturbation and minimizes its sensitivity to it. The algorithm works in two steps. First, it maximizes the loss of the input image by adding a small adversarial perturbation. However, the network trained with a min-max regularizer performed worse on the classification tasks \\cite{Huang2015LearningWA} -- even when trained with rather small perturbations. Due to the additional maximization problem, the algorithm is even more complex to execute, which further increases training time.\n\nSimilar approaches were introduced by \\cite{Miyato2015DistributionalSW,madry2018towards}. The approach of \\cite{Miyato2015DistributionalSW}, denoted \\textit{Virtual Adversarial Training} (VAT), works in a similar way as the min-max optimization, yet it maximizes the Kullback-Leibler divergence \\cite{Kullback51klDivergence} between the probability vectors of the training image and an adversary. The KL-divergence between the original and an adversary can be interpreted as the local smoothness of the decision space. Thus, the network not only learns to classify the training images correctly, but also smoothens the decision boundaries around each of the training points. Miyato et al.~\\cite{Miyato2015DistributionalSW} used \\textit{Projected Gradient Descent} (PGD) to solve the inner maximization problem. The authors could demonstrate a guaranteed robustness even against multi-step adversaries under small $l_\\infty$-norm -- with $l_\\infty = max\\{|x~-~x^{adv}|\\}$ -- where $x$ is the unchanged input image while $x^{adv}$ is the image with additional adversarial perturbation. Multi-step black- and white-box attacks have been conducted under a $l_\\infty\\leq8$-constraint. Even under multi-step white-box attacks, the accuracy of their model is 45\\%, whereas the accuracy of a non-robustified, and thus defenseless, model would be close to zero. However, as there are additional forward-backward passes required, training time takes accordingly longer.\n\n\\section{Crafting Strong and Transferable Adversaries}\n\\label{sec:transfer}\n\\begin{figure}[t!]\n\\centering\n\t\\includegraphics[width=\\columnwidth, trim = 0mm 8mm 0mm 0mm, clip]{figures\/example_pic.png}\n\t\\caption[Random Images from the Dataset]{Exemplary images from the modified \\textit{Large Scale Visual Recognition Challenge 2017} dataset: Seal, Bear, Dragonfly and Rabbit (left to right).}\n\t\\label{fig:example_pic}\n\\end{figure}\n\n\nGoodfellow et al.~\\cite{goodfellow2014} introduced the \\textit{Fast Gradient Sign Method} (FGSM):\n\\begin{align}\n    x^{adv} &= x + \\varepsilon \\cdot \\text{sign} \\left(\\nabla_{x}~L(\\Theta, x, y)\\right) \\label{eq:fgsm}\n\\end{align}\nThe loss function\\footnote{In this work, we refer to the cross-entropy as loss function. Yet, any other function can most likely be used, too.} $L$ of the model $\\Theta$ is differentiated with respect to the input image $x$ and its true class $y$. The sign of the resulting gradient $\\nabla_{x}~L$ is used to calculate adversarial noise which, when applied to an image, increases the model's loss and, thereby, could shift its classification decision. The parameter $\\varepsilon$ is used to control the amount of perturbations. Note that a larger $\\varepsilon$ does not necessarily lead to a higher chance of fooling a model as demonstrated by Figure \\ref{fig:loss}. Instead, tuning $\\varepsilon$ is highly important. The resulting adversary is a single-step one.\n\nOne can execute the FGSM multiple times with a small $\\varepsilon$-value to converge towards the adversarial sub-space iteratively. The resulting adversaries are then referred to as multi-step ones. In order to find multi-step adversaries under limited amount of perturbations and close to the original image, several methods such as \\cite{MoosaviDezfooli15,Carlini017,madry2018towards,KurakinGB16a} have been introduced. When the perturbation exceeds a certain limit -- usually the $l_\\infty$- or $l_2$-norm of the adversarial perturbation is used -- the perturbation is projected back into the search space, e.g., by using pixel-wise clipping.\n\nAs pointed out by \\cite{TabacofV15,tramer2017space}, adversaries lie in sub-spaces which are in close proximity to true training examples and which often intersect between models. One can argue that when crafting adversaries with multi-step methods, these adversaries often do not lie in intersecting sub-spaces because the resulting adversary may be overfitted to the particular model as its loss is maximized. To overcome this issue, \\cite{dong2018boosting} introduced the (i) \\textit{Momentum Iterative Gradient Based Method} (MI-FGSM) and (ii) the \\textit{ensemble in logits}. The former method uses a momentum to overcome local maxima and the latter one combined the logits of an ensemble of models to find a common and intrinsic direction. Attacking an ensemble of models increases the likelihood of transferable adversaries as these adversaries are not optimized for a single model but for several models simultaneously. Hence, the adversaries lie more likely in intersecting adversarial sub-spaces. \n\nIn contrast to \\cite{dong2018boosting}, we used the combined gradient of an ensemble of models to craft adversaries which lie in intersecting sub-spaces. We call our method \\textit{gradient ensemble attack}. We found that these adversaries are likely to transfer between models of the same architecture -- and frequently transfer to other architectures as well (see Section~\\ref{sec:results}). To ensure that every model's gradient has the same impact on the resulting adversarial perturbation, we normalize each gradient to length one:\n\\begin{align}\n    \\hat{\\nabla}_{x}~L(\\Theta, x, y) &= \\frac{\\nabla_{x}~L(\\Theta, x, y)}{||\\nabla_{x}~L(\\Theta, x, y)||_2}. \\label{eq:grad_norm}\n\\end{align}\nAfterwards, the different gradients are summed up to find a common direction of intersecting adversarial sub-spaces. Similar to \\cite{KurakinGB16a}, we approximated the true direction by using the sign-method of the summed and normalized gradients to the image in an iterative process:\n\\begin{eqnarray}\n x^{adv}_{i} &=& x_{i-1} + \\lambda \\cdot \\text{sign} \\left(\\sum_{n=0}^N~\\hat{\\nabla}_{x}~L(\\Theta_n, x_{i-1}, y) \\right) \\label{eq:adv_methode}\\\\\n  \\text{s.t.} && ||x^{adv}_{i} - x||_{\\infty}~\\leq~\\varepsilon \\quad \\text{and} \\quad x_0 = x.\\nonumber\n\\end{eqnarray}\nTo ensure that the magnitude of the perturbations stays within a given limit, we used pixel-wise clipping. Further, we used a learning rate of $\\lambda=1$ (to slowly convert towards the adversarial spot) and set the number of steps to $I=\\min(\\varepsilon + 4, 1.25\\cdot\\varepsilon)$ as proposed by \\cite{KurakinGB16a} to craft adversarial images. In addition, we chose $\\varepsilon\\in\\{4,8,16\\}$ to limit the amount of perturbations in order to test and compare different magnitudes of adversarial perturbations.\n\n\\section{Proposed Defense Strategies}\nAs demonstrated by \\cite{TramerKPGBM18} and \\cite{athalye18a}, adversarial training may lead to \\emph{gradient masking}. This term describes a phenomenon in which the decision boundaries are roughly the same, yet the corresponding gradient is damaged or obfuscated in a way that white-box attacks fail. However, the model does not become more resilient against adversarial examples in black-box attack scenarios or against transferable adversaries as these attacks are based on surrogate models.\n\nIn order to avoid the risk of gradient masking, we propose a method that does not require gradient calculations and still flattens the decision space. In addition, we avoid the expensive optimization of an inner maximization problem as done in VAT or PGD. We found that while random noise is mostly ignored by the model, superimposing two pictures does distract a model. Therefore, as illustrated in Figure~\\ref{fig:alpha_example}, we designed training images $\\tilde{x}_i$ by placing a randomly selected image $x_r$ on top of the original training image $x_i$:\n\\begin{align}\n    \\tilde{x}_i = (1-\\alpha) \\cdot x_i + \\alpha \\cdot x_r. \\label{eq:alpha_method}\n\\end{align}\nThe parameter $\\alpha \\in (0, 0.5)$ controls the impact of image $x_r$ on the generated image $\\tilde{x}_i$. As the majority of the image originates from the original image $x_i$, the generated image $\\tilde{x}_i$ will inherit the class label $y_i$ of the original image. Our proposed approach thus allows to (i) generate \\emph{more training examples}, and (ii) create images that are closer to the decision boundaries of at least two classes -- and thus \\emph{harder to distinguish} -- as $\\tilde{x}_i$ contains properties of two image classes. Thereby, the space between the different classes is flattened and the boundaries are regularized. Thereafter, the networks will be trained by minimizing the loss $L(\\Theta,\\tilde{x}_i, y_i)$.\nAs the results depend on the choice of $\\alpha$, we considered three different approaches:\n\\begin{compactenum}\n\\item Using a fixed $\\alpha$-value.\n\\item Following the idea of \\cite{Miyato2015DistributionalSW}, i.e., first predicting the classes $y_i$ and $\\tilde{y}_i$ based on $x_i$ and $\\tilde{x}_i$ (and using a fixed $\\alpha$ as in 1.), and then minimizing the loss of the unmodified training examples \\emph{and} the KL-Divergence between the predicted classes of the unmodified and modified training examples:\n\\[\\underset{\\Theta}{\\text{minimize}} \\quad L(\\Theta, x_i, y_i) + \\lambda \\cdot KL(\\hat{y}_i~||~\\tilde{y}_i).\\]\n\\item Similarly to 2., but this time drawing $\\alpha$ from a beta-distribution instead of a fixed $\\alpha$-value; $\\alpha \\sim B(p,q)$ with $p=2$ and $q \\in \\{4,6,10\\}$.\n\\end{compactenum}\nNote that while the first method does not require any additional passes, the latter two methods require a second forward-pass to calculate the KL-Divergence -- but no additional backward-pass. As stated by \\cite{Miyato2015DistributionalSW}, the KL-divergence can be interpreted as the local smoothness of the decision space. If the divergence is high, the predictions of both input images are dissimilar, implying that the decision space between the two activations is not flattened. By minimizing the divergence, the models learn to find similar activations for both images and, thereby, flatten the decision space between the two activations.\n\nAs a side effect, our method allows to generate a multitude of training images (using Equation~(\\ref{eq:alpha_method})), which in turn allows to train a larger area.\n\\begin{figure}[t!]\n\\centering\n\t\\includegraphics[width=0.95\\columnwidth]{figures\/acc_over_time.png}\n\t\\caption[Validation Accuracy during training]{Visualization of the validation accuracy during training. The \\textit{Projected Gradient Descent} \\cite{madry2018towards} substantially reduces the overall accuracy compared to the \\textit{Base Model}, our proposed method, or the \\textit{Virtual Adversarial Training} \\cite{Miyato2015DistributionalSW}. Furthermore, the model $\\alpha \\sim B(2, 4)$, $KL_{\\lambda=10}$ achieves an accuracy of more than 70\\% with fewer methods than the \\textit{Base Model}.}\n\t\\label{fig:acc_druing_training}\n\\end{figure}\n\n\\section{Methodology and Experiments}\n\\label{sec:experiments}\nIn the following, we briefly outline the models' topologies and used training data.\n\n\\subsection{Dataset}\nWithin our experiments, we used the data from the second challenge of the \\textit{Large Scale Visual Recognition Challenge 2017}\\footnote{\\url{http:\/\/image-net.org\/challenges\/LSVRC\/2017\/}} \\cite{ILSVRC15} for training and validating our models. \nYet, as down-scaling the whole images to the required size was not reasonable, we first used bounding boxes to cut out the objects within each of the images. Then, as the bounding boxes were of different sizes, the cropped images were down-scaled to the desired resolution of 128 $\\times$ 128 pixels (cf. Figure \\ref{fig:example_pic}).\n\nAs a result, our dataset contains roughly 400,000 images. Unfortunately, the images are not uniformly distributed across all 200 classes. As the mismatch between the classes is too large to use oversampling, we selected 100 classes with an identical number of images per class. For training and validation purposes, we performed holdout with a 80-20-split. Thereby, we created a dataset which is larger and has a higher resolution than CIFAR10\/100~\\cite{krizhevsky2009learning} while (at the same time) being significantly smaller and less complex than ImageNet~\\cite{ILSVRC15}. In addition, by using balanced classes, we eliminate any side-effects which may occur from unbalanced datasets and may influence our results.\n\n\\subsection{Models}\n\\begin{table}[!t]\n    \\centering\n    \\caption{Measured training Time for different models (all models were trained on a single Nvidia Quadro RTX 6000). Training a single epoch with our proposed method takes only slightly longer than the \\textit{Base Model} without any defense. In addition, $\\alpha \\sim B(2, 4)$, $KL_{\\lambda=10}$ reaches 60\\% and 70\\% Validation Accuracy about 30 minutes earlier than the \\textit{Base Model} as it converges faster (cf. Figure \\ref{fig:acc_druing_training}). Training models using the \\textit{Projected Gradient Descent} \\cite{madry2018towards} or \\textit{Virtual Adversarial Training} \\cite{Miyato2015DistributionalSW} takes noticeably longer per epoch, and they take even longer to converge.}\n    \\label{tab:training_time}\n   \n   \n        \\begin{tabular}{l|cccc}\n                       & Median Time & Time until & Time until & Time until \\\\\n            Model      & p. Epoch    & 50\\% Acc.  & 60\\% Acc.  & 70\\% Acc. \\\\\n            \\midrule\n            \\textit{Base} &  \\cellcolor{gray!25}\\textbf{6:43m} & \\cellcolor{gray!25}\\textbf{00:59h} & 02:15h & 03:17h \\\\\n            $\\alpha=0.4$   & 6:55m & 02:18h & 03:14h & -\/- \\\\\n            $\\alpha \\sim B(2, 4)$ & 6:51m & 01:01h & \\cellcolor{gray!25}\\textbf{01:42h} & \\cellcolor{gray!25}\\textbf{02:44h} \\\\\n            VAT ($\\varepsilon=15$) & 14:45m & 02:12h & 03:55h & 08:05h \\\\\n            PGD ($I=7$) & 22:40m & 11:46h & 15:56h & -\/- \\\\\n            PGD ($I=3$) & 11:37m & 03:53h & 07:45h & -\/- \n        \\end{tabular}\n   \n   \n\\end{table}\nTo investigate the effects of adversarial perturbations, we trained multiple ANNs. First, an ``unprotected'' model -- denoted \\textit{base model} in the following -- is trained. For this purpose, we took a VGGNet~\\cite{SimonyanZ14}, as it provides a straightforward and uncluttered design of convolutions, pooling and dense layers. In addition, we modified the design by using a \\textit{Global-Average-Pooling} layer \\cite{LinCY13} and extended each CNN layer with a \\textit{Batch Normalization} layer \\cite{Ioffe15}. Afterwards, we compared our proposed methods to different ResNets \\cite{HeZRS16} to verify our findings.\n\nIn order to compare different methods to defend against adversarial attacks, we trained several models with and without different defense methods (see Section~\\ref{sec:results} for more details). All models were trained on a single Nvidia RTX 6000 or a single Nvidia V100 GPU. We found, that by using VAT \\cite{Miyato2015DistributionalSW} or PGD \\cite{madry2018towards} the training speed is reduced, significantly. \nEach training epoch not only takes longer, but the models also converge more slowly towards the optimum. \nIn contrast, our proposed method is more time efficient as it only requires an additional forward pass and it even converges faster (cf. Figure \\ref{fig:acc_druing_training} and Table \\ref{tab:training_time}).\n\n\\subsection{Assessing a Network's Resilience against Adversarial Perturbations}\n\\label{sec:assessment}\nTo assess the robustness of a network against adversaries, we trained six models per considered network architecture. We then used an ensemble of one, three or five of the six ANNs\\footnote{Note that our method applied to an ensemble of one model is identical to the i-FGSM.} (see Section~\\ref{sec:results} for details) to extract a set of adversarial images using Equation~(\\ref{eq:adv_methode}). More precisely, an image is considered being adversarial, if it is misclassified by \\emph{all} of the ensemble's networks -- note that for simplicity the networks do not have to agree on the same wrong class. The set of the extracted adversarial images is then classified by the sixth model and its accuracy is taken as quality indicator of the respective network architecture.\n\n\\section{Results and Discussion}\n\\label{sec:results}\n\\begin{table*}[!t]\n    \\centering\n    \\caption{Accuracy of different networks on the sets of original and adversarial images. All adversarial datasets are crafted using our proposed \\textit{gradient ensemble attack} approach with one, three or five (vulnerable) VGGNet13 models. Our method is equivalent to \\cite{KurakinGB16a,madry2018towards} if only one model is used. The shown results are for $\\varepsilon=16$. When training a model with PGD, the performance on the original images is significantly lower in comparison to the \\textit{Base Model}. In contrast, VAT and our proposed method slightly increase the validation accuracy. Across all adversarial perturbations, VAT models achieved the highest accuracy. Noteworthy values are highlighted.\n    }\n    \\label{tab:results}\n   \n   \n        \\begin{tabular}{r|cc|cc|cc|cc}\n        &\\multicolumn{2}{c|}{\\bf Accuracy on} & \\multicolumn{6}{c}{\\bf Accuracy on Adversarial Dataset based on} \\\\\n        &\\multicolumn{2}{c|}{\\bf Original Data} & \\multicolumn{2}{c|}{\\bf One Model} & \\multicolumn{2}{c|}{\\bf Three Models} & \\multicolumn{2}{c}{\\bf Five Models} \\\\\n        Parametrization \\, & \\, Train &Val. \\, & \\, True& Adv. \\, & \\, True \\, & \\, Adv. \\, & \\, True \\, & \\, Adv.\\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Base Model}}\\\\\n         \\, & \\, 91.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{73.4}\\% \\, & \\, 89.9\\% \\, & \\, \\cellcolor{gray!25}\\textbf{23.8}\\% \\, & \\, 95.3\\% \\, & \\, \\cellcolor{gray!25}\\textbf{4.1}\\% \\, & \\, 96.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{0.9}\\% \\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Projected Gradient Descent (PGD) with} $\\varepsilon = 8$ \\textit{and} $\\lambda = 2$}\\\\\n        $I = 3$ \\, & \\, 86.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{66.2}\\% \\, & \\, 79.6\\% \\, & \\, 79.0\\% \\, & \\, 84.6\\% \\, & \\, 83.6\\% \\, & \\, 86.6\\% \\, & \\, 86.5\\% \\\\\n        $I = 7$ \\, & \\, 77.8\\% \\, & \\, \\cellcolor{gray!25}\\textbf{62.4}\\% \\, & \\, 75.6\\% \\, & \\, 75.1\\% \\, & \\, 80.7\\% \\, & \\, 80.0\\% \\, & \\, 82.8\\% \\, & \\, 81.8\\% \\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Virtual Adversarial Training (VAT) with} $I = 3$ \\textit{and} $\\lambda = 1$}\\\\\n        $\\varepsilon = \\phantom{1}5$ \\, & \\, 97.7\\% \\, & \\, 76.2\\% \\, & \\, 90.6\\% \\, & \\, 85.1\\% \\, & \\, 95.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{87.2}\\% \\, & \\, 96.6\\% \\, & \\, 86.9\\% \\\\\n        $\\varepsilon = 15$ \\, & \\, 99.0\\% \\, & \\, 74.6\\% \\, & \\, 88.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{86.0}\\% \\, & \\, 93.0\\% \\, & \\, 86.0\\% \\, & \\, 94.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{90.3}\\% \\\\\n        $\\varepsilon = 25$ \\, & \\, 96.9\\% \\, & \\, 76.5\\% \\, & \\, 91.1\\% \\, & \\, 83.1\\% \\, & \\, 95.5\\% \\, & \\, 82.8\\% \\, & \\, 96.8\\% \\, & \\, 80.2\\% \\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Our Proposed Method} with $KL_{\\lambda=10}$}\\\\\n        $\\alpha \\sim B(2, \\phantom{1}4)$ \\, & \\, 90.6\\% \\, & \\, 74.3\\% \\, & \\, 88.8\\% \\, & \\, 71.9\\% \\, & \\, 93.5\\% \\, & \\, 62.1\\% \\, & \\, 95.1\\% \\, & \\, 53.7\\% \\\\\n        $\\alpha \\sim B(2, \\phantom{1}6)$ \\, & \\, 84.7\\% \\, & \\, 72.4\\% \\, & \\, 87.1\\% \\, & \\, 74.2\\% \\, & \\, 91.8\\% \\, & \\, 70.0\\% \\, & \\, 93.7\\% \\, & \\, 66.0\\% \\\\\n        $\\alpha \\sim B(2, 10)$ \\, & \\, 86.9\\% \\, & \\, 71.4\\% \\, & \\, 85.8\\% \\, & \\, 76.9\\% \\, & \\, 90.4\\% \\, & \\, 75.0\\% \\, & \\, 92.4\\% \\, & \\, 72.4\\% \\\\\n        \\end{tabular}\n   \n       \n\\end{table*}\n\nAt first, adversaries have been generated as described in Section~\\ref{sec:assessment}. For the gradient alignment, we considered an ensemble of one, three and five models, respectively. In a first analysis, we investigated the impact of the perturbation parameter for the values $\\varepsilon \\in\\{4,8,16\\}$. Interestingly, we observed that the success rate for crafting adversaries is not sensitive to the tested values for $\\varepsilon$. Therefore, we are only referring to the adversaries with $\\varepsilon=16$ as they are the most powerful ones. If a single unprotected model -- which is fully identical to the \\textit{Base Model} in terms of topology and training execution -- is used to calculate multi-step adversaries, the \\textit{Base Model's} classification accuracy is still $23.8\\%$ as shown in Table~\\ref{tab:results}. However, aligning the gradients of an ensemble of three or five models, the \\textit{Base Model's} accuracy on these adversaries decreases to $4\\%$ and $0.9\\%$, respectively.\n\nNext, we trained multiple models with \\textit{Virtual Adversarial Training (VAT)}, \\textit{Projected Gradient Descent Training (PGD)} and the three different defense methods proposed in this work (see Section~\\ref{sec:counteractions}). For VAT we used $I=3$, $\\lambda=1$ and $\\varepsilon\\in\\{5,10,15,20,25\\}$. Miyato et al. \\cite{Miyato2015DistributionalSW} recommended\nusing $I=1$ and $\\lambda=1$ as they found it to be sufficient. We increased the \\textit{power of iterations} to $I=3$ to ensure a better conversion (cf. Miyato et al. \\cite{Miyato2015DistributionalSW} for more details). As tuning $\\varepsilon$ is most important, we tried several different values and compared them to each other. Next, we adapted the default parameter for PGD as proposed by \\cite{madry2018towards}: $\\varepsilon=8$, $\\lambda=2$ and $I\\in\\{3, 7\\}$ as the number of iterations. $\\varepsilon=8$, $\\lambda=2$ and $I=\\{7\\}$ are the default settings used for on CIFAR10 by \\cite{madry2018towards}. In addition, we used $I=3$ to speed up training.\n\nTable \\ref{tab:results} shows the results of different methods based on their accuracy on our crafted adversarial images. As indicated by the base model's accuracy values on the adversarial data, the more models are used for our proposed \\textit{gradient ensemble attack}, the higher is the success rate of transferring the adversaries to other models. This demonstrates that our adversaries, crafted from an ensemble of models, are likely transferable to other networks. Moreover, the VAT models seem to perform best on adversarial images.\n\nTo test the generalizability of our approach, we additionally assessed our adversaries on ResNet32 and Res\\-Net50~\\cite{HeZRS16}. As shown in Table~\\ref{tab:transfer}, when applying a \\textit{gradient ensemble attack} on VGGNet13 and the ResNet models together, the resulting adversaries likely transfer between both topologies. The accuracy of unprotected (base) models on our combined adversarial dataset is 0.01\\% for the VGGNet13 network and about 26 to 27\\% for both ResNet models -- although all three models have an accuracy of over 90\\% on the original images. Even adversaries that were originally crafted for a different topology reduce a model's accuracy noticeably. \n    \nWe further tested our method on different ResNets. As shown in the bottom half of Table~\\ref{tab:transfer}, we found that not only the originally considered VGGNet13 models, but also ResNet32 and ResNet50 became more resilient against the transferable adversaries.\n\nHowever, comparing the performance of adversarial defense methods merely based on the model's accuracy or on the success rate for crafting adversaries is problematic. Adversarial sub-spaces may occur a little aside of the original ones or gradient masking could prevent gradient-based attack methods from being successful. Therefore, we do not only refer to a model's accuracy on strong and transferable adversaries, but also investigate the surrounding decision space, as well as its gradient.\n\n\\begin{figure*}[p]\n    \\centering\n    \\centering\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_base.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (a) \\textit{Base Model}\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_vat.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (b) VAT ($\\varepsilon=15$) \\cite{Miyato2015DistributionalSW}\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_pgd_07.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (c) PGD ($I=7$) \\cite{madry2018towards}\n    \\end{minipage}\\\\\n    \\medskip\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_vgg_alpha_04.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (d) $\\alpha=0.4$\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/vgg_vialpha_4_10.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (e) $\\alpha=0.4$, $KL_{\\lambda=10}$\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_vibeta_204_10.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (f) $\\alpha \\sim B(2, 4)$, $KL_{\\lambda=10}$\n    \\end{minipage}\n   \n    \\caption{Visualization of the decision space of six different VGGNet13 models in two adversarial directions of the same input image $X$. The loss is plotted using $x^* = x + \\varepsilon_1 \\cdot \\text{sign} \\left(\\nabla_{x}~L_1\\right) + \\varepsilon_2 \\cdot \\text{sign} \\left(\\nabla_{x}~L_2\\right)$ (cf. \\cite{TramerKPGBM18}). The red peak is an adversarial spot, which corresponds to a very high loss. The magnitude of the related model's gradients $\\text{sign}\\left(\\nabla_{x}~L_2\\right)$ is controlled by $\\varepsilon_2\\in[-8,32]$, whereas $\\varepsilon_1\\in[-16,16]$ controls the magnitude of an unprotected model's gradient $\\text{sign}\\left(\\nabla_{x}~L_1\\right)$. The six images visualize how the loss values change when the input $x$ is moved in one of these two directions. The images in (d) to (f) correspond to our proposed models.}\n    \\label{fig:gradient}\n    \n    \\bigskip\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_vgg_base.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (a) \\textit{Base Model}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_vat_15.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (b) VAT ($\\varepsilon=15$) \\cite{Miyato2015DistributionalSW}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_pgd_07.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (c) PGD ($I=7$) \\cite{madry2018towards}\n        \\end{minipage}\\\\\n        \\medskip\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_vibeta_204_10.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (d) VGGNet13 (Ours)\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_resnet32.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (e) ResNet32 (Ours)\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_resnet50.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (f) ResNet50 (Ours)\n        \\end{minipage}\n       \n        \\caption{Impact of different images $X$ with additional perturbation on the loss value. The magnitude of the loss is controlled by varying $\\varepsilon$. In comparison to the models from the literature (top row), our approach flattens the decision space significantly as demonstrated by the low loss values -- even for very large values of $\\varepsilon$. Note that our proposed models (bottom row) were trained using $\\alpha \\sim B(2, 4)$ and $KL_{\\lambda=10}$.}\n        \\label{fig:loss}\n\\end{figure*}\n\n\\begin{table*}[!t]\n    \\caption{Accuracy of different networks on the sets of original and adversarial images. All adversarial datasets are crafted using our \\textit{gradient ensemble attack} approach with five unprotected VGGNet13, ResNet32 and ResNet50 models. For the last dataset (last column), we combined three VGG13, ResNet32 and ResNet50 models. The table shows the accuracy of three unprotected models on different adversarial datasets. \n    As one can see, multi-step adversaries do transfer between topologies. Further, our proposed method provides resilience against these transferable adversaries.}\n    \\label{tab:transfer}\n   \n    \\centering\n    \\begin{small}\n   \n        \\begin{tabular}{r|cc|cc|cc|cc|cc}\n        &\\multicolumn{2}{c|}{\\bf Accuracy on} & \\multicolumn{8}{c}{\\bf Accuracy on Adversarial Dataset}\\\\\n        &\\multicolumn{2}{c|}{\\bf Original Data} & \\multicolumn{2}{c|}{\\bf VGGNet13} & \\multicolumn{2}{c|}{\\bf ResNet32} & \\multicolumn{2}{c|}{\\bf ResNet50} & \\multicolumn{2}{c}{\\bf Combined} \\\\\n        &Train &Val. & True& Adv. & True & Adv. & True & Adv. & True & Adv.\\\\\n        \\midrule \\multicolumn{11}{l}{\\textit{Base Models}}\\\\\n        VGG13 \\, & \\, 91.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{73.4}\\% \\, & \\, 96.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{0.9}\\% \\, & \\, 96.2\\% \\, & \\, 28.3\\% \\, & \\, 96.9\\% \\, & \\, 50.2\\% \\, & \\, 98.4\\% \\, & \\, \\cellcolor{gray!25}\\textbf{0.01}\\% \\\\\n        ResNet32 \\, & \\, 89.4\\% \\, & \\, 66.7\\% \\, & \\, 87.9\\% \\, & \\, 67.6\\% \\, & \\, 95.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{16.0}\\% \\, & \\, 95.3\\% \\, & \\, 44.0\\% \\, & \\, 96.1\\% \\, & \\, \\cellcolor{gray!25}\\textbf{26.1}\\% \\\\\n        ResNet50 \\, & \\, 87.5\\% \\, & \\, 62.8\\% \\, & \\, 84.1\\% \\, & \\, 64.9\\% \\, & \\, 92.2\\% \\, & \\, 24.6\\% \\, & \\, 94.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{30.9}\\% \\, & \\, 94.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{26.7}\\% \\\\\n        \\midrule \\multicolumn{11}{l}{Our Proposed Method with $\\alpha \\sim B(2, 4)$ and $KL_{\\lambda=10}$}\\\\\n        VGG13 \\, & \\, 90.6\\% \\, & \\, \\cellcolor{gray!25}\\textbf{74.3}\\% \\, & \\, 95.1\\% \\, & \\, \\cellcolor{gray!25}\\textbf{53.7}\\% \\, & \\, 96.6\\% \\, & \\, 69.6\\% \\, & \\, 97.1\\% \\, & \\, 81.5\\% \\, & \\, 98.1\\% \\, & \\, \\cellcolor{gray!25}\\textbf{32.4}\\% \\\\\n        ResNet32 \\, & \\, 83.1\\% \\, & \\, 59.8\\% \\, & \\, 80.3\\% \\, & \\, 67.8\\% \\, & \\, 88.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{38.5}\\% \\, & \\, 89.5\\% \\, & \\, 56.9\\% \\, & \\, 90.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{45.1}\\% \\\\\n        ResNet50 \\, & \\, 80.4\\% \\, & \\, 62.4\\% \\, & \\, 83.8\\% \\, & \\, 70.3\\% \\, & \\, 91.6\\% \\, & \\, 44.1\\% \\, & \\, 92.9\\% \\, & \\, \\cellcolor{gray!25}\\textbf{56.4}\\% \\, & \\, 93.6\\% \\, & \\, \\cellcolor{gray!25}\\textbf{46.9}\\% \\\\\n        \\end{tabular}\n   \n   \n    \\end{small}\n\\end{table*}\n\nFigure \\ref{fig:gradient} illustrates the loss of six different models based on the decision space of a single input image in two adversarial directions -- one taken from an unprotected model and the other one in direction of the related model. As depicted by Figure~\\ref{fig:gradient}~(a), i.e., the image of the \\textit{Base Model}, one can see an adversarial sub-space in close proximity of the input image as indicated by the high loss value (shown in red). Thus, it only takes a small change in  $\\varepsilon$ ($\\varepsilon=6$ is optimal in this particular case) to push the input image $X$ deep into this adversarial sub-space. A similar adversarial sub-space occurs in case of the VAT model as shown in Figure~\\ref{fig:gradient}~(b). Here, the distance is greater, but the sub-space still exists. So, it takes a larger $\\varepsilon$ to fool the VAT model.\n\nIn case of our proposed methods, using a fixed $\\alpha$-value is not sufficient either as the blind-spot still exists, as illustrated in Figures~\\ref{fig:gradient}~(d) and (e). Although our proposed methods with a fixed $\\alpha$-value weaken the adversarial spot, they do not eliminate it. Probably, the $\\alpha$-value is chosen too high in this case as there is another adversarial spot behind the first one. We found that sampling $\\alpha \\sim B(p,q)$ is essential to flatten the decision space. As demonstrated by Figure~\\ref{fig:gradient}~(f) the decision space of our method with varying $\\alpha$-values is significantly more flat than all the others. In fact, it is even more flat than the one of the PGD model -- especially for large $\\varepsilon$-values as illustrated in Figures~\\ref{fig:gradient}~(c) and (f), respectively. \n\nTo verify these findings, we generated 128 adversarial images $X$ (using the FGSM) for each defense method and compared the loss values based on varying $\\varepsilon\\in[0,128]$ (c.f. Figure \\ref{fig:loss}). As for the VAT model, adversarial attacks can be highly successful. In fact, $\\varepsilon=26$ provides the greatest success-rate for single-step adversaries while for the \\textit{Base Model} $\\varepsilon=8$ works best (see Figures \\ref{fig:loss}~(a) and (b)). This proves once again, that the adversarial spots are just a little further away compared to the \\textit{Base Model}. This may be an explanation, on why the VAT model performs rather good on our generated adversaries. However, the VAT model is not substantially more robust against adversarial attacks than the unprotected base model -- it only requires a larger $\\varepsilon$ to fool it.\n\nIn contrast, the PGD model and our proposed method clearly flatten the decision space and thereby strongly reduce the risk of adversarial spots (cf. Figure \\ref{fig:loss}~(c) and (d)). As one can see, adversarial examples occur in greater distance compared to the \\textit{Base Model} and occur significantly less frequent as the lower loss values demonstrate. However, there are adversarial sub-spaces which cause a high distraction of the PGD models. In contrast to the PGD model, our model (Virtual Alpha with $p=2$, $q=4$ and $\\lambda=10$) provides a significantly more flattened space, i.e., high loss values rarely occur at all.  \n\nNoticeably, for $\\varepsilon<16$ the loss values of our model are significantly higher than the ones of the PGD model. In addition, loss values grow rapidly for small $\\varepsilon$-values. The high average loss values for small $\\varepsilon$-values may explain \\textit{why} our model has a lower accuracy on our generated adversarial datasets than the PGD model as the loss values do not need do be maximal in order to indicate misclassification.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis work investigates the effects of adversarial attacks on deep learning networks. It analyzes different strategies for increasing a model's resilience and, thus, countervailing malicious attacks. The performance of the different defense strategies is compared across large sets of transferable, carefully generated adversaries. Next, three new approaches to improve resilience against such perturbations were first introduced and then compared against the state-of-the-art techniques \\textit{Virtual Adversarial Training} and \\textit{Projected Gradient Descent}. In addition, a novel adversarial attack method called \\textit{gradient ensemble attack} has been introduced. Further, this work has demonstrated the transferability of adversaries, which have been crafted using our proposed method.\n\nWithin our investigations we have observed that VAT does not provide substantial resilience against adversarial perturbations as the adversarial sub-spaces are just pushed a little further away. However, the incidence of these spaces is similar to an unprotected model. Further, PGD trained models reduce the frequency of adversarial sub-spaces and strongly increase the distance to them. Yet, these sub-spaces still occur. Our proposed method, superimposing two images\nand minimizing the KL-Divergence between the two activations, reduces the risk of adversarial sub-spaces with high loss. In fact, our results demonstrate that these spaces rarely occur. However, the average loss value is significantly higher which explains why our models performed worse on our adversarial test sets. Nevertheless, our proposed method is very promising as it (i) is easily executable (it only requires an additional forward pass), and (ii) still provides a noticeable regularized decision space. \n\n\n\nOur ideas for future work are two-fold: (i) we will compare additional methods to further decrease the overall loss of our proposed method and thereby improve its performance on adversaries; (ii) we will investigate the effects of our \\textit{gradient ensemble attack} for crafting strong and transferable adversaries in a wider context -- especially applying it to different white- and black-box attack scenarios.\n\n\\section*{Acknowledgment}\n\nAll three authors acknowledge support by the \\href{https:\/\/www.ercis.org}{\\emph{European Research Center for Information Systems (ERCIS)}}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Prediction of Hemispheric Asymmetry}\n\nIt is well known, starting already with the researches of Rudolf Wolf in the\nnineteenth century, that the sunspot cycle is not strictly periodic. Significant\ncycle-to-cycle variability is observed in both the amplitude and duration\nof the cycle. The hemispheric asymmetry in sunspot coverage\nwas already noticed by \\cite{Waldmeier1955}, who also pointed out that such\nasymmetry could be sustained for several years.\n\\cite{Babcock1959} reported that this asymmetry appears in the case of the polar field reversal as well.\nBased on observational data for cycles 12-23, \\cite{Norton2010} analyzed the asymmetry and proposed 20\\% as upper limit of sunspot area asymmetry. Regarding the phase lag asymmetry for the same period of time, \\cite{McIntosh2013} found a roughly four cycles long periodicity. \\cite{Zolotova2010} investigated the phase difference of sunspot cycles in the hemispheres for a longer duration, back to the Maunder Minimum and found secular variation. On the other hand, this periodicity appears only in the sign of the phase lag but not in its magnitude \\citep{Norton2014}.\n\\cite{Hathaway2016} predicted the hemispheric asymmetry of cycle 25 by extrapolating the polar fields of cycle 24 using their Advective Flux Transport model.\nAccording to their results, the Southern hemisphere should dominate the North.\nHowever, from a purely statistical point of view, the available polar field data (eg. \\citealt{Munoz2012}) is insufficient to infer a significant\ncorrelation from past cycles.\n\n\\cite{Belucz2013} investigated the hemispheric asymmetry generated\nin a 2D flux transport dynamo model by inserting a second meridional circulation cell on the southern hemisphere with different amplitude, latitude and depth. They found significant hemispheric asymmetry depending the properties of the second cell. Using the same model, \\cite{Shetye2015} focused on the effects of the meridional inflow towards the activity belts. According to their results, the intense inflow in one hemisphere leads to stronger toroidal fields and this asymmetry is sustained for more than one cycle.\n\n\\cite{Karak2017}, using their 3D surface flux transport and Babcock-Leighton solar dynamo model (\\citealt{Miesch2014}; \\citealt{Miesch2016}),\ninvestigated the influence of the tilt angle distribution by adding random scatter on Joy's law and a tilt-angle saturation was also added to the model (on this point\nsee also \\citealt{Lemerle2017}).\nThe SpotMaker algorithm they use places new BMRs in each hemispheres, with a set time\ndelay in order to avoid artificially imposing hemispheric symmetry. One of their results is the hemispheric asymmetry appearing in the polar fluxes\nwas only weakly correlated to the toroidal flux of the subsequent cycle. Due to the\nstrong diffusive coupling between the hemispheres in the model, the asymmetry reduces quickly.\n\n\nIn the present paper \\fix{we extend the work of \\citep{Nagy2017} on\nthe hemispheric asymmetry triggered by rogue BMRs. We present a detailed analysis of how rogue BMRs affect the hemispheric asymmetry of the polar cap flux, including prediction of the asymmetry level of the subsequent cycle based on the polar field asymmetry. We also investigate whether the asymmetry in the model shows periodicity or temporal persistence in its characterizing parameters.\nFollowing \\citep{Nagy2017}, our analysis is based on simulations carried out\nusing the recent $2\\times2$D dynamo model of \\citealt{Lemerle2015, Lemerle2017}.}\n\n\n\\section{Rogue BMRs in the $2\\times2$D Dynamo Model}\n\nThe \\cite{Lemerle2017} solar cycle model invokes differential rotation shear\nand the regeneration of the solar dipole via surface decay of active regions\n(the so-called Babcock-Leighton mechanism) as its primary inductive mechanisms.\nThis mean-field-like kinematic model\ncouples a 2D surface flux transport module\n(SFT) with a\n2D axisymmetric flux transport dynamo (FTD). The SFT component provides the azimutally averaged radial field\nserving as the upper boundary condition for the FTD\nsimulation, while the FTD module drives the SFT through the emergence\nof new bipolar magnetic region (BMR). This step is based\non a semi-empirical emergence function that sets the\nprobability of a BMR emerging (radially)\nas a function of toroidal magnetic field $B_t$ at\nthe bottom of the convective zone in the FTD module.\n\\fix{Motivated by the modelling of the destabilization and buoyant rise of thin\nmagnetic flux ropes initially located immediately beneath the base of the\nsolar convection zone \\citep[see, e.g.,][and references therein]{Fan2009},\na lower threshold\non $B_t$ is introduced, below which the emergence probability\nvanishes. The presence of a threshold implies that the dynamo is not self-excited,\nas the internal magnetic field must remain above threshold for regeneration of\nthe surface dipole to take place. Above this threshold, the proportionality constant\n$K$ between emergence probability and\n$B_t^{1.5}$ acts as the dynamo number for the model, since it effectively\nsets the mean surface dipole growth rate for a given internal toroidal\nfield strength.}\nProperties of the\nnew BMR --- emergence latitude, longitude, flux, angular separation, tilt --- are randomly drawn\nfrom distribution functions for these quantities built from observed\nstatistics of solar active regions during solar cycle 21, as described in Appendix A of\n\\citealt{Lemerle2015}.\n\nBecause the model is kinematic and includes a steady quadrupole-like meridional\nflow in the FTD module, the only physical mechanism available to couple\nthe two hemispheres is diffusive transport, operating in both the SFT and FTD modules.\n\nThe only amplitude-limiting nonlinearity included in the model\nis a\nreduction of the average BMR tilt angle $\\alpha$ as a function of the deep-seated\ntoroidal field strength $B_t$, parametrized\naccording to the following ad hoc algebraic formula:\n\\begin{equation}\\label{eq:tiltquenching}\n    \\alpha_q = \\frac{\\alpha(\\theta)}{1 + (B_t\/B_q)^2},\n\\end{equation}\nwhere $B_q$ is the quenching field amplitude, and $\\alpha$ is the reference\ntilt variation with latitude, i.e., Joy's law \\citep{McClintock2013}.\n\\fix{Lacking any reliable information on the manner in which the emerging flux ropes\nproducing BMRs disconnect from the underlying toroidal magnetic flux system,\nwe do not implement any flux reduction in the FTD module when emergences\nare introduced in the SFT module.}\n\nThe main advantages of the $2\\times2$D model are its high numerical efficiency\nand the fact that it is calibrated to follow accurately the\nstatistical properties of the real Sun. The complete\nlatitude--longitude representation of the simulated solar surface in\nthe SFT component further makes it possible to achieve high spatial\nresolution and account for the effect of individual active region\nemergences.\n\nThe reference solar cycle solution presented in \\citet{Lemerle2017} is\ndefined by 11 adjustable parameters, for which optimal values were obtained\nvia a formal optimization by a\ngenetic algorithm. The algorithm was designed to minimize the differences between the\nspatiotemporal distribution of emergences produced by the model, and\nthe observed sunspot butterfly diagram during cycle 21.\n\nFigure \\ref{fig:refsolution} shows a portion of the reference dynamo solutions\nused for the analyses presented in what follows. The solid lines on the\ntop and middle panels\nshow time series of hemispheric pseudo-sunspot number and polar cap flux, \\fix{which is calculated here as the surface integral of the radial magnetic field over a latitudinal extent of $20^{\\circ}$ from the poles.}\nThe bottom panel shows the corresponding time-latitude ``butterfly''\ndiagram for the spatial density of emerging BMRs, encoded as a grayscale.\nThe red dot indicates the time and latitude at which a large ``rogue''\nBMR emerged in this simulation,\nits properties being listed in the first row of Table \\ref{tab:BMRs}.\nArtificially removing this single BMR from the simulation leads to\na markedly different subsequent evolution of the dynamo solution,\nas shown by the dashed time series on panels (a) and (b)\nof Figure \\ref{fig:refsolution}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{Refcase_SSN.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{Refcase_PC.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{Refcase_Bfly.pdf}\\\\\n  \\caption{A short segment of the reference dynamo solution used for the analyses\ndiscussed in the present paper.\nPanel (a) shows the pseudo sunspot number time series separately for the hemispheres.\nThe solid lines shows the reference simulation run.\nOn panel (b) hemispheric time series of the polar cap flux are plotted, in\nred and blue for the Northern and Southern hemispheres, respectively.\nOn both of these panels dashed lines\npertain to a experiment\nin which a single large ``rogue'' BMR was removed from the simulation at the time indicated\nby the vertical black dashed line.\nPanel (c) shows the pseudo-sunspot butterfly diagram of the reference simulation plotted\nas solid lines in panels (a) and (b).\nThe gray scale encodes the density of emerging BMRs, and\nthe red dot indicates the position of the rogue BMR removed from the simulation\nto yield the dashed time series in (a) and (b).\n}\\label{fig:refsolution}\n\\end{figure}\n\nThe reference solution plotted on Figure \\ref{fig:refsolution}\nis the same as adopted in the numerical experiments of\n\\citet{Nagy2017} where the impact of individual ``rogue'' BMRs were analyzed. These peculiar active regions were identified based on their contribution to the global dipole moment defined as follows:\n\\begin{equation}\n \\delta D_{\\mathrm{BMR}} \\approx F \\,d \\, \\sin\\alpha \\,  \\sin\\theta,\n \\label{eq:thenumber}\n\\end{equation}\nwhere $F$ is magnetic flux, $d$ is the\nangular separation of the two polarities, $\\alpha$ is the tilt angle, $\\theta$\nis the colatitude. According to this expression, BMRs with high flux content, tilt angle and angular separation, close to the equator influence the most the building up dipole moment, and therefore the strength of the subsequent cycle \\fix{as suggested by \\citet{Jiang2015} as an explanation for the low amplitude of Cycle 24}.\n\n\\begin{figure}[t!]\n  \\centering\n  \\begin{minipage}{0.55\\textwidth}\n  \\includegraphics[width=0.95\\linewidth]{DeltaTheta_andothers.pdf}\n  \\end{minipage}\n  \\begin{minipage}{0.44\\textwidth}\n  \\caption{Average effect of varying the properties of a BMR (2nd data\nrow of Table \\ref{tab:BMRs}), inserted in the simulations at cycle\nmaximum, on the amplitude of the subsequent cycle. Variations in BMR\nflux (green), tilt angle (blue) and  polarity separation (orange) are\nconverted to the contribution to the dipole moment according to\nEquation (\\ref{eq:thenumber}), while the varying colatitudes (red) are\nshown on the top axis (adapted from Figure 11 in \\citealt{Nagy2017}).}\\label{fig:summarize}\n  \\end{minipage}\n\\end{figure}\n\n\\citet{Nagy2017} carried out\nseveral numerical experiments aiming to study how the parameters of BMRs affect the next, or even the ongoing cycle. A selected ``test'' BMR\nwith specified properties ($F$, $d$, etc.)\nwas inserted manually into simulations while the parameters of the active region were varied separately during each experimental series. The test-BMR emerged spontaneously during the reference simulation with parameters listed in the second row of Table \\ref{tab:BMRs}. The experiments were performed for three cycles with average, below average and above average amplitudes, respectively.\nIn each case two series of experiments were carried out with Hale (anti-Hale) test-BMR in order to increase (decrease) the dipole moment of the examined cycle. The characteristics of the\ntest-BMR -- emergence time and latitude, flux, tilt angle and angular separation -- were changed one by one in order to map the impact of each property on the subsequent simulated cycle.\nThe results of these experiments are summarized in Figure \\ref{fig:summarize}.\nBy jointly varying the flux, tilt angle or separation of the test-BMR, similar results were obtained for the amplitude of the upcoming cycle. This is because\nthese quantities have a combined effect in the form of Equation (\\ref{eq:thenumber}). The impact of the emergence latitude is also shown in Figure \\ref{fig:summarize} (red curve, top axis).\nThe effect of the BMR decreases as a function of the emergence latitude, but is still significant $20^{\\circ}$ away from the Equator. The emergence epoch is also an important factor; the strongest impact on the subsequent cycle is expected when the test-BMR appears at cycle maximum,\nand diminishes gradually as the rogue BMR is forced to emerge later and later in the descending\nphase of the cycle.\nWhen the emergence occurs during the rising phase of the perturbed cycle, it modifies the ongoing cycle as well.\n\n\\begin{table}[t!]\n    \\center\n  \\caption{Parameters of active regions discussed in the\npaper.  Colatitudes $\\theta_{\\mathrm{lead}}$ and\n$\\theta_{\\mathrm{trail}}$ are the\n  latitudinal positions of leading and trailing polarities;  $F$ is\nthe flux of the trailing polarity ($F_{\\mathrm{trail}} =\n-F_{\\mathrm{lead}}$); $\\alpha$ is the tilt angle and $d$ is the\nangular separation of leading and trailing polarities. $\\delta\nD_{\\mathrm{BMR}}$, the contribution of the BMR to the global dipole\nmoment, is defined according to Equation (\\ref{eq:thenumber}). J\/H\nindicates whether the active region is (anti-)Joy\/(anti-)Hale. In the\ncase of the second row a J\/H (J\/a-H) test-BMR increases (decreases)\nthe dipole moment during the experiments detailed in Section 5 of\n\\citep{Nagy2017}. }\\label{tab:BMRs}\n  \\begin{tabular}{ c  c  c  c  c  c  c }\n  \\hline\n  $\\theta_{\\mathrm{lead}}$ & $\\theta_{\\mathrm{trail}}$ & $F $ [$10^{23}$ Mx] & $\\alpha$  & $d$ & $\\delta D_{\\mathrm{BMR}}$[$10^{23}$ Mx] & J\/H  \\\\\n  \\hline\n  112$^{\\circ}$           & 118.6$^{\\circ}$           & 4.39              & $-11.08^{\\circ}$ & 34.08$^{\\circ}$   & $-0.5124$ &  J\/H\\\\\n  89.5$^{\\circ}$           & 82.1$^{\\circ}$            & --1.39              & 13.98$^{\\circ}$  & 30.97$^{\\circ}$   & --0.1810 & J\/H  \\\\\n                           &                           &                     &                  &                   &          & J\/a-H \\\\\n  \\hline\n  \\end{tabular}\n\n\\end{table}\n\n\n\\section{Hemispheric Asymmetry due to Rogue BMRs}\n\nAs proposed by \\cite{Hathaway2016} in the context of variation in the surface\nmeridional flow, the hemispheric asymmetry of a solar cycle originates from the polar cap flux asymmetry during the preceding cycle. \\citet{Nagy2017} analyzed whether the polar cap flux asymmetry is a good indicator of the upcoming simulated cycle's asymmetry in the $2\\times2$D model.\n\nComparison of the solid and dashed lines on Figure \\ref{fig:refsolution}(a) and (b) indicates that a single, large BMR with unusual characteristics\ncan have a large and persistent impact on\nhemispheric asymmetry in the resulting dynamo solution.\n\\fixx{The top panel on Figure \\ref{fig:synopticRogue} shows a synoptic magnetogram extracted from a simulation in which the rogue BMR listed in the first row of Table \\ref{tab:BMRs} was inserted at simulation time $t = 1992.72$, one cycle before the strongly asymmetric cycle that we will discuss in the next section.\nThis snapshot is extracted six months after emergence of the rogue BMR.\nThis\nis an anti-Hale BMR, with polarity ordering opposite to that which should\ncharacterize its hemisphere, so that the emergence impedes the build up\nof the Southern polar fields. The poleward surge of positive polarity (red)\ncan be seen quite clearly.\nFor comparison, the bottom\npanel shows a second synoptic magnetogram, extracted at the same time in\na parent simulation without artificial insertion of the rogue BMR.\nComparing the two snapshots reveals how\nthe positive trailing flux of the decaying rogue has strongly\naltered\nthe pattern of magnetic flux transport to the Southern polar regions.\nThis pattern is qualitatively similar to that highlighted by\n\\citet{Upton2018}, with the large\nactive region AR12192\nemerging in October 2014 and producing a poleward stream of positive\nmagnetic polarity which weakened the buildup of the southern polar cap\nnegative magnetic field.\n}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{FrameT-400rogue5952-6950.pdf}\n  \\includegraphics[width=\\textwidth]{FrameTnorogue5952-6950.pdf}\n  \\caption{\\fixx{On the top we plot the synoptic magnetogram of the rogue BMR listed in the first row of Table \\ref{tab:BMRs} at central longitude $\\phi\\sim360^{\\circ}$ and simulation time $t = 1992.72$, six months after the time of emergence (see the corresponding time series in Figure \\ref{fig:refsolution} and \\ref{fig:magnetogr}).\nThe bottom panel shows the magnetic field distribution without the emergence of this active region at the same epoch.  Here (and on Fig.~\\ref{fig:magnetogr} following), the color scale is strongly saturated to make the weaker magnetic fields visible.\n}}\\label{fig:synopticRogue}\n\\end{figure}\n\nFigure \\ref{fig:magnetogr}\noffers another example of this effect, for a different rogue event\nand now in the form of time-latitude maps\nof the zonally-averaged\nsurface radial magnetic field component. The top panel corresponds to the\nreference solution, while the bottom panel shows the magnetogram resulting from the\nartificial removal of a single large BMR at the time indicated by the vertical dashed line.\nNote in particular how the reversal of the polar field occurs almost\nsimultaneously in both hemispheres when the rogue BMR is removed, while in the\noriginal reference solution the southern polar cap reverses polarity some two years\nprior to the northern hemisphere.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{Refcase_magnetogr.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{Refcase_magnetogr_norogureBMR.pdf}\\\\\n  \\caption{The top panel shows a synoptic magnetogram of the surface radial magnetic\nfield component in the reference solution of Figure \\ref{fig:refsolution}. The bottom panel\nshows the synoptic magnetogram resulting from removal of the rogue BMR at the\ntime indicated by the vertical dashed line, corresponding to the dashed time\nseries on panels (a) and (b) of Figure \\ref{fig:refsolution}.}\\label{fig:magnetogr}\n\\end{figure}\n\\fix{Note that here the polar cap flux peaks 2--3 years prior to SSN minimum,\nwhile in\nthe case of the sun this peak usually occurs somewhat closer to cycle minimum.\nHowever, in our model as in the sun,\nthe peak polar cap flux does turn out\nto be a good predictor of the SSN amplitude of the subsequent cycle, so we retain\nit as a measure of cycle dipole strenth in all analyses that follow.}\n\nTo quantify the level of asymmetry, the normalized asymmetry of the peak polar cap flux ($\\Delta_{\\Phi}$) produced during the cycles is\ncompared to two asymmetry measures characterizing the subsequent cycles. These measures are the asymmetry of the total number of emergences in each hemisphere ($\\Delta_{\\mathrm{SSN}}$), and the time delay between the epochs when the new cycle BMRs first\nstart to emerge in the North and the South ($\\Delta_{T}$).\n\nThe asymmetry of the polar cap flux at a given cycle is defined as follows:\n        \\begin{equation}\n        \\Delta_{\\Phi} = \\frac{|\\Phi_{\\mathrm{N},\\mathrm{max}}| - |\\Phi_{\\mathrm{S},\\mathrm{max}}|} {( |\\Phi_{\\mathrm{N},\\mathrm{max}}| + |\\Phi_{\\mathrm{S},\\mathrm{max}}| )\/2},\n        \\end{equation}\n    \\noindent where $\\Phi_{\\mathrm{N},\\mathrm{max}}$ ($\\Phi_{\\mathrm{S},\\mathrm{max}}$) is the northern (southern) polar cap flux maximum.\nThe asymmetry of the activity level is defined similarly, but in\nterms of the pseudo-sunspot number constructed from the model output:\n        \\begin{equation}\n        \\Delta_{\\mathrm{SSN}} = \\frac{\\Sigma \\mathrm{SSN}_{\\mathrm{N}} - \\Sigma \\mathrm{SSN}_{\\mathrm{S}} } {( \\Sigma \\mathrm{SSN}_{\\mathrm{N}} + \\Sigma \\mathrm{SSN}_{\\mathrm{S}} )\/2},\n        \\end{equation}\n    \\noindent where $\\Sigma \\mathrm{SSN}_{\\mathrm{N}}$ ($\\Sigma \\mathrm{SSN}_{\\mathrm{S}}$) is the total number of emergences in the northern (southern) hemisphere.\nFinally, the time lag between the hemispheres is defined as:\n        \\begin{equation}\n          \\Delta_{T} = \\frac{t_{\\mathrm{N}} - t_{\\mathrm{S}}} {( T_{\\mathrm{N}} + T_{\\mathrm{S}} )\/2},\n        \\end{equation}\n    \\noindent where $t_{\\mathrm{N}}$ ($t_{\\mathrm{S}}$) is the beginning epoch of the cycle, while $T_{\\mathrm{N}}$ ($T_{\\mathrm{S}}$) is the duration of the cycle on the North (South).\n\nUpon calculating these asymmetry measures for 540 simulated cycles, \\citet{Nagy2017} found strong anticorrelations between the polar cap flux asymmetry of cycle $i$ and time delay, $\\Delta_{\\mathrm{T}}$ ($r = -0.7174$)\nduring cycle $i+1$.\nIn the case of asymmetry in number of emergences of cycle $i+1$, $\\Delta_{\\mathrm{SSN}}$  the correlation coefficient is $r = 0.7430$ as it is shown in Figure \\ref{fig:asymmetry500cycles}. This result shows that in the model the asymmetry of a cycle can be predicted via the asymmetry of the polar cap flux built up during the previous cycle.\n\n\n    \\begin{figure}[t!]\n      \\centering\n      \\begin{minipage}[b]{1\\textwidth}\n          \\includegraphics[width=0.5\\textwidth]{HemisphericAsymm_crossSSN.pdf}\n          \\includegraphics[width=0.5\\textwidth]{HemisphericAsymm_crossT.pdf}\n      \\end{minipage}\n      \\begin{minipage}[b]{1\\textwidth}\n        \\caption{\\small{ Two-dimensional histograms of the asymmetry of the hemispheric total\n     pseudo-SSN (\\textbf{left}) and the time lag between North and South (\\textbf{right})\n     in pseudo-solar cycle $i$ against the polar cap flux asymmetry during the\n     previous cycle for 540 simulated cycles. Some outlier data have been removed.\n     The number of cases (cycles) in each bin are\n     indicated by the colour codes. The correlation coefficient is\n     $r_{\\Delta SSN} = 0.7430$ and $r_{\\Delta T} = -0.7174$, respectively (adapted from Figure 6 in \\citealt{Nagy2017}).\n     }}\\label{fig:asymmetry500cycles}\n      \\end{minipage}\n    \\end{figure}\n\nIn order to assess the persistence of hemispheric asymetry, we separate the\nsimulated cycles in two groups according to hemispheric dominance, as measured\nby the quantity $\\Delta_{\\rm SSN}$ introduced above. For each group, we then\nconstruct the histograms of $\\Delta_{\\rm SSN}$ values characterising the\ncycle following each member of the groups. The resulting histograms are plotted\nin Figure\n\\ref{fig:PeriodHist}. Both are very well fit with Gaussian centered on\n$\\Delta_{\\rm SSN}=0$, with deviation from zero at the $10^{-2}$ level\nand standard deviation $\\sim 0.4$. This indicates that the probability of\nfinding a given hemispheric dominance in cycle $n+1$ is independent of\nhemispheric dominance in cycle $n$, and thus that hemispheric dominance is\ndetermined by processes operating on inter-cycle timescales.\n\\begin{figure}[t!]\n  \\centering\n  \\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\linewidth]{hist_SSN_afterN.pdf}\n  \\includegraphics[width=0.4\\linewidth]{hist_SSN_afterS.pdf}\n  \\end{minipage}\n  \\begin{minipage}{\\textwidth}\n  \\caption{ Strength asymmetry histograms of cycles following a North (South) dominated cycle on the left (right) panel. The histograms show the distribution of asymmetry\nparameters for all cycles following a Northern-dominated cycle (left) or\nSouthern-dominated cycle (right). The distributions are approximately Gaussian,\nwith means and standard deviations are respectively\n$0.012$ and $0.36$ in the left panel, and $-0.008$ and $0.41$ for the right panel.\nThis indicates that hemispheric dominance shows no significant\npersistence from one cycle to the next, at least in the (optimal) parameter regime\nof this simulation run.\n}\\label{fig:PeriodHist}\n  \\end{minipage}\n\\end{figure}\nWe repeated this exercise, this time constructing the distribution of asymetry\nparameters two cycles in the future instead of one, in order to possibly detect\npersistence of hemispheric asymmetry associated with one magnetic polarity\ndominating over the other for a few subsequent cycles, a features sometimes\nobserved in the \\cite{Lemerle2017} dynamo solutions. Once again the mean\nof the distribution are very close to zero, with standard deviations $\\sim 0.4$,\nindicating that hemispheric asymmetry does not persist beyond\none cycle in this model.\n\nBased on their numerical experiments, \\citet{Nagy2017} identified another interesting effect triggered by the emergence of a rogue BMRs. After such emergences in one cycle, the next cycle tends to be strongly asymmetric. This phenomenon was analyzed using a new test-BMR described in the first line of Table \\ref{tab:BMRs}, emerged on the southern hemisphere at cycle maximum, indicated by the vertical dashed line in Figure \\ref{fig:casestudy}. According to the previous results, the AR's impact on the upcoming cycle is the strongest at this epoch. On the other hand, the original position of the BMR is a bit far from the Equator.\nFor this reason, during the experimental runs the test region was replaced to emerge closer, about $15^{\\circ}$ far from the equator, within the region where significant effect was observed during the first experimental series.\nAt this position the active region's flux was decreased from about $4\\cdot10^{23}$ Mx down to $2.19\\cdot10^{23}$ Mx.\nThe black solid line in the top panel of Figure \\ref{fig:casestudy} shows the reference case when the BMR emerged at the original position, while black dashed line shows the case when the BMR was removed from the simulation. The coloured dashed lines show how the asymmetry changed for various values of the test-BMR's flux, as color-coded.\nOne can see that the asymmetry is changing according to the flux of the test region. There are slight changes in the amplitude of the northern hemisphere as well due to the\ndiffusive hemispheric coupling in the model.\nThe bottom panel of Figure \\ref{fig:casestudy} shows that the hemispheric asymmetry already appears in the form of polar cap flux asymmetry during the cycle within which\nthe test-BMR emerges.\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{cycle_03_hemSSN_BMRonS.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{cycle_03_PC_BMRonS.pdf}\\\\\n  \\caption{The \\textbf{top} panel shows how a test-BMR can modify the amplitude of the subsequent cycle, separately in each hemisphere. The properties of this region are listed in the first row in Table \\ref{tab:BMRs}.\n  Black solid curves indicate the reference solution, with a ``rogue'' BMR emerging\n$25^{\\circ}$ away from the Equator, at the time indicated by the vertical dashed line.\nThe black dashed curves refer to a\nmodified simulation in which this ``rogue'' BMR is removed. Colored curves indicate the results with test-BMRs with different flux, $15^{\\circ}$ far from the Equator, as labeled.\nOn the \\textbf{bottom} panel the polar cap flux is shown. Solid and dot-dashed lines indicates the southern polar cap flux while dashed lines correspond to the northern polar cap flux.\n}\\label{fig:casestudy\n\\end{figure}\n\nThe correlation between the polar cap flux asymmetry triggered by the test-BMR and the asymmetry parameters of the subsequent cycle is plotted in Figure \\ref{fig:asymmetry_seed512}. Besides the results for both hemispheres of the reference cycle, we plot results for five more cycles that were studied using the same test-BMR emerging $15^{\\circ}$ far from the equator at cycle maximum.\nWhen the BMR is inserted in the northern hemisphere, its tilt and polarity are set to obey Joy's and Hale's law. Considering all the six experimental runs, the correlation coefficient between the polar cap flux asymmetry during the perturbed cycle ($\\Delta_{\\Phi,i-1}$) and the asymmetry of the number of emerged BMRs during the next cycle ($\\Delta_{\\mathrm{SSN},i}$) is 0.8431. In the case of the time lag between the hemispheres ($\\Delta_{T,i}$) this correlation is $-0.8029$.\n\n     \\begin{figure}[t!]\n      \\centering\n      \\begin{minipage}[b]{1\\textwidth}\n          \\includegraphics[width=0.51\\textwidth]{crossplot_AsymmCorr_SSN.pdf}\n          \\includegraphics[width=0.51\\textwidth]{crossplot_AsymmCorr_TL.pdf}\n      \\end{minipage}\n      \\begin{minipage}[b]{1\\textwidth}\n        \\caption{\\small{ Correlation plots between the asymmetry of the hemispheric total\n     pseudo-SSN (\\textbf{left}) and the time lag between North and South (\\textbf{right}) in\n     pseudo-solar cycle $i$ against the polar cap flux asymmetry during the\n     previous cycle. Colored markers correspond to the example shown in Figure \\ref{fig:casestudy} and also for the experimental series for the northern hemisphere (red markers). Gray markers show the same series for five more cycles in the same simulation. The correlation coefficients are\n     $0.9463$ and $-0.9158$, respectively, and $0.8431$ and $-0.8029$ when using\ndata from the combined six experiments.  }}\\label{fig:asymmetry_seed512}\n      \\end{minipage}\n    \\end{figure}\n\n\nDiffusive transport is the only cross-equatorial coupling mechanism operating\nin the \\cite{Lemerle2017} dynamo model used to carry out the various experiments\ndescribed above. The leading member of a BMR emerging $15^\\circ$ from the equator\nwill diffuse to the equator on a timescale $\\tau=(\\pi R\/12)^2\/\\eta_R\\simeq 2\\,$yr\nfor the surface diffusivity value $\\eta_R=6\\cdot 10^{12}\\,$cm$^2\\,$s$^{-1}$\nused in the SFT module. On the other hand, the internal toroidal field\nat 15 degrees will diffuse to the equator on a timescale controlled\nby the internal magnetic diffusivity $\\eta_t=10^{12}\\,$cm$^2\\,$s$^{-1}$ of the\nFTD module,\nleading to a timescale $\\tau\\simeq 12\\,$yr. The first timescale\nindicates that a rogue\nBMR emerging close to the equator can induce a polar cap asymmetry in the ongoing\ncycle, in agreement with the experimental results displayed\non Fig.~\\ref{fig:asymmetry_seed512};\nwhile the second timescale reveals that this asymmetry,\nonce it has built up in the internal toroidal field, can persist over\na full cycle before being diffusively balanced. Periodicity in hemispheric\nasymmetry cannot be driven or sustained by such diffusive coupling alone, and\nindeed is not observed in our dynamo simulations. Dynamical backreaction on\nlarge-scale flows, namely meridional circulation and differential rotation\nwould be the most likely candidate mechanism that could lead to such behavior.\n\n\\section{Conclusion}\n\nThe hemispheric asymmetry triggered by rogue active regions was studied using simulated data of the $2\\times2$D solar dynamo model. The flux of a selected test-BMR was changed while its position was fixed to $15^{\\circ}$ far from the Equator either\non the northern or southern hemispheres. The emergence epoch was the cycle maximum, while the polarity was set to increase the building up dipole moment. Experimental series were carried out in the case of six simulated cycles of varying amplitudes.\n\nIn contrast to the results of \\cite{Karak2017} we found strong correlation between the hemispheric asymmetry of the polar cap flux of cycle $i$ and the asymmetry of hemispheric activity levels\nduring the subsequent cycle $i+1$. The time lag between the hemispheric\nlag in the onset of cycle $i+1$ is also strongly correlated to the asymmetry of the\npolar cap flux of the preceding cycle. These results can be understood in terms\nof diffusive coupling of the magnetic field across the equatorial plane.\n\nOur results thus demonstrate that\nthe polar cap flux asymmetry at the end of a cycle can be determined\nby a single peculiar active region, emerging relatively close to the equator.\nThis offers an alternate scenario to that suggested by \\cite{Hathaway2016},\nbased on hemispheric variations in the surface meridional flow, which,\nin our kinematic dynamo model, remains strictly constant.\nIn view of the relatively strong\nhemispheric asymmetry observed in cycle 24, the unfolding of cycle 25 may\nallow to discriminate between these two scenarios.\n\n\\section*{Acknowledgements}\nM.N.'s research is currently supported by the \\'UNKP-18-3 New National Excellence Program of the Ministry of Human Capacities.\nP.C. and A.L. are supported through the Discovery Grant Program of the\nNatural Sciences and Engineering Research Council of Canada.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{&Intro}}\n\nThe symmetries of the Euler-Lagrange equations of motion\nwere recently used to study the constrained dynamics of singular\nLagrangians \\cite{ADS2020}. The focus was on almost regular\nLagrangians \\cite{Got1978, Got1979, Got1980, Car1990a}, and it was\nfound that for these Lagrangians the Euler-Lagrange equations of motion  \nadmit a generalized Lie symmetry (also known as a local gauge\nsymmetry). The generators $\\mathcal{S}\\hbox{ym}$ of this symmetry group\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ were determined in the Lagrangian\nphase space approach to Lagrangian mechanics, and were found to lie\nin the kernel of the Lagrangian two-form $\\mathbf{\\Omega}_L$. While it\nis well-known that the solutions $\\mathbf{X}_E$ of the energy equation,  \n\\begin{equation}\n  0=\\mathbf{d}E-i_{\\mathbf{X}_E}\\mathbf{\\Omega}_L,\n  \\label{EnergyE}\n\\end{equation}\nis not unique for almost regular Lagrangians, it was shown in \\cite{ADS2020} that\nthe action of $\\mathcal{S}\\hbox{ym}$ on a general solution  \nto this equation\\textemdash and in particular, on the \\textbf{second-order,\n  Lagrangian vector field} (SOLVF)\\textemdash will result in a vector\nfield that is no longer a solution of Eq.~$(\\ref{EnergyE})$. Thus, not all\nsolutions of the energy equation have\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ as a symmetry group. It is,\nhowever, possible to construct \nsolutions to Eq.~$(\\ref{EnergyE})$ for whom $\\mathcal{S}\\hbox{ym}$\ndoes generate a group of symmetry transformations\n\\cite{ADS2020}. These vector fields are called \\textbf{second-order,\n  Euler-Lagrange vector fields} (SOELVFs). As the \nevolution of the dynamical system for singular Lagrangians must lie on\nLagrangian constraint surfaces \\cite{Car1990a}, a Lagrangian constraint\nalgorithm for SOELVFs was also introduced in \\cite{ADS2020} to\nconstruct such solutions to the energy equation. It was\nthen shown that these SOELVFs, along with the dynamical structures\nin the Lagrangian phase space needed \nto describe and determine the motion of the dynamical system, are projectable to the\nHamiltonian phase space. In particular, the primary\nHamiltonian constraints can be constructed from vectors that lie in the kernel of\n$\\mathbf{\\Omega}_L$, and the Lagrangian constraint\nalgorithm for the SOELVF is equivalent to the stability analysis of \nthe total Hamiltonian (we follow the terminology found in\n\\cite{Hen1992}; see also \\cite{Dir1950, Mun1989, Lus2018}) obtained\nusing constrained Hamiltonian mechanics. Importantly, the end result\nof this stability analysis gives a Hamiltonian vector field that is the \nprojection of the SOELVF obtained from the Lagrangian constraint\nalgorithm. The Lagrangian and Hamiltonian formulations of mechanics\nfor almost regular Lagrangians were thereby shown to be equivalent. \n\nWhile \\cite{ADS2020} focused on the generalized Lie symmetries of the\nEuler-Lagrange equations of motion and whether the dynamical\nstructures constructed in the Lagrangian phase space are projectable to\nthe Hamiltonian phase space, in this paper the focus is on the symmetries\nof the action itself and the impact these symmetries have on the\nevolution of dynamical systems. This impact is found to be quite broad,\nsurprisingly restrictive, and unexpectedly subtle. Indeed, even the\nseemingly reasonable expectation that any generalized Lie symmetry of\nthe Euler-Lagrange equations of motion should be a reflection of the\nsymmetries of the action itself is not borne out.\n\nWe find that if the action has a generalized Lie\nsymmetry, then its Lagrangian is necessarily singular; the converse\nneed not be true, as we show through a specific example. We also find\nthat the generators of the generalized Lie symmetry of the action form\na Lie sub-algebra of the generators of the\ngeneralized Lie symmetry of the Euler-Lagrange equation of motion;\nonce again, the converse is not true. We give an example of a dynamical\nsystem for which the Euler-Lagrange equations of motion has a\ngeneralized Lie symmetry, while its action does not. Most importantly,\nfor systems where the Lagrangian is almost regular and for \nwhich the two-form $\\mathbf{\\Omega}_L$ has constant rank, we show that\neach generalized Lie symmetry of the action contributes one\narbitrary constant to the SOELVF. The dimensionality of the space of\nsolutions to the energy equation that have\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ as a symmetry group is thus at\nleast as large as the number of generalized Lie symmetries of the\naction. Moreover, if second- or higher-order Lagrangian\nconstraints are introduced during the application of the Lagrangian\nconstraint algorithm, these additional constraints cannot be due to the\ngeneralized Lie symmetry of the action.      \n      \nSymmetries of Lagrangian systems have been studied before. However,\nsuch analyses have been focused on time-dependent Lagrangians \\cite{Pri1983, Pri1985, \nCra1983, Car1991, Car1988b, Car1992, Car1993, Car2003}; on systems of\nfirst-order evolution equations \\cite{Car1990b, Mar1992, Gra2002, Gra2005,\n  Pop2017}; or on general solutions of Eq.~$(\\ref{EnergyE})$ \n\\cite{deL1995} (see also \\cite{Dim2016}). Importantly, the great majority of these\nstudies have been done using first-order prolongations on\nfirst-order jet bundles with a focus on the Lie symmetries of first-order\nevolution equations. Our interest is in the symmetries of the action, which\nnaturally leads us to consider generalized Lie symmetries and\nsecond-order prolongations. To our knowledge, such symmetry analysis\nof the action has not been done before. (The framework for\n$k^{\\hbox{\\textit{th}}}$-order prolongations on\n$k^{\\hbox{\\textit{th}}}$-order jet bundles have been introduced before\n\\cite{Car1993, deL1995, Car2003, Pop2009, Pop2011}, but they were not applied\nto the action or to the Euler-Lagrange equations of motion.)\n\nThe rest of the paper is arranged as follows. In \\textbf{Section\n  \\ref{&Sym}} the conditions under which the action for a dynamical\nsystem, and the conditions under which the Euler-Lagrange equations of\nmotion for this action, have a generalized Lie symmetry are\ndetermined. To compare the conditions for each, the analysis for the\ntwo are done separately, with each self-contained. In\n\\textbf{Section \\ref{&review}} properties of the Lagrangian phase\nspace are reviewed, and the notation used here established. The\ngenerators of the generalized Lie symmetry group for the\nEuler-Lagrange equations of motion were determined in \\cite{ADS2020},\nand a summary of the results found therein that are needed here is\ngiven. In \\textbf{Section \\ref{&A-S}} the generators of the\ngeneralized Lie symmetry group for the action is found within the\nLagrangian phase space approach, and their relation to the\ngenerators for the symmetry group of the Euler-Lagrange equations of\nmotion is determined. The impact of the symmetries of the action on\nthe SOELVF is then analyzed by applying the \nLagrangian constraint algorithm introduced in \\cite{ADS2020} to these\nSOELVF. The results obtained in this paper is then applied to three different\ndynamical systems in \\textbf{Section \\ref{&Exam}}. In particular, an\nexample of a dynamical system that has no generalized Lie symmetries\nand yet is still singular, and another example where the action has no\nsymmetries and yet the Euler-Lagrange equations of motion do, are\ngiven. Concluding remarks can be found in \\textbf{Section \\ref{&Conc}}.      \n\n\\section{Generalized Lie symmetries and Lagrangian Mechanics\\label{&Sym}}\n\nIn this section we determine the conditions under which the action of\na dynamical system, and the conditions under which the Euler-Lagrange\nequations of motion for this system, has a generalized Lie\nsymmetry. While the determination for both is done within Lagrangian\nmechanics, the analysis for the action is completed separately from that of\nthe equations of motion\\textemdash with each self-contained\\textemdash so \nthat the two conditions can be compared. We will later show that every\ngenerator of the generalized Lie symmetry of the action is a\ngenerator of a generalized Lie symmetry of the Euler-Lagrange\nequations of motion. Interestingly, the converse is not true. \n\n\\subsection{Symmetries of the Action \\label{&A-Sym}}\n\nWe begin with Lagrangian mechanics, and an analysis of the\ngeneralized Lie symmetry \\cite{Olv1993} of the action \n\\begin{equation}\n  S := \\int_{t_1}^{t_2}L\\left(q(t),\\dot{q}(t)\\right)dt,\n  \\nonumber\n\\end{equation}\nfor a dynamical\nsystem on a $D$-dimensional configuration space $\\mathbb{Q}$. Here,\n$L\\left(q(t), \\dot{q}(t)\\right)$ is the Lagrangian along a path $q(t)\n= \\left(q^1(t), \\dots, q^D(t)\\right)$ on $\\mathbb{Q}$ with end\npoints given by $Q_1:=q(t_1), Q_2:= q(t_2)$. These points are \nchosen at the same time the choice of $S$ is made, and are fixed.\n\nAs $L\\left(q(t),\\dot{q}(t)\\right)$ depends on both the position $q(t)$\nand the velocity $\\dot{q}(t)$ of the path, we consider a generalized\nLie symmetry that is generated by \n\\begin{equation}\n  \\mathbf{g}_L := \\rho_L(q, \\dot{q})\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\,}{\\mathbf{\\partial} q}, \n  \\nonumber\n\\end{equation}\nwhere $\\rho_L(q,\\dot{q})$ does not depend explicitly on\ntime. Evolution along the path gives the total time derivative \n\\begin{equation}\n  \\frac{\\mathbf{d}\\>\\>\\>}{\\mathbf{d}t} := \\dot{q}\\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} +\\ddot{q}\\cdot\n  \\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} \\dot{q}}.\n  \\label{Dt}\n\\end{equation}\nThis in turn gives $\\dot{\\rho}_L:=\\mathbf{d}\\rho_L\/\\mathbf{d}t$, and the second-order\nprolongation vector \\cite{Olv1993}, \n\\begin{equation}\n  \\hbox{\\textbf{pr }}\\mathbf{g}_L := \\rho_L \\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} \n  + \\dot{\\rho}_L \\cdot \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} \\dot{q}} +\n  \\ddot{\\rho}_L\\cdot \\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial}\n    \\ddot{q}},\n  \\label{prog}\n\\end{equation}\non the second-order jet space $\\mathbb{M}^{(2)}=\\{(t, q,\n\\dot{q},\\ddot{q})\\}$ where this $\\hbox{\\textbf{pr\n}}\\mathbf{g}_L\\in\\mathbf{T}\\mathbb{M}^{(2)}$.\n\nUnder this generalized Lie symmetry, the action varies by\n\\begin{equation}\n  \\delta S=\\int_{t_1}^{t_2}\\hbox{\\textbf{pr }}\\mathbf{g}_L \\Big[L(q(t),\n  \\dot{q}(t))\\Big]dt,\n  \\nonumber\n\\end{equation}\nwith the requirement that \n$\\rho_L(q(t_1), \\dot{q}(t_1))=0=\\rho_L(q(t_2),\n\\dot{q}(t_2))$. Then after an integration by parts,\n\\begin{equation}\n  \\delta S=\\int_{t_1}^{t_2} \\rho_L\\cdot\\left[\\frac{\\partial L}{\\partial\n    q} - \\frac{d\\>\\>\\>}{dt}\\left(\\frac{\\partial L}{\\partial\n    \\dot{q}}\\right)\\right]dt.\n  \\label{Ae1}\n\\end{equation}\nIt is important to realize that the action may be evaluated along any path on\n$\\mathbb{Q}$. As such, if $\\mathbf{g}_L$ generates a symmetry of the action, then\nEq.~$(\\ref{Ae1})$ must vanish for \\textit{all} paths $q(t)$ on\n$\\mathbb{Q}$, and not just for those that minimize the action.\n\nTo make connection with the Lagrangian phase space approach used in\nthe rest of the paper, we make use of\n\\begin{equation}\nE\\left(  q,\\dot{q}\\right)  :=\\dot{q}^{a}\\frac{\\partial L\\left(  q,\\dot\n  {q}\\right)  }{\\partial\\dot{q}^{a}}-L\\left(  q,\\dot{q}\\right)  ,\n\\nonumber\n\\end{equation\nalong with\n\\begin{equation}\nM_{ab}\\left(  q,\\dot{q}\\right)  :=\\frac{\\partial^{2}L\\left(  q,\\dot{q}\\right)\n}{\\partial\\dot{q}^{a}\\partial\\dot{q}^{b}}, \\quad \\hbox{and }\\quad \nF_{ab}\\left(  q,\\dot{q}\\right)  := \\frac{\\partial^{2}L\\left(\n  q,\\dot{q}\\right)  }{\\partial\\dot{q}^{a}\\partial q^{b}} -\n\\frac{\\partial^{2}L\\left(  q,\\dot{q}\\right)}{\\partial\\dot{q}^{b}\\partial \n  q^{a}},\n\\nonumber\n\\end{equation}\nto express Eq.~$(\\ref{Ae1})$ as\n\\begin{equation}\n  \\delta S=-\\int_{t_1}^{t_2} \\rho^a_L\\left(\\frac{\\partial E}{\\partial\n    q^a} +F_{ab}(q,\n  \\dot{q})\\dot{q}^b+M_{ab}(q,\\dot{q})\\ddot{q}^b\\right)dt.\n  \\label{e2}\n\\end{equation}\nHere, Latin indices run from $1$ to $D$, and Einstein's summation\nconvention is used. We then arrive at our first result.\n\n\\begin{lemma} \\label{Action-Sym} An action $S$ of a dynamical\n  system has a generalized Lie symmetry generated by $\\mathbf{g}_L$ if\n  and only if there exists a $\\rho_L\\in \\hbox{ker }M_{ab}$ such that \n  \\begin{equation}\n    0=\\rho_L^a(q,\\dot{q})\\left(\\frac{\\partial E}{\\partial\n      q^a}+F_{ab}(q,\\dot{q})\\dot{q}^b\\right),\n    \\label{e3}\n  \\end{equation}\n  on $\\mathbf{T}\\mathbb{Q}$.\n  \n  \\begin{proof}\n    If $\\mathbf{g}_L$ generates a generalized Lie symmetry of $S$, then\n    Eq. $(\\ref{e2})$ must vanish for all paths on $\\mathbb{Q}$. For an\n    arbitrary path on $\\mathbb{Q}$ the\n    curvature of the path $\\ddot{q}$ will not depend on\n    either the $q(t)$ or the $\\dot{q}(t)$ for the path, however. As such, \n    for $\\delta S=0$, it must be that $\\rho^a_LM_{ab}\\ddot{q}^b=0$ for any choice of\n    $\\ddot{q}$, and thus $\\rho_L^a\\in\\hbox{ker }M_{ab}$. The remaining\n    terms in Eq.~$(\\ref{e2})$ gives the condition\n    Eq.~$(\\ref{e3})$. \n  \\end{proof}\n\\end{lemma}\n\nThe set of all vector fields $\\mathbf{g}_L$ that satisfy \\textbf{Lemma\n  $\\mathbf{\\ref{Action-Sym}}$} is denoted by  $\\mathfrak{g}_L$, while \n$\\hbox{\\textbf{pr }}\\mathfrak{g}_L := \\{\\hbox{\\textbf{pr\n}}\\mathbf{g}_L\\ \\vert \\ \\ \\mathbf{g}_L\\in \\mathfrak{g}_L\\}$ is the set of\ntheir prolongations. This $\\hbox{\\textbf{pr }}\\mathfrak{g}_L$ is involutive\n\\cite{Olv1993}, and the conditions under which $\\hbox{\\textbf{pr }}\\mathfrak{g}_L$\ngenerates a generalized Lie symmetry group are given in\n\\cite{Olv1993}.\n\nWe see from \\textbf{Lemma \\ref{Action-Sym}} that if the\naction has a generalized Lie symmetry, then the Lagrangian is\nnecessarily singular, and as such the Lagrangian two-form\n$\\mathbf{\\Omega}_L$ will not have maximum rank. It is also important\nto note that while equations of the form\nEq.~$(\\ref{e3})$ often appear in the Lagrangian phase space description of \nmechanics \\cite{ADS2020}, they appear as Lagrangian\nconstraints, conditions that must be imposed for evolution under the Euler-Lagrange\nequations to be well defined. Here, Eq.~$(\\ref{e3})$ is not a \nconstraint. Rather, because the action must have this symmetry for\n\\textit{all} possible paths on $\\mathbb{Q}$, and since the set of all possible \npaths cover $\\mathbb{Q}$, Eq.~$(\\ref{e3})$ is a condition on $\\rho_L$\nthat must be satisfied identically on \\textit{all} of\n$\\mathbf{T}\\mathbb{Q}$\\textemdash and \nthus, on the Lagrangian phase space\\textemdash for $\\mathbf{g}_L$ to\nbe a generator of the symmetry group. We will see that not all the vectors in\n$\\hbox{ker } M_{ab}$ satisfy the identity Eq.~$(\\ref{e3})$,\nhowever, and thus not all of these vectors will generate a generalized Lie\nsymmetry of the action.   \n  \n\\subsection{Symmetries of the Euler-Lagrange\n  Equations of Motion\\label{&EL-Sym}}\n\nWhile in \\textbf{Section \\ref{&A-Sym}} the focus was on arbitrary paths on the\nconfiguration space $\\mathbb{Q}$ and the symmetries of the action,\nin this section the focus is on the trajectories that minimizes the\naction and the generalized Lie symmetries of them. These trajectories\nare solutions of the Euler-Lagrange equations of motion, and for almost\nregular Lagrangians such solutions form a family of curves. It is, in fact, the\npresence of this family of curves that gives rise to the generalized\nLie symmetry. The treatment here follows closely to that given in \\cite{ADS2020}. \n\nFor almost regular Lagrangians the solutions of the Euler-Lagrange\nequations of motion \n\\begin{equation}\nM_{ab}(q, \\dot{q})\\ddot{q}^{b}=-\\frac{\\partial E}{\\partial\n  q^{a}}-F_{ab}(q, \\dot{q})\\dot{q}^{b}, \n\\label{2ndEL1\n\\end{equation}\nare not unique. While for these Lagrangians the rank of\n$M_{ab}\\left(q, \\dot{q} \\right) =D-N_{0}$\\textemdash with $N_0=\\hbox{dim\n}\\left(\\hbox{ker }M_{ab}(q, \\dot{q})\\right)$\\textemdash is constant, this rank\nis not maximal, and thus Eq.~$(\\ref{2ndEL1})$ does not have a unique\nsolution for $\\ddot{q}$. Instead, for a chosen set of initial data\n$\\left(q_{0}=q(t_0),\\dot{q}_{0}=\\dot{q}(t_0)\\right)$, the solution to\nEq.~$(\\ref{2ndEL1})$ results in a family of solutions that evolve from\nthis $(q_0, \\dot{q}_0)$. As with the paths in \\textbf{Section \\ref{&A-Sym}}, these\nsolutions are related to one another through a generalized Lie \nsymmetry \\cite{Olv1993}.\n\nFollowing \\cite{Olv1993}, the collection of functions\n\\begin{equation}\n  \\Delta_a(q,\\dot{q}, \\ddot{q}) := \\frac{\\partial E(q, \\dot{q})}{\\partial q^a} +\n  F_{ab}(q, \\dot{q})\\dot{q}^b + M_{ab}(q, \\dot{q}) \\ddot{q}^b,  \n\\label{delta}\n\\end{equation}\ndefines a set of  surfaces $\\Delta_a(q,\\dot{q}, \\ddot{q})=0$ on\n$\\mathbb{M}^{(2)}$, while the family of solutions to Eq.~$(\\ref{2ndEL1})$ \n\\begin{equation}\n\\mathcal{O}\\left(q_0, \\dot{q}_0\\right)  :=\\big\\{q\\left(  t\\right)\n\\ \\vert \\ \\\n\\Delta_a(q,\\dot{q}, \\ddot{q}) =0 \\hbox{ with }\nq\\left(  t_{0}\\right)\n=q_{0},\\ \\dot{q}\\left(  t_{0}\\right)  =\\dot{q}_{0}\\big\\}  ,\n\\nonumber\n\\end{equation}\nthat evolve from the same initial data $(q_0, \\dot{q}_0)$ gives the collection\nof trajectories that lie on these surfaces. Indeed, for any two such \nsolutions $q^a(t)$ and $Q^a(t)$ there exists a \n$\\mathfrak{z}(q, \\dot{q})\\in\\hbox{ker } \nM_{ab}(q, \\dot{q})$ such that $\\ddot{Q}^a-\\ddot{q}^a =\n\\mathfrak{z}^a$. Importantly, because $\\mathfrak{z}^a$ depends on both $q$\nand $\\dot{q}$, the symmetry group that maps one member of\n$\\mathcal{O}$ to another must be a generalized Lie symmetry. We\ntherefore take the generator of this symmetry group to be \n\\begin{equation}\n  \\mathbf{g} := \\rho(q, \\dot{q})\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\>}{\\mathbf{\\partial} q},\n  \\nonumber\n\\end{equation}\nwith the corresponding the second-order prolongation vector\nfor $\\mathbf{g}$ being,\n\\begin{equation}\n  \\hbox{\\textbf{pr }}\\mathbf{g} := \\rho \\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} \n  + \\dot{\\rho} \\cdot \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} \\dot{q}} +\n  \\ddot{\\rho}\\cdot \\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} \\ddot{q}},\n  \\nonumber\n\\end{equation}\nwith this $\\hbox{\\textbf{pr }}\\mathbf{g}\\in\\mathbf{T}\\mathbb{M}^{(2)}$. As with\nthe above, the total time \nderivative is given by Eq.~$(\\ref{Dt})$, but unlike the analysis in\n\\textbf{Section \\ref{&A-Sym}}, the evolution of the path\\textemdash and indeed,\nfor all the trajectories in $\\mathcal{O}(q_0,\\dot{q}_0)$\\textemdash here is\ngiven by the Euler-Lagrange equations of motion.\n\nThe action of this\nprolongation on $\\Delta_a$ on the $\\Delta_a=0$ surface gives,   \n\\begin{equation}\n  \\hbox{\\textbf{pr }}\\mathbf{g}\\left[\\Delta_a(q,\n    \\dot{q},\\ddot{q})\\right] = -\\frac{\\partial \n    \\ddot{q}^b}{\\partial q^a}M_{bc}(q, \\dot{q})\\rho^c +\n  \\frac{d\\>\\>\\>}{dt}\\left[F_{ab}(q, \\dot{q})\\rho^b + M_{ab}(q,\n    \\dot{q})\\dot{\\rho}^b\\right]. \n  \\nonumber   \n\\end{equation}\nSince $N_0>0$, $\\ddot{q}$ is not unique on this surface, and yet \n$\\mathbf{g}$ must generate the same symmetry group for all the\ntrajectories in $\\mathcal{O}(q_0, \\dot{q}_0)$. Necessarily, \n$\\rho(q, \\dot{q})\\in\\hbox{ker }M_{ab}(q, \\dot{q})$. It then follows\nthat $\\hbox{\\textbf{pr g}}[\\Delta_a(q,\\dot{q}, \\ddot{q})] =0$\nif and only if (iff) there are constants $b_a$ such that $b_a = \nF_{ab}\\rho^b + M_{ab}\\dot{\\rho}^b$. The solutions in $\\mathcal{O}(q_0,\n\\dot{q}_0)$ all have the same initial data, however, and thus\nnecessarily $\\rho(q_0, \\dot{q}_0)=0=\\dot{\\rho}(q_0,\\dot{q}_0)$. We\nconclude that $b_a=0$. The following result, first proved in\n\\cite{ADS2020}, then follows.  \n\n\\begin{lemma} \\label{GS} If $\\mathbf{g}$ is\n  a generalized infinitesimal symmetry of $\\Delta_a$, then\n  $\\rho^a(q, \\dot{q})\\in\\hbox{ker } M_{ab}(q, \\dot{q})$, and\n  $\\dot{\\rho}^a(q, \\dot{q})$ is a solution of\n    \\begin{equation}\n    0=F_{ab}(q, \\dot{q})\\rho^b(q, \\dot{q}) +\n    M_{ab}(q, \\dot{q})\\dot{\\rho}^b(q, \\dot{q}).\n    \\label{sol}\n   \\end{equation}\n\\end{lemma}\n\nAs before, we denote the set of all vector fields $\\mathbf{g}$ that satisfy \\textbf{Lemma\n  $\\mathbf{\\ref{GS}}$} by  $\\mathfrak{g}$, while \n$\\hbox{\\textbf{pr }}\\mathfrak{g} := \\{\\hbox{\\textbf{pr\n}}\\mathbf{g}\\ \\vert \\ \\ \\mathbf{g}\\in \\mathfrak{g}\\}$ is the set of\ntheir prolongations. Once again $\\hbox{\\textbf{pr }}\\mathfrak{g}$ is\ninvolutive, and the conditions under which $\\hbox{\\textbf{pr\n}}\\mathfrak{g}$ \ngenerates a generalized Lie symmetry group are given in\n\\cite{Olv1993}. Note, however, that while $\\rho=0$ and\n$\\dot{\\rho}=\\mathfrak{z}$ for any $\\mathfrak{z}\\in\\hbox{ker\n}M_{ab}(q, \\dot{q})$ is a solution of \n  Eq.~$(\\ref{sol})$, we require that $\\dot{\\rho} =\n  \\mathbf{d}\\rho\/\\mathbf{d}t$; these solutions cannot be generators of the\n  generalized Lie symmetry. Next, if $\\dot{\\rho}$ is a solution of\n  Eq.~$(\\ref{sol})$, then \n$\\dot{\\rho}^a+\\mathfrak{z}$ is a solution of Eq.~$(\\ref{sol})$ as\n  well, and thus these solutions are not unique. This, along with the\n  previous observation, leads us to generators that are constructed from\n  equivalence classes of prolongations. Finally, Eq.~$(\\ref{delta})$\n  gives for any $\\mathfrak{z} \\in\\hbox{ker }M_{ab}(q, \\dot{q})$,\n\\begin{equation}\n  0=\\mathfrak{z}^a\\left(\\frac{\\partial E}{\\partial\n    q^a}+F_{ab}(q,\\dot{q})\\dot{q}^b\\right),\n  \\label{e}\n\\end{equation}\non the solution surface $\\Delta_a(q,\\dot{q},\\ddot{q})=0$. If\nEq.~(\\ref{e}) does not hold identically, it must be\nimposed, leading to Lagrangian constraints\n\\cite{Car1990a}. More importantly, because each\n$q(t)\\in\\mathcal{O}(\\mathfrak{u}_0)$   \nmust lie on the Lagrangian constraint submanifold, any symmetry transformation of\n$q(t)$ generated by $\\mathbf{pr }\\>\\> \\mathbf{g}$ must give a path\n$Q(t)$ that also lies on the constraint submanifold.\n\nNot all vectors in $\\mathbf{pr}\\> \\mathfrak{g}$ will be generators of\nthe generalized Lie symmetry group for\n$\\mathcal{O}(\\mathfrak{u}_0)$. Determining which of these vectors are,\nand the relationship between the generators of symmetries of\nthe Euler-Lagrange equations of motion and those of \nthe action, is best done within the Lagrangian phase space\nframework. To accomplish this, we will need the following generalization of\n\\textbf{Lemma \\ref{GS}}.\n\nConsider the vector\n\\begin{equation}\n  \\mathbf{k}:=c\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}\n  +\\dot{c}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n    \\dot{q}},\n    \\nonumber\n\\end{equation}\nwith a $c\\in\\hbox{ker }M_{ab}(q, \\dot{q})$ along with the quantity\n\\begin{equation}\n  l_a:=F_{ab}c^b(q, \\dot{q})+M_{ab}\\dot{c}^b(q,\\dot{q}).\n  \\nonumber\n\\end{equation}\nAfter an integration by parts,\n\\begin{eqnarray}\n    l_a&=& c^b(q,\n    \\dot{q})\\left\\{F_{ab}(q, \\dot{q})-\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}\n    \\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n        \\dot{q}^a\\mathbf{\\partial} \\dot{q}^b}\\right\\},\n      \\nonumber\n      \\\\\n      &=& c^b(q,\n    \\dot{q})\\left\\{F_{ab}(q, \\dot{q})-\\left[\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}, \\frac{\\partial\n        \\>\\>\\,}{\\mathbf{\\partial} \\dot{q}^a}\\right]\\frac{\\mathbf{\\partial} L}{\\mathbf{\\partial}\n      \\dot{q}^b}- \\frac{\\mathbf{\\partial}\\>\\>\\,}{\\mathbf{\\partial}\n      \\dot{q}^a} \\left(\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}\n    \\frac{\\mathbf{\\partial} L}{\\mathbf{\\partial}\n      \\dot{q}^b}\\right)\\right\\}. \n    \\nonumber\n\\end{eqnarray}\nUsing Eq.~$(\\ref{Dt})$ we have\n\\begin{equation}\n  \\left[\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}, \\frac{\\mathbf{\\partial}\n      \\>\\>\\,}{\\mathbf{\\partial} \\dot{q}^a}\\right]\\frac{\\partial L}{\\partial \\dot{q}^b} =\n  -\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial} q^a\\mathbf{\\partial}\n    \\dot{q}^b}-\n  \\frac{\\mathbf{\\partial}\n    \\ddot{q}^c}{\\mathbf{\\partial}\n    \\dot{q}^a}\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n    \\dot{q}^c\\mathbf{\\partial} \\dot{q}^b}.\n  \\nonumber\n\\end{equation}\nAs $q(t)$ is  a solution of the Euler-Lagrange equations of\nmotion, we find that\n\\begin{equation}\nl_a= c^b(q, \\dot{q})\\left\\{F_{ab}(q, \\dot{q})\n+\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial} q^a\\mathbf{\\partial}\n    \\dot{q}^b}-\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n      \\dot{q}^a\\mathbf{\\partial}q_b} +\n  \\frac{\\mathbf{\\partial}\n    \\ddot{q}^c}{\\mathbf{\\partial}\n    \\dot{q}^a}\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n    \\dot{q}^c\\mathbf{\\partial} \\dot{q}^b}\n  \\right\\}.\n    \\nonumber\n\\end{equation}\nThis last expression vanishes after the definition of $F_{ab}(q,\n\\dot{q})$ is used along with the requirement that $c\\in\\hbox{ker\n}M_{ab}(q,\\dot{q})$. We then have the following result. \n\n\\begin{lemma} \\label{AllS} For any vector \n\\begin{equation}\n  \\mathbf{k}=c\\cdot\\frac{{\\partial}\\>\\>\\>}{{\\partial} q}\n  +\\dot{c}\\cdot\\frac{{\\partial}\\>\\>\\>}{{\\partial}\n    \\dot{q}},\n  \\nonumber\n\\end{equation}\nsuch that $c\\in\\hbox{ker }M_{ab}$, \n\\begin{equation}\n  0=F_{ab}c^b(q, \\dot{q})+M_{ab}\\dot{c}^b(q,\\dot{q}).\n  \\nonumber\n\\end{equation}.\n\\end{lemma}\n\n\\section{Generators of the Generalized Lie Symmetry for the\n  Euler-Lagrange Equations of Motion\\label{&review}}\n\nThe generators of the generalized Lie symmetry for both the\nEuler-Lagrange equations of motion and the action are best found using\nthe Lagrangian phase space approach to mechanics. This phase space\nand its concomitant mathematical structure provide the tools needed to\ndetermine both the generators of the symmetry and the solutions to the\nenergy equation on which they act. For the Euler-Lagrange equations of \nmotion this determination was done in \\cite{ADS2020}. In this\nsection we will review the Lagrangian phase space approach, establish\nthe notation used in this paper, and summarize the results obtained in\n\\cite{ADS2020} that are needed here. (We will also take the\nopportunity to correct typographical errors made in \\cite{ADS2020}.)\nProofs of the majority of the assertions listed in this section will\nnot be given; the reader is instead referred to \\cite{ADS2020} where the\nproofs and the context of their development can be found.\n\n\\subsection{The Lagrangian Phase space\\label{&phase}}\n\nFor a configuration space $\\mathbb{Q}$ the \\textbf{Lagrangian phase\n  space} $\\mathbb{P}_L$ is the tangent space\n$\\mathbb{P}_L=\\mathbf{T}\\mathbb{Q}$, with the coordinates on\n$\\mathbb{P}_L$ denoted as $\\mathfrak{u}=(q^1, \\dots, q^D, v^1, \\dots \nv^D)$. Integral flows on $\\mathbb{P}_L$,\n$t\\in[t_0,\\infty)\\to\\mathfrak{u}(t)\\in\\mathbb{P}_L$ \n\\cite{Abr1978}, for a set of initial data $\\mathfrak{u}_0=(q_0,\nv_0)$ are given as solutions to  \n\\begin{equation}\n\\frac{d\\mathfrak{u}}{dt}:=\\mathbf{X}  (\\mathfrak{u}),\n\\nonumber\n\\end{equation}\nwhere $\\mathbf{X}$ is a smooth vector field in \n$\\mathbf{T}\\mathbb{P}_L=\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$. The two\ntangent spaces $\\mathbf{T}\\mathbb{Q}$ and $\\mathbf{T}\\mathbb{P}_L$\nhave the bundle projections: $\\tau_{\\mathbb{Q}}:\\mathbf{T}\\mathbb{Q}\\to  \n\\mathbb{Q}$ and \n$\\tau_{\\mathbf{T}\\mathbb{Q}}:\\mathbf{T}(\\mathbf{T}\\mathbb{Q})\n\\to\\mathbf{T}\\mathbb{Q}$. They can be used to\nconstruct two other projection maps: $\\tau_{\\mathbb{Q}}\\circ \n\\tau_{\\mathbf{T}\\mathbb{Q}}:\\mathbf{T}(\\mathbf{T}\\mathbb{Q})  \n\\to\\mathbb{Q}$ and the prolongation of $\\tau_{\\mathbf{T}\\mathbb{Q}}$\nto $\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$ (see \\cite{Got1979} and\n\\cite{Abr1978}). This prolongation is the map \n$\\mathbf{T}\\tau_{\\mathbb{Q}}:\\mathbf{T}(\\mathbf{T}\\mathbb{Q})\\to\\mathbf{T}\\mathbb{Q}$,\nand is defined by requiring that the two maps\n$\\tau_{\\mathbb{Q}}\\circ \\tau_{\\mathbf{T}\\mathbb{Q}}$ and \n$\\tau_{\\mathbb{Q}}\\circ \\mathbf{T}\\tau_{\\mathbb{Q}}$ map any point in\n$\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$ to the same point in\n  $\\mathbb{Q}$. The \\textbf{vertical subbundle}\n$[\\mathbf{T}\\mathbb{P}_L]^v$ of $\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$\nis $[\\mathbf{T}\\mathbb{P}_L]^v = \\hbox{ker\n}\\mathbf{T}\\tau_{\\mathbb{Q}}$ \\cite{Got1979}; a $\\mathbf{X}^v\\in\n[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L]^v$ above a point\n$\\mathfrak{u}\\in\\mathbb{P}_L$ is called a \\textbf{vertical vector\n  field}. The \\textbf{horizontal subbundle} \n$[\\mathbf{T}\\mathbb{P}_L]^q$ of\n$\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$ is \n$[\\mathbf{T}\\mathbb{P}_L]^q = \\hbox{Image\n}\\mathbf{T}\\tau_{\\mathbb{Q}}$; a\n$\\mathbf{X}^q\\in[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L]^q$ is called a \n\\textbf{horizontal vector field}. Consequently, each  \n$\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$ consists of a\n$\\mathbf{X}^q \\in\n\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^q$ and a\n$\\mathbf{X}^v \\in\n\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v$ with $\\mathbf{X}\n= \\mathbf{X}^q + \\mathbf{X}^v$. In terms of local coordinates,  \n\\begin{equation}\n\\mathbf{X}^{q}\n:=X^{qa}\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}q^{a}} ,\\quad\n\\hbox{and}\\quad\\mathbf{X}^{v} :=X^{va}\n\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v^{a}}.\n\\nonumber\n\\end{equation}\nOf special interest is the second order Lagrangian vector field\n$\\mathbf{X}_L$. This vector field is the particular solution of\nEq.~$(\\ref{EnergyE})$ for which \n$\\mathbf{T}\\tau_{\\mathbb{Q}}\\circ\\mathbf{X}_L$ is the identity on\n$\\mathbf{T}\\mathbb{Q}$ (see \\cite{Abr1978}). In terms of local\ncoordinates\n\\begin{equation}\n  \\mathbf{X}_L = v^a \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q^a} +\n  X^{va} \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v^a}.\n  \\nonumber\n\\end{equation}\n\nThe space of one-forms on\n$\\mathbb{P}_L$ is the cotangent space\n$\\mathbf{T}^{*}\\mathbb{P}_L$. For a one-form $\\mathbf{\\alpha}\\in \n\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L$, and a vector\nfield $\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$, the\ndual prolongation map $\\mathbf{T}^{*}\\tau_{\\mathbb{Q}}$ is defined as\n\\begin{equation}\n  \\langle \\mathbf{\\alpha}\\vert\n  \\mathbf{T}\\tau_{\\mathbb{Q}}\\mathbf{X}\\rangle = \\langle\n  \\mathbf{T}^{*}\\tau_{\\mathbb{Q}} \\mathbf{\\alpha}\\vert\n  \\mathbf{X}\\rangle,\n  \\nonumber\n\\end{equation}\nafter a useful adaptation of Dirac's bra and ket notation. In\naddition, for a general $k$-form \\textbf{$\\mathbf{\\omega}$} in the\n$k$-form bundle $\\mathbf{\\Lambda}^{k}\\left(\\mathbb{P}_L\\right)$, \n\\begin{equation}\n\\mathbf{\\omega}\\left(  x\\right)  :\\mathbf{Y}_{1}\\otimes\\cdots\\otimes\n\\mathbf{Y}_{k}\\rightarrow\\left\\langle \\left.  \\mathbf{\\omega}\\left(  x\\right)\n\\right\\vert \\mathbf{Y}_{1}\\otimes\\cdots\\otimes\\mathbf{Y}_{k}\\right\\rangle\n\\in\\mathbb{R},\n\\nonumber\n\\end{equation}\nwith\n$\\mathbf{Y}_{j}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$ for $j = 1,\n\\dots, k$. The \\textbf{vertical one-form subbundle}\n$[\\mathbf{T}^{*}\\mathbb{P}_L]^v$ of $\\mathbf{T}^{*}\\mathbb{P}_L$ is \n$[\\mathbf{T}^{*}\\mathbb{P}_L]^v :=\\hbox{ker }\n\\mathbf{T}^{*}\\tau_{\\mathbb{Q}}$; a\n$\\mathbf{\\alpha}_v\\in [\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L]^v$ is\ncalled a \\textbf{vertical one-form}. The \\textbf{horizontal\none-form subbundle} $[\\mathbf{T}^{*}\\mathbb{P}_L]^q$ of\n$\\mathbf{T}^{*}\\mathbb{P}_L$ is \n$[\\mathbf{T}^{*}\\mathbb{P}_L]^q=\\hbox{Image }\n\\mathbf{T}^{*}\\mathbb{P}_L$; a\n$\\mathbf{\\alpha}_q\\in[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{Q}]^q$ is\ncalled a \\textbf{horizontal one-form}. Each one-form $\\mathbf{\\varphi}\\in\n\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L$ consists of a\n$\\mathbf{\\varphi}_{q} \\in \\left[\\mathbf{T}_{\\mathfrak{u}}^{*}\\mathbb{P}_L\\right]^q$ and\na $\\mathbf{\\varphi}_{v} \\in \n\\left[\\mathbf{T}_{\\mathfrak{u}}^{*}\\mathbb{P}_L\\right]^q$\nsuch that $\\mathbf{\\varphi}=\\mathbf{\\varphi}_{q}\n+\\mathbf{\\varphi}_{v}$. In terms\nof local coordinates \n$\\mathbf{\\varphi}_{q} :=\\varphi_{qa} \\ \\mathbf{d}q^{a}$ and \n$\\mathbf{\\varphi}_{v} :=\\varphi_{va} \\mathbf{d}v^{a}$. \n\nFollowing \\cite{Got1979, Got1980}, the Lagrangian two-form is defined\nas $\\mathbf{\\Omega}_L := -\\mathbf{d}\\mathbf{d}_JL$, \nwhere $\\mathbf{d}_J$ is the vertical derivative (see\n\\cite{Got1979}). This two-form can be expressed as\n$\\mathbf{\\Omega}_{L}:=\\mathbf{\\Omega}_{F}+\\mathbf{\\Omega}_{M}$ such\nthat for any $\\mathbf{X},\n\\mathbf{Y}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$. \n\\begin{equation}\n\\mathbf{\\Omega}_F(\\mathbf{X},\\mathbf{Y}) :=\n\\mathbf{\\Omega}_L(\\mathbf{T}\\tau_{\\mathbb{Q}}\\mathbf{X},\n\\mathbf{T}\\tau_{\\mathbb{Q}}\\mathbf{Y}),\n\\nonumber\n\\end{equation}\nand is thus the \\textbf{horizontal two-form} of $\\mathbf{\\Omega}_L$. As\n$\\mathbf{\\Omega}_M(\\mathbf{X},\\mathbf{Y})=\\mathbf{\\Omega}_L(\\mathbf{X},\\mathbf{Y})- \n\\mathbf{\\Omega}_F(\\mathbf{X},\\mathbf{Y})$, $\\mathbf{\\Omega}_M$ is then\na \\textbf{mixed two-form} of \n$\\mathbf{\\Omega}_L$. In terms of local coordinates,  \n\\begin{equation}\n\\mathbf{\\Omega}_{L}=-\\mathbf{d{\\theta}}_{L},\\quad \\hbox{where} \\quad\n\\mathbf{\\theta}_{L}:=\\frac{\\partial L}{\\partial v^{a\n}\\mathbf{d}q^{a},\n\\nonumber\n\\end{equation}\nwhile\n\\begin{equation}\n\\mathbf{\\Omega\n}_{F}:=\\frac{1}{2}F_{ab}\\mathbf{d}q^{a}\\wedge\\mathbf{d}q^{b\n,\\ \\hbox{and} \\ \\mathbf{\\Omega}_{M}:=M_{ab}\\mathbf{d}q^{a}\\wedge\\mathbf{d}v^{b}.\n\\nonumber\n\\end{equation}\n\nFor regular Lagrangians $\\mathbf{X}_L$ is the unique solution of\nEq.~$(\\ref{EnergyE})$. For almost regular Lagrangians, on the other\nhand, this solution is not unique, but instead depends on \n\\begin{equation}\n\\ker\\>  \\mathbf{\\Omega}_{L}\\left(  \\mathfrak{u}\\right) \n:=\\left\\{  \\mathbf{K}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L\n\\ \\vert\\ \\ i_{\\mathbf{K}}\\mathbf{\\Omega}_{L}=0\\right\\}.\n\\nonumber\n\\end{equation}\nFrom \\textbf{Section \\ref{&Sym}} we expect this kernel to\nplay a role in determining the generators of the generalized Lie \nsymmetry of both the Euler-Lagrange equations of motion and the\naction. Indeed, consider the natural isomorphism $iso: (t, q, \\dot{q}, \n\\ddot{q}) \\in \\mathbb{M}^{(2)} \\to (t, q, v, X^{va}_{L})$ defined in\n\\cite{ADS2020}, and the prolongation $\\hbox{\\textbf{pr} }\\mathbf{g}$ \nof a generator $\\mathbf{g}\\in\\mathbf{\\mathfrak{g}}$ of a \ngeneralized Lie symmetry of the Euler-Lagrange equations of\nmotion. This $\\hbox{\\textbf{pr }}\\mathbf{g}$ contains the vector  \n\\begin{equation}\n  \\mathbf{k}=\\rho\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} q} +\n  \\dot{\\rho}\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial}\\dot{q}}.\n  \\nonumber\n\\end{equation}\nThe collection of all such vectors has been shown to be\ninvolutive (see \\cite{ADS2020}). The isomorphism maps $iso:\n\\mathbf{k}\\to\\mathbf{k}'$ where \n\\begin{equation}\n  \\mathbf{k}'=\\rho\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} q} +\n  \\dot{\\rho}\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} v}.\n  \\nonumber\n\\end{equation}\nThen $\\mathbf{k}'\\in\\mathbf{T}\\mathbb{P}_L$, and from\n\\textbf{Lemma \\ref{GS}}, $\\mathbf{k}'\\in\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$ as well. A similar result holds for\nthe generators in $\\mathbf{\\mathfrak{g}}_L$ after \n\\textbf{Lemma \\ref{Action-Sym}} and \\textbf{Lemma \\ref{AllS}} are used.\n\nThe two-form $\\mathbf{\\Omega}_L$ gives the lowering map\n$\\mathbf{\\Omega}_L^{\\flat}:\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L}\\rightarrow \n\\mathbf{T}_{\\mathfrak{u}}^{\\ast}\\mathbb{P}_{L}$, with\n$\\Omega_L^{\\flat}\\mathbf{X}:=i_{\\mathbf{X}}\\mathbf{\\Omega}_L$.\nThis map consists of\n$\\Omega_{L}^{\\flat}=\\Omega_{F}^{\\flat}+\\Omega_{M}^{v\\flat}+\\Omega_{M}^{q\\flat}$, \nwith $\\Omega_{F}^{\\flat}:\n\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L \\to\n\\left[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^q$; \n$\\Omega_{M}^{q\\flat}:\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L \n\\to\\left[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^q$; and\n$\\Omega_{M}^{v\\flat}:\n\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L \\to\n\\left[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v$. In terms of local\ncoordinates, \n$\\Omega_{F}^{\\flat}\\mathbf{X} = F_{ab}X^{qa}\\mathbf{d}q^b$, \n$\\Omega_{M}^{q\\flat}\\mathbf{X}= -M_{ab}X^{va}{}\\mathbf{d}q^{b}$, and\n$\\Omega_{M}^{v\\flat}\\mathbf{X}= M_{ab}X^{qa}\\mathbf{d}v^{b}$.\n\nFor almost regular Lagrangians $\\ker\\>\n  \\Omega_{M}^{v\\flat} = \\mathcal{C}\\oplus\n  \\left[ \\mathbf{T}_{\\mathfrak{u}} \\mathbb{P}_{L}\\right]  ^{v}$ \n  while $\\ker\\>  \\Omega_{M}^{q\\flat}  =\\left[\n    \\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L}\\right]  ^{q}\\oplus  \n  \\mathcal{G}$. Here\n  \\begin{equation}\n    \\mathcal{C}:=\\left\\{\\mathbf{C}\\in[\\mathbf{T}_q\\mathbb{P}_L]^q\n      \\ \\vert \\ i_{\\mathbf{C}}\\mathbf{\\Omega}_M =0\\right\\},\n      \\nonumber\n  \\end{equation}\n  and\n  \\begin{equation}\n    \\mathcal{G}:=\\left\\{\\mathbf{G}\\in\n    [\\mathbf{T}_q\\mathbb{P}_L]^v \\ \\vert\n    \\ i_{\\mathbf{G}}\\mathbf{\\Omega}_M =0\\right\\}.\n    \\nonumber\n  \\end{equation}\nAs $M_{ab}(\\mathfrak{u})$ has constant rank on\n$\\mathbb{P}_L$, there exists a basis,  \n\\begin{equation}\n\\Big\\{\n\\mathbf{\\mathfrak{z}}_{\\left(  n\\right)  }\\left(  \\mathfrak{u}\\right) =\\left(  \\mathfrak{z\n_{\\left(  n\\right)  }^{1}\\left(  \\mathfrak{u}\\right)  ,\\ldots,\\mathfrak{z\n_{\\left(  n\\right)  }^{D}\\left(  \\mathfrak{u}\\right)  \\right)  \\vert\n\\ M_{ab}\\left(  \\mathfrak{u}\\right)  \\mathfrak{z}_{\\left(  n\\right)  \n^{b}\\left(  \\mathfrak{u}\\right)  =0,\nn=1,\\ldots,N_{0} \\Big\\}  ,\n\\nonumber\n\\end{equation}\nfor $\\ker  M_{ab}\\left(  \\mathfrak{u}\\right) $ at each\n$\\mathfrak{u}\\in\\mathbb{P}_{L}$. Spans of both\n\\begin{eqnarray}\n  \\mathcal{C}&=& \\hbox{span }\n  \\left\\{\\mathbf{U}^q_{(n)}=\\mathbf{\\mathfrak{z}}_{(n)}\\cdot\n    \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}, n=1,\n    \\dots, N_0\\right\\}, \\hbox{and}\\>\n    \\nonumber\n    \\\\\n  \\mathcal{G}&=& \\hbox{span }\n  \\left\\{\\mathbf{U}^v_{(n)}=\\mathbf{\\mathfrak{z}}_{(n)}\\cdot\n    \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v}, n=1,\n    \\dots, N_0\\right\\},\n    \\nonumber\n\\end{eqnarray}\ncan then be constructed. Importantly, $\\mathcal{G}$ is involutive\n\\cite{Car1990a}, and when the rank of\n$\\mathbf{\\Omega}_L(\\mathfrak{u})$ is constant on $\\mathbb{P}_L$,\n$\\hbox{ker } \\mathbf{\\Omega}_L(\\mathfrak{u})$ is involutive as well.\n\nCorresponding to $\\mathbf{U}^q_{(n)}$ and $\\mathbf{U}^v_{(n)}$ we have\nthe one-forms  \n$\\mathbf{\\Theta}^{(m)}_q$ and $\\mathbf{\\Theta}^{(m)}_v$\nwhere $\\langle\\mathbf{\\Theta}^{(m)}_q\\vert\n\\mathbf{U}^q_{(n)} \\rangle= \\delta_{(n)}^{(m)}$ and\n$\\langle\\mathbf{\\Theta}^{(m)}_v\\vert \\mathbf{U}^v_{(n)} \\rangle=\n\\delta_{(n)}^{(m)}$. Then $\\left[  \\mathbf{T\n_{\\mathfrak{u}}\\mathbb{P}_{L}\\right]  ^{q}\n\\mathcal{C\n\\oplu\n\\mathcal{C\n_{\\perp}$ and $\\left[  \\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L}\\right]  ^{v}\n\\mathcal{G\n\\oplu\n\\mathcal{G\n_{\\perp}$, where \n\\begin{eqnarray\n\\mathcal{C\n_{\\perp}:=\\bigg\\{  \\mathbf{X}\\in\\left[  \\mathbf{T}_{\\mathfrak{u}\n  \\mathbb{P}_{L}\\right]  ^{q}\\ &\\vert&\\\n\\left\\langle \\left.  \\mathbf{\\Theta}_{q}^{(n)}\\right\\vert \\mathbf{X\n\\right\\rangle =0,\\>\\>  n=1,\\dots,N_{0}  \\ \\bigg\\}  ,\n\\hbox{and\n\\nonumber\n\\\\\n\\mathcal{G}_{\\perp}:=\\bigg\\{  \\mathbf{X}\\in\\left[\n  \\mathbf{T}_{\\mathfrak{u}}\n  \\mathbb{P}_{L}\\right]  ^{v}\\ &\\vert&\\\n\\left\\langle \\left.  \\mathbf{\\Theta}_{v}^{(n)}\\right\\vert \\mathbf{X\n\\right\\rangle =0, \\>\\>   n=1,\\dots,N_{0}  \\ \\bigg\\}.\n\\nonumber\n\\end{eqnarray}\nThe vectors that lie in $\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$ can be determined by using the reduced\nmatrix $ {F}_{nm}:=\\mathfrak{z}_{\\left(  n\\right)\n}^{a}F_{ab}\\mathfrak{z}_{\\left( \nm\\right)  }^{b}$ to define\n\\begin{equation}\n\\overline\n\\mathcal{C\n}:=\\left\\{  \\overline{\\mathbf{C}}\\i\n\\mathcal{C\n\\ \\bigg\\vert\\ \\sum_{m=1}^{N_{0}}\\bar{F}_{nm}\\overline{C}^{\\left(  m\\right)\n}=0\\right\\}  \\subse\n\\mathcal{C}\n\\nonumber\n\\end{equation}\nThen,\n\n\\begin{theorem}\n\\label{@NVGen}The vectors $\\mathbf{K=K}^{q}+\\mathbf{K}^{v}\\in\\ker\n\\mathbf{\\Omega}_{L}$ are given by\n\\begin{equation}\n\\mathbf{K}^{q}=\\overline{\\mathbf{C}}\\mathbf{,\\ \\ K}^{v}=\\mathbf{G+\n\\widehat{\\mathbf{C}},\n\\nonumber\n\\end{equation}\nwhere, $\\overline{\\mathbf{C}}\\in\\overline\n\\mathcal{C\n}$, $\\mathbf{G}\\i\n\\mathcal{G\n$, and $\\widehat{\\mathbf{C}}\\i\n\\mathcal{G\n_{\\perp}$ is the unique solution of $M_{ab}\\widehat{C}^{b}=-F_{ab}\\overline\n{C}^{b}$.\n\\end{theorem}\nWe found in \\cite{ADS2020} that $\\dim\\>  \\left(\\ker\n\\mathbf{\\Omega}_{L}\\left(  \\mathfrak{u}\\right)\\right) \n  =N_{0}+\\bar{D}, $ where $\n\\ \\bar{D}  :=\\dim \\>\\overline{\\mathcal{C}}\\le N_0$ (see \\cite{ADS2020}\nfor proof). However, the results of \\textbf{Lemma \\ref{AllS}} show\nthat we can construct from any vector $\\mathbf{U}^q\\in \\mathcal{C}$ a\nvector that lies in the $\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})$,\nand as dim $(\\mathcal{C})=N_0$, it follows that dim $(\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u}))=2N_0$. \n\n\\subsection{First-order Lagrangian constraints\\label{&LCon}}\n\nFor singular Lagrangians solutions of the energy equation\n$\\mathbf{X}_E$ are not unique. It is well known that they also do not,\nin general, exist throughout $\\mathbb{P}_L$, but are instead confined\nto a submanifold of the space given by Lagrangian constraints. \n\nWith $\\mathbf{X}_{E}=\\mathbf{X}^q_{E}+\\mathbf{X}^v_{E}$, it is\nconvenient to use the one form $\\mathbf{\\Psi}$\n\\begin{equation}\n  \\Omega_{M}^{q\\flat}\\mathbf{X}_{E}^{v}=\\mathbf{\\Psi}.\n\\nonumber\n\\end{equation}\nconstructed from the energy equation. The \\textbf{first-order\n  constraint functions} are then $\\gamma_{n}^{\\left[\n    1\\right]  }:=\\left\\langle \n\\left.  \\mathbf{\\Psi}\\right\\vert \\mathbf{U}_{\\left(  n\\right)\n}^{q}\\right\\rangle=0$ for $n=1,\\ldots,N_{0}$. In terms of local coordinates,\n\\begin{equation}\n\\gamma_{n}^{\\left[  1\\right]  }= U_{\\left(  n\\right)  \n^{qa}\\left(  \\frac{\\partial E}{\\partial q^{a}}+F_{ab}v^{b}\\right).\n\\nonumber\n\\end{equation}\nThey may also be expressed \\cite{Got1979,\n  Got1980} as $\\gamma^{[1]}_n = \\langle\n\\mathbf{d}E\\vert\\mathbf{P}_{(n)}\\rangle=\\mathbf{P}_{(n)}E$ for any\nbasis $\\{\\mathbf{P}_{(n)}\\}$ of ker $\\mathbf{\\Omega}_L(\\mathfrak{u})$\nfor which $\\langle\\mathbf{\\Theta}^{(m)}_q\\vert\n\\mathbf{P}_{(n)}\\rangle=\\delta^{(m)}_{(n)}$. \nIn general, $\\gamma_{n}^{\\left[  1\\right]  }\\ne0$ on\n$\\mathbb{P}_L$. Instead, the condition $\\gamma_{n}^{\\left[  1\\right]\n}=0$ must be imposed, and this in turn defines a set of submanifolds\nof $\\mathbb{P}_L$ given by the collection $\\hbox{C}_{L}^{\\left[  1\\right]  }:=\\left\\{\n\\gamma_{1}^{\\left[  1\\right]  },\\ldots,\\gamma_{N_{0}}^{\\left[  1\\right]\n}\\right\\}  $. The collection of these\nsurfaces, \n$\\mathbb{P}_{L}^{\\left[  1\\right]  }:=\\left\\{ \n\\mathfrak{u}\\in\\mathbb{P}_{L}\\ \\vert\\ \\ \\gamma_{n}^{\\left[  1\\right]\n}\\left(  \\mathfrak{u}\\right)  =0\\, ,  n=1,\\ldots,N_{0}\\ \\right\\}$ is\ncalled the \\textbf{first-order Lagrangian constraint submanifold}, and\nhas $\\dim \\mathbb{P}_{L}^{\\left[1\\right]  } =2D-I_{\\left[\n    1\\right]}$. Here $I_{\\left[1\\right]  }$ is the\nnumber of independent functions in $\\hbox{C}_{L}^{\\left[  1\\right]\n}$ with $I_{\\left[ 1\\right]  }=\\hbox{rank }  \\left\\{\\mathbf{d}\\gamma_{n}^{\\left[\n      1\\right]  }\\right\\} \\leq N_{0}$. \n\nThe \\textbf{constraint\n  one-form}  \n\\begin{equation}\n  \\mathbf{\\beta}[\\mathbf{X}_E] :=\n  \\mathbf{d}E-i_{\\mathbf{X}_E}\\mathbf{\\Omega}_L,\n\\nonumber\n\\end{equation}\nwas introduced in \\cite{ADS2020} with the condition \n$\\mathbf{\\beta}[\\mathbf{X}_E]=0$ giving both the  \nsolution of the energy equation and the submanifold\n$\\mathbb{P}_L^{[1]}$. As $\\langle\\mathbf{\\beta} \\vert \\mathbf{U}^q_{(n)}\\rangle=\n\\gamma_n^{[1]}$, this $\\mathbf{\\beta}[\\mathbf{X}_E]$ can also\nbe expressed as \n\\begin{equation}\n  \\mathbf{\\beta}[\\mathbf{X}_E]=\\sum_{n=1}^{N_0}\n  \\gamma_n^{[1]}\\mathbf{\\Theta}^{(n)}_q.\n  \\label{beta}\n\\end{equation}\n\n\\subsection{The Generalized Lie Symmetry Group for the Euler-Lagrange\n  Equations of motion\\label{&GenEL-Sym}} \n\nThe generalized Lie symmetry group for $\\mathcal{O}(\\mathfrak{u}_0)$\nis determined using\n\\begin{equation}\n  \\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})} :=\n  \\{\\mathbf{P} \\in \\hbox{ker } \\mathbf{\\Omega}_L(\\mathfrak{u})\\\n  \\vert \\ [\\mathbf{G},\\mathbf{P}]\n  \\in\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v \\>\\>\\forall\n  \\>\\>\\mathbf{G}\\in\\mathcal{G}\\},  \n\\end{equation}\nalong with the following collection of functions on $\\mathbb{P}_L$,  \n\\begin{equation}\n  \\overline{\\mathcal{F}} := \\{f\\in C^\\infty \\hbox{on } \\mathbb{P}_L\n  \\ \\vert\\ \\ \\mathbf{G}f = 0 \\>\\>\\forall\\>\\> \\mathbf{G} \\in\n  \\mathcal{G}\\}.\n\\nonumber\n\\end{equation}\nThis $\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is also\ninvolutive.\n\nThe following results were proved in \\cite{ADS2020}.\n\n\\begin{lemma} \\label{basic} Let  \n  $\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$ \n  and $\\mathbf{G}\\in\\mathcal{G}$ such that $[\\mathbf{G},\n    \\mathbf{X}]\\in\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})$. Then \n  $[\\mathbf{G},\n    \\mathbf{X}]\\in\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v$\n  iff $[\\mathbf{G}, \\mathbf{X}]\\in\\mathcal{G}$.\n\\end{lemma}\nIt then follows that $[\\mathbf{G},\\mathbf{P}]\\in\\mathcal{G}$ for all\n$\\mathbf{P}\\in \\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$. As $\\mathcal{G}$ is involutive\nand as $\\mathcal{G}\\subset \\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})$,\n$\\mathcal{G}\\subset \\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ as well, and thus   \n$\\mathcal{G}$ is an ideal of $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$.  \n\n\\begin{lemma}\n  \\label{basis}\n  There exists a choice of basis for ker\n  $\\mathbf{\\Omega}_L(\\mathfrak{u})$ that is \n  also a basis of $\\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}$.\n\\end{lemma}\n\nAs $\\mathcal{G}$ is an ideal of $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$, we may define for\nany $\\mathbf{P}_1, \\mathbf{P}_2\\in\\overline{\\hbox{ker \n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ the equivalence relation:\n$\\mathbf{P}_1\\sim\\mathbf{P}_2$ iff\n$\\mathbf{P}_1-\\mathbf{P}_2\\in\\mathcal{G}$. The equivalence class,\n\\begin{equation}\n  \\left[\\mathbf{P}\\right]:=\n  \\{\\mathbf{Y}\\in \\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}\\ \\vert \n  \\ \\mathbf{Y}\\sim\\mathbf{P}\\},  \n\\end{equation}\ncan be constructed along with the quotient space\n$\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$.   \n(For the sake of notational clarity  we will suppress\nthe square brackets for equivalence classes when there\nis no risk of confusion.) This space is a collection of\nvectors that lie in the kernel of $\\mathbf{\\Omega}_L$, but with the\nvectors in $\\mathcal{G}$ removed; $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$ thereby addresses the\nfirst two observations listed at the end of \\textbf{Section\n  \\ref{&EL-Sym}}. \n\nWe now turn our attention to the third observation. Because the\nintegral flow $\\mathfrak{u}_{\\mathbf{X}}(t)$ of any solution \n$\\mathbf{X}$ of the energy equation must lie on $\\mathbb{P}_L^{[1]}$, a \nsymmetry transformation of $\\mathfrak{u}_{\\mathbf{X}}(t)$ must result\nin an integral flow $\\mathfrak{u}_{\\mathbf{Y}}(t)$ of another solution\n$\\mathbf{Y}$ of the energy equation, which must also lie on\n$\\mathbb{P}_L^{[1]}$. Implementing this condition is done through\n$\\mathbf{\\beta}[\\mathbf{X}_E]$. \n\nAs $\\langle\\mathbf{\\beta}[\\mathbf{X}_E]\\vert \\mathbf{G}\\rangle=\\langle\n\\mathbf{d}E\\vert\\mathbf{G}\\rangle= \\mathbf{G}E=0$ for all  \n$\\mathbf{G}\\in\\mathcal{G}$ on $\\mathbb{P}_L^{[1]}$, the Lie derivative\n$\\mathfrak{L}_{\\mathbf{G}}$ of $\\mathbf{\\beta}$ along $\\mathbf{G}$ is, \n\\begin{equation}\n  \\mathfrak{L}_{\\mathbf{G}}\\mathbf{\\beta}[\\mathbf{X}_E]= \\sum_{n=1}^{N_0}\n  \\left(\\mathbf{G}\\gamma_n^{[1]}\\right) \\mathbf{\\Theta}^{(n)}_q.\n  \\nonumber\n\\end{equation}\nGiven a\n$\\mathbf{P}_{(n)} \\in\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ such that $\\mathbf{P}_{(n)} = \n\\mathbf{U}_{(n)}^q + \\widehat{\\mathbf{U}}_{(n)} +\\mathbf{G}'$ with\n$\\mathbf{G}'\\in\\mathcal{G}$, $\\mathbf{G}\\gamma_n^{[1]}=\n       [\\mathbf{G}, \\mathbf{P}_{(n)}]E + \n\\mathbf{P}_{(n)}\\mathbf{G}E$. But $\\mathcal{G}$ is an ideal of\n$\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}$, and thus\n$\\mathbf{G}\\gamma_{n}^{[1]}=0$ on the first-order constraint\nmanifold. It follows that $\\mathfrak{L}_{\\mathbf{G}}\\mathbf{\\beta}\n=0$ on $\\mathbb{P}^{[1]}_L$. The collection of vectors,     \n\\begin{equation}\n  \\mathcal{S}\\hbox{ym} := \\big\\{\\mathbf{P}\\in \\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}\n  \\ \\vert\\ \\\n  \\mathfrak{L}_{\\mathbf{P}}\\mathbf{\\beta}[\\mathbf{X}_E] =\n  \\mathbf{d}\\langle \\mathbf{\\beta}[\\mathbf{X}_E]\\vert \\mathbf{P}\\rangle \\hbox{ on }\n  \\mathbb{P}_L^{[1]}\\big\\}, \n  \\nonumber\n\\end{equation}\nis therefore well defined, and is involutive. It follows that\n$\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}$ iff    \n$\\langle \\mathbf{d}\\mathbf{\\beta}[\\mathbf{X}_E]\\vert \\mathbf{P}\\otimes\n\\mathbf{X}\\rangle=0$ for all $\\mathbf{X}\\in\n\\mathbf{T}\\mathcal{P}_L$. We are then able to construct from each\n$\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}$ a one-parameter subgroup\n$\\mathbf{\\sigma}_{\\mathbf{P}}(\\epsilon,x)$ defined as the solution\nto  \n\\begin{equation}\n  \\frac{d\\mathbf{\\sigma}_{\\mathbf{P}}}{d\\epsilon} :=\n  \\mathbf{P}\\left(\\mathbf{\\sigma}_{\\mathbf{P}}\\right), \n  \\nonumber\n\\end{equation}\nwhere $\\sigma_{\\mathbf{P}}(0,\\mathfrak{u}) = \\mathfrak{u}$ for\n$\\mathfrak{u}\\in\\mathbb{P}_L$. The collection of such \nsubgroups with give the Lie group\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$.\n\n\\subsection{Euler-Lagrange Solutions of the Energy\n  Equation\\label{&Sol}}\n\nWe denote the set of \\textbf{general solutions} to the energy equation as\n\\begin{equation}\n\\mathcal{S}\\hbox{ol}\n:=\\{\\mathbf{X}_{E}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\n\\ \\vert\\ \\ i_{\\mathbf{X}_{E}}\\mathbf{\\Omega}_L = \\mathbf{d}E\n\\hbox{ on } \\mathbb{P}_L^{[1]}\\}.\n\\nonumber\n\\end{equation}\nIf $\\mathfrak{u}(t)$ is the integral flow of a vector in\n$\\mathcal{S}\\hbox{ol}$ whose projection onto\n$\\mathbb{Q}$ corresponds to a solution of\nthe Euler-Lagrange equations of motion, then\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ must map one of\nsuch flows into another one. However, while\n$\\mathfrak{L}_{\\mathbf{G}}\\mathbf{X}_{L} = \n[\\mathbf{G}, \\mathbf{X}_{L}]\\in \\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$, in general\n$\\mathfrak{L}_{\\mathbf{G}}\\mathbf{X}_{L} \\notin \\mathcal{G}$. The \naction of $\\sigma_{\\mathbf{P}}$ on the flow\n$\\mathfrak{u}_{\\mathbf{X}_{L}}$ will in general result in a flow\n$\\mathfrak{u}_{\\mathbf{Y}}$ generated by a $\\mathbf{Y}$ that is\n\\textit{not} a SOLVF. It need not even be a solution of the energy\nequation. By necessity, general solutions of the energy equation must\nbe considered, leading us to consider the collection of solutions\n\\begin{equation}\n\\overline{\\mathcal{S}\\hbox{ol}}\n:=\\{\\overline{\\mathbf{X}}_{EL}\\in\\mathcal{S}\\hbox{ol}  \n\\ \\vert \\ [\\mathbf{G},\n  \\overline{\\mathbf{X}}_{EL}]\\in\n\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v \\>\\>\n\\>\\> \\forall \\mathbf{G}\\in\n\\mathcal{G}\\}.  \n\\nonumber\n\\end{equation}\nThis collection generates the family of integral flows  \n\\begin{equation}\n  \\mathcal{O}_{EL}(\\mathfrak{u}_0) := \\bigg\\{\\mathfrak{u}(t) \\ \\bigg\\vert \\\n  \\frac{d\\mathfrak{u}}{dt}=\\overline{\\mathbf{X}}_{EL}(\\mathfrak{u}),\n  \\overline{\\mathbf{X}}_{EL}\\in\\overline{\\mathcal{S}\\hbox{ol}},\n  \\>\\>\\hbox{and }\n  \\mathfrak{u}(t_0)=\\mathfrak{u}_0\\bigg\\}.\n  \\nonumber\n\\end{equation}\nImportantly, if $\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}$, then \n\\begin{equation}\n  i_{[\\mathbf{X}_E,\n      \\mathbf{P}]}\\mathbf{\\Omega}_L=i_{\\mathbf{P}}\\mathbf{d}\\mathbf{\\beta}[\\mathbf{X}_E]=0.\n  \\nonumber\n\\end{equation}\nAs such, we find that \n\n\\begin{lemma} \\label{X_EL} $[\\overline{\\mathbf{X}}_{EL}, \\mathbf{P}] \\in\n  \\overline{\\hbox{ker } \\mathbf{\\Omega}_L(\\mathfrak{u})}$ for all\n  $\\mathbf{P}\\in \\mathcal{S}\\hbox{ym}$.  \n\\end{lemma}\n\n\\noindent It then follows that\n\n\\begin{theorem}\\label{Group-EL} $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ forms a group of\n  symmetry transformations of $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$.\n\\end{theorem}\n\n\\noindent Proof of both assertions can be found in \\cite{ADS2020}.\n\nThe generators of the generalized Lie symmetry for\n$\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ are \nthus given by $\\mathcal{S}\\hbox{ym}$. The corresponding solutions\nto the Euler-Lagrange equations that have this symmetry are given by\n$\\overline{\\mathcal{S}\\hbox{ol}}$, and a vector\n$\\overline{\\mathbf{X}}_{EL} \\in \\overline{\\mathcal{S}\\hbox{ol}}$ is\ncalled a \\textbf{second-order, Euler-Lagrange vector field (SOELVF)}. It has\nthe general form,\n\\begin{equation}\n  \\overline{\\mathbf{X}}_{EL}=\\overline{\\mathbf{X}}_{L} +\n  \\sum_{m=1}^{N_0} u^m(\\mathfrak{u}) \\left[\\mathbf{P}_{(m)}\\right],\n  \\label{EL}\n\\end{equation}\nwhere $u^m(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$ and\n$\\{[\\mathbf{P}_{(n)}], n=1, \\dots, N_0\\}$ is a choice of basis \nfor $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$. The vector field\n$\\overline{\\mathbf{X}}_L$ is constructed from the second order\nLagrangian vector field $\\mathbf{X}_L$ and vectors in $\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$ by requiring\n$\\overline{\\mathbf{X}}_L\\in\\overline{\\mathcal{S}\\hbox{ol}}$. This\nconstruction is described in \\cite{ADS2020}; we will only need the\nexistence of such a vector field in this paper.\n\n\\section{Generalized Lie Symmetries of the Action and its Impact on\n  Dynamics\\label{&A-S}} \n\nWe now turn our attention to the generators of the\ngeneralized Lie symmetry of the action, and the impact this symmetry\nhas on the evolution of dynamical systems.\n\n\\subsection{The Generalized Lie Symmetry of the Action\\label{SandA}}\n\nIn determining the conditions (as listed in \\textbf{Lemma \\ref{Action-Sym}}) under\nwhich the action admits a generalized Lie symmetry, the understanding\nthat the action must have this symmetry for all possible paths on\n$\\mathbb{Q}$ played an essential role. By necessity, these\nconditions could only be placed on $\\rho_L$, and not on \n$\\dot{\\rho}_L$; unlike $\\rho_L$, $\\dot{\\rho}_L$ depends explicitly on the\nevolution of a particular path, while the symmetry must hold\nfor all paths. We note, however, that the family $\\mathcal{O}_{EL}$ of\ntrajectories determined by the Euler-Lagrange equations of motion also consists of\npaths on $\\mathbb{Q}$, and as such the generalized Lie symmetry of the\naction is also a symmetry of $\\mathcal{O}_{EL}$. Importantly, how these\ntrajectories evolve with time is known, and as such, the $\\dot{\\rho}_L$ for a\ngiven $\\rho_L$ is also known for these trajectories. With this\nunderstanding, and after comparing \\textbf{Lemma \\ref{Action-Sym}} and\nthe results of \\textbf{Lemma \\ref{AllS}} with \\textbf{Lemma \\ref{GS}},\nwe conclude that the generators of the generalized Lie symmetry of the\naction must also be generators of the generalized Lie symmetry of the\nEuler-Lagrange equations of motion. This leads us to consider the\nfollowing collection of vectors.  \n\\begin{equation}\n  \\mathcal{S}\\hbox{ym}\\mathcal{L} = \\{\\mathbf{P}\\in \\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}\\ \\vert\\\n  \\gamma_{\\mathbf{P}}^{[1]}=\\langle \\mathbf{\\beta}\\vert\\mathbf{P}\\rangle\n    =0 \\hbox{ on }\\mathbb{P}_L \\}.\n    \\nonumber\n\\end{equation}\nWe will also need  $N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}=\\hbox{dim\n}(\\mathcal{S}\\hbox{ym}\\mathcal{L})$ in the following. \n\n\\begin{lemma}\\label{subset}\n  $\\mathcal{S}\\hbox{ym}\\mathcal{L}\\subset\\mathcal{S}\\hbox{ym}$.\n  \n  \\begin{proof}\n    Let $\\{\\mathbf{P}_{(l)}, l = 1, \\dots, N_0 \\}$ be a basis of\n    $\\overline{\\hbox{ker\n      }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$ such that\n    $\\mathbf{P}_{(l)}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$ for\n    $l= 1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$. We may choose the basis of\n    $\\mathcal{C}$ such that $\\langle \\mathbf{\\Theta}^{(m)}_q\\vert\n    \\mathbf{P}_{(l)}\\rangle=\\delta^{(m)}_{(l)}$. Then for any\n    $\\mathbf{P}_{(n)}\\in\\mathcal{S}\\hbox{ym}$, we see from\n    Eq.~$(\\ref{beta})$ that, \n    \\begin{equation}\n      \\langle\\mathbf{d}\\mathbf{\\beta}\\vert \\mathbf{P}_{(n)}\\otimes\\mathbf{Y}\\rangle= \\sum_{m=1}^{N_0}\n      \\left(\n      \\langle\\mathbf{d}\\gamma_m^{[1]}\\vert\n      \\mathbf{P}_{(n)}\\rangle \\langle \\mathbf{\\Theta}^{(m)}_q\\vert\n      \\mathbf{Y}\\rangle-\n      \\langle\\mathbf{d}\\gamma_m^{[1]}\\vert\\mathbf{Y}\n      \\rangle \\langle \\mathbf{\\Theta}^{(m)}_q\\vert\n      \\mathbf{P}_{(n)}\\rangle+\n      \\gamma_m^{[1]}\\langle\\mathbf{d}\\mathbf{\\Theta}^{(m)}_q\\vert\n      \\mathbf{P}_{(n)}\\otimes\\mathbf{Y}\\rangle \n      \\right),\n      \\nonumber\n    \\end{equation}\n    for any $\\mathbf{Y}\\in \\mathbf{T}\\mathbb{P}_L$. The last term\n    vanishes on the first-order constraint manifold\n    $\\mathbb{P}_L^{[1]}$, while for the second term, \n    $\\langle\\mathbf{d}\\gamma_m^{[1]}\\vert\\mathbf{Y} \\rangle\n    \\langle \\mathbf{\\Theta}^{(m)}_q\\vert \\mathbf{P}_{(n)}\\rangle=\n    \\langle\\mathbf{d}\\gamma_n^{[1]}\\vert\\mathbf{Y} \\rangle\n    \\delta ^{(m)}_{(n)}$. But as\n    $\\mathbf{P}_{(n)}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$,\n    $\\gamma_n^{[1]}=0$ on $\\mathbb{P}_L$, and this term\n    vanishes as well. Finally, for the first term,\n    $\\langle\\mathbf{d}\\mathbf{\\gamma}_m^{[1]}\\vert\n    \\mathbf{P}_{(n)}\\rangle= \\mathbf{P}_{(n)}\\mathbf{P}_{(m)}E=\n    [\\mathbf{P}_{(n)},\\mathbf{P}_{(m)}]E+\\mathbf{P}_{(m)}\\mathbf{P}_{(n)}E$. But\n    $\\gamma_n^{[1]}=\\mathbf{P}_{(n)}E=0$ on $\\mathbb{P}_L$, while\n    $\\overline{\\hbox{ker }\\mathbf{\\Omega}(\\mathfrak{u})}$ is\n    involutive. There then exists a $\\mathbf{P}_{(nm)}\\in\n    \\overline{\\hbox{ker }\\mathbf{\\Omega}(\\mathfrak{u})}$ such that\n    $\\mathbf{P}_{(nm)}=[\\mathbf{P}_{(n)}, \\mathbf{P}_{(m)}]$. As\n    $\\mathbf{P}_{(nm)}E:=\\gamma_{(nm)}^{[1]}$, this\n    $\\gamma_{(nm)}^{[1]}$ must be a linear\n    combination of first-order constraint functions, and they also vanish\n    on $\\mathbb{P}_L^{[1]}$. It then follows that\n    $\\langle\\mathbf{d}\\mathbf{\\beta}\\vert\n    \\mathbf{P}_{(n)}\\otimes\\mathbf{Y}\\rangle=0$ on\n    $\\mathbb{P}_L^{[1]}$, and\n    $\\mathbf{P}_{(n)}\\in\\mathcal{S}\\hbox{ym}$.\n  \\end{proof}\n\\end{lemma}\n\nIf $\\mathbf{P}_1, \\mathbf{P}_2 \\in \\mathcal{S}\\hbox{ym}\\mathcal{L}$, then\n$\\gamma^{[1]}_{[\\mathbf{P}_1,\\mathbf{P}_2]} =\\mathbf{P}_1\\mathbf{P}_2E-\n\\mathbf{P}_2\\mathbf{P}_1E=\\mathbf{P_1}\\gamma_{\\mathbf{P}_2}-\n\\mathbf{P_2}\\gamma_{\\mathbf{P}_1}=0$, and thus\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is involutive. Then for  \neach $\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$ we once again have the\none-parameter subgroup\n$\\sigma^{\\mathcal{S}\\hbox{ym}\\mathcal{L}}_{\\mathbf{P}}(\\epsilon,\\mathfrak{u})$\ndefine as the integral flow of\n\\begin{equation}\n  \\frac{\\mathbf{d}\\sigma^{\\mathcal{S}\\hbox{ym}\\mathcal{L}}_{\\mathbf{P}}}{\\mathbf{d}\\epsilon}\n  :=\\mathbf{P},\n  \\nonumber\n\\end{equation}\nwith\n$\\sigma^{\\mathcal{S}\\hbox{ym}\\mathcal{L}}_{\\mathbf{P}}(0,\\mathfrak{u})=\\mathfrak{u}$\nfor $\\mathfrak{u}\\in\\mathbb{P}_L$. The collection of such subgroups\ngives the Lie group $\\hbox{Gr}_{\\mathcal{S}{ym}\\mathcal{L}}$. As\n$\\mathcal{S}{ym}\\mathcal{L}\\subset \\mathcal{S}{ym}$,\n$\\hbox{Gr}_{\\mathcal{S}{ym}\\mathcal{L}}$ is a Lie subgroup of\n$\\hbox{Gr}_{\\mathcal{S}{ym}}$. It then follows from \\textbf{Theorem \\ref{Group-EL}}\nthat $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ also forms a group of\n  symmetry transformations of $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$. As\n  the family $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ of trajectories are\n  paths on $\\mathbb{Q}$, and as the symmetry transformation of the\n  action must be the same for all paths on $\\mathbb{Q}$, it also follows that, \n\n\\begin{theorem}\\label{Group-L}\n  $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ forms the group of\n  symmetry transformations of the action $S$.\n\\end{theorem}\n\n\\subsection{Symmetries and Dynamics\\label{SandD}}\n\nWhile $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ gives the family of integral\nflows on which both $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ and\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ act, a general flow in\n$\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ need not be confined to $\\mathbb{P}_{L}^{\\left[1\\right]\n}$, and yet this is  the submanifold on which the solutions \n$\\overline{\\mathbf{X}}_{EL}\\in mathcal{S}{\\mathrm{ol}}$ of the energy\nequations exist. In such cases it is necessary to jointly choose a\nSOELVF $\\overline{\\mathbf{X}}_{EL}$ and a submanifold \nof $\\mathbb{P}_{L}^{\\left[  1\\right]  }$ on which the resultant flow\n$\\mathfrak{u}_{\\overline{\\mathbf{X}}_{EL}}$ will be confined. This is\ndone through the implementation of a constraint algorithm, one of\nwhich was proposed in \\cite{ADS2020}. In that paper the product of this\nalgorithm was the most that could be said about the general structure \nof SOELVFs that have integral flow fields which lie on\n$\\mathbb{P}_L^{[1]}$. Here, with the results obtained in\n\\textbf{Section \\ref{SandA}}, we can say much more, and we will see\nthat the presence of a generalized Lie symmetry of the action\ngreatly restricts the structure of the SOELVFs that such systems can have.\n\nFollowing \\cite{ADS2020}, we introduce for a \n$\\overline{\\mathbf{X}}_{EL}\\in\\overline{\\mathcal{S}\\hbox{ol}}$ the\nnotation \n\\begin{equation}\n\\overline{\\mathbf{X}}_{EL}^{[1]} := \\overline{\\mathbf{X}}_{EL}, \\>\n\\overline{\\mathbf{X}}_{L}^{[1]} := \\overline{\\mathbf{X}}_{L}, \\>\n\\mathbf{P}^{[1]}_{(n)} := \\mathbf{P}_{(n)}, \\> u_{[1]}^m := u^m, \\>\nN_0^{[1]} := N_0,\n\\nonumber\n\\end{equation}\nwhen the constraint algorithm is implemented, with the superscript $[1]$ denoting\nthe first iteration of this algorithm. (This notation is only used in\nthis section.) In addition, we choose \n$\\mathbf{P}^{[1]}_{(n)}\\in \\mathcal{S}\\hbox{ym}\\mathcal{L}$ for\n$n=1,\\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$.\n\nFor the integral flow field of $\\overline{\\mathbf{X}}_{EL}$ to lie on \n$\\mathbb{P}_L^{[1]}$,\n\\begin{equation}\n  \\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\mathbf{\\beta}=0,\n  \\label{stable}\n\\end{equation}\nwhich reduces to\n$\\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\gamma_n^{[1]}=0$ on $\\mathbb{P}_L^{[1]}$.\nThis is called the \\textbf{constraint condition}. \nAs both $u_{[1]}^n,\n\\gamma^{[1]}_n\\in \\overline{\\mathcal{F}}$,\n$\\left[\\mathbf{P}_{(n)}^{[1]}\\right]\\gamma^{[1]}_m = \n\\mathbf{P}_{(n)}\\gamma^{[1]}_m$, and after making use of the general\nform of a SOELVF given in Eq.~$(\\ref{EL})$, Eq.~$(\\ref{stable})$ reduces to\n\\begin{equation}\n  \\sum_{m=1}^{N_0} \\Gamma^{[1]}_{nm} u^m_{[1]} =\n  -\\left\\langle \\mathbf{d} \\gamma^{[1]}_n\\Big\\vert\n  \\overline{\\mathbf{X}}^{[1]}_{L}\\right\\rangle, \\>\\hbox{with }\n  \\Gamma^{[1]}_{nm} := \\left\\langle\n  \\mathbf{d}\\gamma^{[1]}_n\\Big\\vert \\mathbf{P}^{[1]}_{(m)}\\right\\rangle.\n  \\label{first-order}\n\\end{equation}\nSince $\\left\\langle\n\\mathbf{d}\\gamma^{[1]}_n\\Big\\vert\n\\mathbf{P}^{[1]}_{(m)}\\right\\rangle=\\mathbf{P}^{[1]}_{(m)}\\mathbf{P}^{[1]}_{(n)}E=\n       [\\mathbf{P}^{[1]}_{(m)}, \\mathbf{P}^{[1]}_{(n)}]E\n+\\Gamma^{[1]}_{mn}$.\nBut $\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is\ninvolutive, and thus $[\\mathbf{P}^{[1]}_{(m)},\n  \\mathbf{P}^{[1]}_{(n)}]E$ is a linear combination of first-order\nLagrangian constraints. As these constraints vanishes on\n$\\mathbb{P}_L^{[1]}$, \n$\\Gamma^{[1]}_{nm}=\\Gamma^{[1]}_{mn}$ on the first-order constraint\nmanifold.\n\nNext, when $n=1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$,\n$\\mathbf{P}_{(n)}^{[1]}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$, and\n$\\gamma_n^{[1]}=0$. Thus, $\\Gamma^{[1]}_{nm}=0$ when $n=1, \\dots,\nN_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$, and as $\\Gamma^{[1]}_{nm}$ is a\nsymmetric matrix on $\\mathbb{P}_L^{[1]}$, $\\Gamma^{[1]}_{mn}=0$ for\nthese values of $n$ as well. Thus while\n$\\Gamma^{[1]}_{nm}$ is a $N_0\\times N_0$ matrix, \nthe only nonzero components of this matrix lie in the\n$\\left(N_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}\\right)\\times\n\\left(N_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}\\right)$ submatrix\n$\\bar{\\Gamma}^{[1]}_{\\overline{n}\\,\\overline{m}}:=\n\\left\\langle\\mathbf{d}\\gamma^{[1]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\bigg\\vert\n  \\mathbf{P}^{[1]}_{\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\right\\rangle$\nwhere $\\overline{n}, \\overline{m} = 1, \\dots,\nN_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$. As $\\left\\langle \\mathbf{d}\n\\gamma^{[1]}_n\\Big\\vert\n\\overline{\\mathbf{X}}^{[1]}_{L}\\right\\rangle=0$ as well when $n=1,\n\\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$, Eq.~$(\\ref{first-order})$\n  reduces to  \n\\begin{equation}\n  \\sum_{\\overline{m}=1}^{N_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\n  \\bar{\\Gamma}^{[1]}_{\\overline{n}\\bar{m}}\n  u^{\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}_{[1]} = -\\left\\langle \\mathbf{d}\n  \\gamma^{[1]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\Big\\vert \n  \\overline{\\mathbf{X}}^{[1]}_{L}\\right\\rangle.\n  \\label{red-first-order}\n\\end{equation}\nIt is then readily apparent that the\n$N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions $u^m_{[1]}$\nfor $m=1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ are not\ndetermined at this iteration, while $r^{[1]} =\\hbox{rank }\n\\bar{\\Gamma}^{[1]}_{\\overline{n}\\,\\overline{m}}$ of the $u^m_{[1]}$ for\n$m>N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ are. There\nare then $N_0^{[2]} :=N_0^{[1]}-r^{[1]}$\n  \\textbf{second-order Lagrangian constraint functions} \n\\begin{equation}\n  \\gamma^{[2]}_{n_{[2]}} := \\left\\langle\n  \\mathbf{d}\\gamma^{[1]}_{n_{[2]}}\\Big\\vert \n  \\overline{\\mathbf{X}}_{L}^{[1]}\\right\\rangle, n_{[2]}=1, \\cdots,\n  N_0^{[2]},\n  \\nonumber\n\\end{equation}\nwith the conditions $\\gamma^{[2]}_{n_{[2]}}=0$ imposed if necessary. In general\nthere will be $I_{[2]}:=  \n\\hbox{rank }\\left\\{\\mathbf{d}\\gamma^{[1]}_{n_{[1]}},\n  \\mathbf{d}\\gamma^{[2]}_{n_{[2]}}\\right\\}$\nindependent functions in $\\hbox{C}^{[2]}_L :=\n\\hbox{C}^{[1]}\\cup \\left\\{\\gamma^{[2]}_{n_{[2]}}\\ \\vert\n\\ n_{[2]} = 1, \\dots, N_0^{[2]}\\right\\}$, and $\\mathbb{P}_L^{[1]}$ is\nreduced to the \\textbf{second-order constraint submanifold}, \n\\begin{equation}\n  \\mathbb{P}_L^{[2]} := \\left\\{\\mathfrak{u}\\in\\mathbb{P}_L^{[1]}\n  \\ \\Big\\vert \\ \\gamma^{[2]}_{[n_2]}(\\mathfrak{u})=0, n_{[2]} = 1, \\dots,\n  N_0^{[2]} \\right\\}, \n  \\nonumber\n\\end{equation}\nwhere dim $\\mathbb{P}^{[2]}_L = 2D-I_{[2]}$. At this point, there are two\npossibilities. If $I_{[2]}=I_{[1]}$ or\n$I_{[2]}=2D$, the iterative process stops, and no new Lagrangian\nconstraints are introduced. If not, the process continues. \n\nFor the second iteration in the constraint algorithm, we choose a basis\n$\\left\\{\\mathbf{P}_{(n)}^{[2]}\\right\\}$ for $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$ and the arbitrary functions\n$\\left\\{u_{[2]}^m\\right\\}$ such that for $m=1, \\dots, N_0^{[2]}$,\n$u_{[2]}^m$ are linear combinations of $u_{[1]}^m$ that lie in the kernel\n$\\Gamma^{[1]}_{nm}$. We once again require that\n$\\mathbf{P}_{(n)}^{[2]}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$\nfor $n=1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$.  \nThen\n\\begin{equation}\n\\overline{\\mathbf{X}}_{EL}^{[2]} = \\overline{\\mathbf{X}}_{L}^{[2]} +\n\\sum_{m=1}^{N_0^{[2]}} u_{[2]}^m \\left[\\mathbf{P}_{(m)}^{[2]}\\right],\n\\nonumber\n\\end{equation}\nwith\n\\begin{equation}\n  \\overline{\\mathbf{X}}_{L}^{[2]} = \\overline{\\mathbf{X}}_{L}^{[1]} +\n  \\sum_{m=N_0^{[2]}+1} ^{N_0^{[1]}}u^m_{[2]}\\left[\\mathbf{P}^{[2]}_{(m)}\\right].\n  \\nonumber\n\\end{equation}\nHere, the functions $u^m_{[2]}$ for $m = N_0^{[2]}+1, \\dots, N_0^{[1]}$ have been\ndetermined through the constraint analysis of $\\gamma^{[1]}_n$. \n\nAs shown in \\cite{ADS2020}, $\\mathbf{G}u_{[1]}^m=0$. Similarly,\n  $\\mathbf{G}\\gamma^{[2]}_n = \\mathfrak{L}_{[\\mathbf{G},\n      \\overline{\\mathbf{X}}_{EL}]}\\mathbf{d}\\gamma^{[2]}_n=0$. Clearly\n  $\\gamma^{[2]}_n\\in\\overline{\\mathcal{F}}$ and we may require\n  $u^m_{[2]}\\in\\overline{\\mathcal{F}}$ as well. It \n  then follows that $\\left[\\mathbf{P}^{[2]}_{(n)}\\right]\\gamma^{[2]}_m \n=\\mathbf{P}^{[2]}_{(n)}\\gamma^{[2]}_m$, and imposing Eq.~$(\\ref{stable})$\non $\\gamma^{[2]}_n $, gives\n\\begin{equation}\n  \\sum_{m=1}^{N_0^{[2]}}\\Gamma^{[2]}_{nm} u^m_{[2]} =\n  -\\left\\langle \\mathbf{d}\\gamma^{[2]}_n\\Big\\vert\n  \\overline{\\mathbf{X}}^{[2]}_{L}\\right\\rangle,\\>\\hbox{where }\n  \\Gamma^{[2]}_{nm} := \\left\\langle\n  \\mathbf{d}\\gamma^{[2]}_n\\Big\\vert \\mathbf{P}^{[2]}_{(m)}\\right\\rangle,\n\\>\\> n=1, \\dots,\nN_0^{[2]}.\n\\label{second-order}\n\\end{equation}\nOnce again, $\\Gamma^{[2]}_{nm}=\\Gamma^{[2]}_{mn}$, but now on the constraint\nmanifold $\\mathbb{P}^{[2]}_L$. Moreover, since\n$\\gamma^{[2]}_n=\\gamma^{[1]}_n=0$ for $n=1, \\dots,\nN_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$,  \n$\\Gamma^{[2]}_{nm}=0=\\Gamma^{[2]}_{mn}$, and $\\left\\langle\n\\mathbf{d}\\gamma_n^{[2]}\\Big\\vert\\overline{\\mathbf{X}}^{[2]}_L\\right\\rangle\n=0$. There is once again a reduction of Eq.~$(\\ref{second-order})$,\nand we are left with \n\\begin{equation}\n  \\sum_{\\overline{m}=1}^{N_0^{[2]}-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\n    \\overline{\\Gamma}^{[2]}_{\\overline{n}\\,\\overline{m}} u^{\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}_{[2]} =\n    -\n    \\langle \\mathbf{d}\\gamma^{[2]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\vert\n  \\overline{\\mathbf{X}}^{[2]}_{L} \\rangle.\n\\nonumber\n\\end{equation}\nwhere\n$\\bar{\\Gamma}^{[2]}_{\\overline{n}\\,\\overline{m}}:=\n\\left\\langle\\mathbf{d}\\gamma^{[2]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\n\\bigg\\vert\\mathbf{P}^{[2]}_{(\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}})}\\right\\rangle$. As\nbefore, the $N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions\n$u^{m}_{[2]}$ are not determined, while $r^{[2]} := \\hbox{rank\n}\\bar{\\Gamma}^{[2]}_{\\overline{n}\\,\\overline{m}}$ of the\nremaining $u^m_{[2]}$ for $m>N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$\nare. There are now \n$N^{[3]}_0=N_0^{[2]}-r^{[2]}$ \\textbf{third-order Lagrangian\n  constraint functions},\n\\begin{equation}\n  \\gamma^{[3]}_{n_{[3]}} =\n  \\left\\langle\\mathbf{d}\\gamma^{[2]}_{n_{[3]}}\\Big\\vert\n  \\overline{\\mathbf{X}}^{[2]}_{L}\\right\\rangle, \\> n_{[3]} = 1, \\dots,\n  N_0^{[3]},\n  \\nonumber\n\\end{equation}\nwith the conditions $\\gamma^{[3]}_{n_{[3]}}=0$ \nimposed if necessary. With \n\\begin{equation}\nI_{[3]} := \\hbox{rank } \\left\\{ \n  \\mathbf{d}\\gamma_{n_{[1]}}^{[1]},\n \\mathbf{d}\\gamma_{n_{[2]}}^{[2]},\n \\mathbf{d}\\gamma_{n_{[3]}}^{[3]}\n \\right\\},\n \\nonumber\n\\end{equation}\nindependent functions in $\\hbox{C}^{[3]}_L :=\n\\hbox{C}_L^{[2]}\\cup\\left\\{\\gamma_{n_{[3]}}^{[3]}, n_{[3]}=1,\n\\dots, N_0^{[3]}\\right\\}$, we now have the \\textbf{third-order\n  constraint submanifold},  \n\\begin{equation}\n  \\mathbb{P}_L^{[3]}:=\\left\\{\\mathfrak{u}\\in \\mathbb{P}^{[2]}_L\n  \\ \\Big\\vert \\ \\gamma^{[3]}_{n_{[3]}}(\\mathfrak{u})=0, n_{[3]}=1,\n  \\dots, N_0^{[3]}\\right\\}.\n  \\nonumber\n\\end{equation}\nOnce again, the process stops when\n$I_{[3]}=I_{[2]}$ or $I_{[3]}=2D$. However, if $I_{[2]}<I_{[3]}<2D$, the\nprocess continues until at the $n_F$-iteration when either\n$I_{[n_F]}=I_{[n_F]-1}$ or $I_{[n_F]}=2D$. \n\nFollowing \\cite{ADS2020}, the end result of this algorithm is \n\\begin{enumerate}\n\\item{A submanifold $\\mathbb{P}^{[n_F]}_L\\subset \\mathbb{P}_L$ on which\n  dynamics takes place.} \n  \\item{A collection\n    $\\hbox{C}_L^{[n_F]}\\subset\\overline{\\mathcal{F}}$ of constraint\n    functions of order $1$ to $n_F$.} \n\\item{A second-order, Euler-Lagrange vector field\n  \\begin{equation}\n    \\overline{\\mathbf{X}}_{EL}^{[n_F]}=\\overline{\\mathbf{X}}_{L}^{[n_F]} +\n    \\sum_{m=1}^{N^{[n_F]}_0}u^m_{[n_F]}(\\mathfrak{u})\\left[\\mathbf{P}_{(m)}^{[n_F]}\\right],  \n    \\nonumber\n  \\end{equation}\n  with $N_0^{[n_F]}\\ge N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions\n  $u^m_{[n_F]}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$ for $m=1,\n  \\dots, N_0^{[n_F]}$, and \n  \\begin{equation}\n    \\overline{\\mathbf{X}}_{L}^{[n_F]} =\n    \\overline{\\mathbf{X}}_{L}^{[1]}+\n    \\sum_{m=N_0^{[n_F]}+1}^{N_0^{[1]}}u^m_{[n_F]}(\\mathfrak{u})\n    \\left[\\mathbf{P}_{(m)}^{[n_F]}\\right],  \n    \\nonumber\n  \\end{equation}\n  where the $N_0^{[1]}-N_0^{[n_F]}$ functions\n  $u^m_{[n_F]}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$, \n  $m=N_0^{[n_F]}+1, \\dots, N_0^{[1]}$, have been uniquely\n  determined through the constraint algorithm.} \n\\end{enumerate}\nWe assume that the rank of $\\Gamma^{[l]}_{nm}$ is constant on\n$\\mathbb{P}_L$ for each $l=1, \\dots, n_F$, and that\n$\\mathbb{P}^{[n_F]}_L$ is non-empty. \n\nThe end result of the constraint algorithm\n$\\overline{\\mathbf{X}}_{EL}^{[n_F]}$ is still a SOELVF, and we define\nthe collection of such vector fields as\n\\begin{equation}\n  \\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L} :=\n  \\{\\overline{\\mathbf{X}}_{EL}\\in\\overline{\\mathcal{S}\\hbox{ol}} \\>\\>\\vert\\>\n  \\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\mathbf{\\beta}=0\\}.\n  \\nonumber\n\\end{equation}\nImportantly, $\\hbox{dim }\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}\\ge\nN_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$. \n\n\\section{The Generalized Lie Symmetries of Three Dynamical\n  Systems\\label{&Exam}}\n\nThree examples of dynamical systems with almost regular Lagrangians\nwere introduced in \\cite{ADS2020}. In that paper the focus of these\nexamples was on the explicit construction of the dynamical structures \nneeded to describe and predict motion in the Lagrangian phase space,\nand to show that these structures are projectable to \nthe Hamiltonian phase space. We return to these examples here, but\nwith the focus now being on the generalized Lie symmetries of each, and the\napplication of the results we have found in this paper. In particular,\nwe are in interested in the dimensionality of the symmetry groups for\neach of the systems as compared to the dimensionality of\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}$ of each. A\nsummary of our results can be found in Table $\\ref{Table}$\n\n\\subsection{A Lagrangian With and Without a Generalized Lie Symmetry\\label{&special}}  \n\nWhether the action\n\\begin{equation}\n  S_{1}:=\\int\\left[\\frac{1}{2}m\\left(\\frac{d\\widehat{q}}{dt}\\right)^2\n    -V(q^a)\\right]dt, \n  \\nonumber\n\\end{equation}\nwith $\\vert q\\vert=\\sqrt{q^a q_a}$ and $\\widehat{q}^a := q^a\/\\vert\nq\\vert$, $a=1, \\dots, D$, has a generalized Lie symmetry depends on the\nchoice of potential $V(q)$. With one choice both the Lagrangian and\nthe Euler-Lagrange equations of \nmotion have a generalized gauge symmetry; with a second choice the equations\nof motion has a generalized Lie symmetry while the Lagrangian\ndoes not; and with a third choice neither the action nor the equations of motion have\na symmetry. Irrespective of the choice of $V(q)$, however, $L$ is\nsingular, demonstrating that while all actions with a \ngeneralized Lie symmetry have a singular Lagrangian, not all singular\nLagrangians have a generalized Lie symmetry. \n\nDefining $\\Pi_{ab}(q):= \\delta_{ab} - \\widehat{q}_a\\widehat{q}_b$, we find\n\\begin{equation}\n  \\mathbf{\\Omega}_M = \\frac{m}{\\vert q\\vert^2}\\Pi_{ab}(q)\n  \\mathbf{d}q^a\\wedge\\mathbf{d}v^b,\\qquad\n  \\mathbf{\\Omega}_F=\\frac{m}{\\vert q\\vert^3}\n  \\left(\\widehat{q}\\cdot\\mathbf{d}q\\right)\\wedge\n  \\left(v\\cdot\\Pi(q)\\cdot\\mathbf{d}q\\right).\n\\nonumber\\end{equation}\nThen $\\mathcal{C}$ and $\\mathcal{G}$ are spanned by $\\mathbf{U}^q_{(1)}\n= \\widehat{q}\\cdot\\mathbf{\\partial}\/\\mathbf{\\partial} q$ and  \n  $\\mathbf{U}^v_{(1)} =\n  \\widehat{q}\\cdot\\mathbf{\\partial}\/\\mathbf{\\partial} v$, \nrespectively, while $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is spanned by\n$\\mathbf{U}^v_{(1)}$ and  \n\\begin{equation}\n  P_{(1)}=\\widehat{q}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} +\n    \\frac{1}{\\vert q\\vert}v\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n      v}.\n\\nonumber\\end{equation}\nThat $\\hbox{dim }(\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G})=1$ then follows.\n\nThe energy is\n\\begin{equation}\n  E=\\frac{1}{2}\\frac{m}{\\vert q\\vert^2}v\\cdot\\Pi(q)\\cdot v + V(q),\n  \\nonumber\n\\end{equation}\nand there is only one first-order Lagrangian constraint,\n\\begin{equation}\n  \\gamma^{[1]}=\\mathbf{U}^q_{(1)} V,\n  \\label{g-con}\n\\end{equation}\nso that $\\mathbf{\\beta}[\\mathbf{X}_{EL}] =\n\\gamma^{[1]}\\mathbf{\\Theta}^{(1)}_q$, where $\\mathbf{\\Theta}^{(1)}_q =\n\\widehat{q}\\cdot \\mathbf{d} q$. Using Eq.~$(\\ref{g-con})$,\n\\begin{equation}\n  \\mathfrak{L}_{\\mathbf{P}_{(1)}} \\mathbf{\\beta} =\n  \\mathbf{d}\\left[\\mathbf{U}^q_{(1)}V\n    \\right]-\\frac{1}{\\vert\n    q\\vert^2}\\widehat{q}\\cdot\\frac{\\partial\\>\\>\\>}{\\partial \n    q}\\left(\\Pi_a^{\\>\\>b}(q)\\frac{\\partial V}{\\partial\\widehat{q}^b}\\right)\n  \\mathbf{d}q^a.\n  \\label{e84}\n\\end{equation}\nWhether or not $\\mathcal{S}\\hbox{ym}$ or\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is empty therefore depends on the\nsymmetries of $V(q)$, as we would expect.  \n\nIt was found in \\cite{ADS2020} that\n\\begin{equation}\n  \\overline{\\mathbf{X}}_{L} = v\\cdot{\\Pi(q)}\\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} +\n  \\frac{(\\widehat{q}\\cdot v)}{\\vert q\\vert}\n  v\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\partial v} -\n  \\frac{\\vert q\\vert^2}{m}\\frac{\\partial V}{\\partial\n    q}\\cdot\\Pi(q)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v},\n\\nonumber\\end{equation}\nand a general SOELVF is given by\n$\\overline{\\mathbf{X}}_{EL}=\\overline{\\mathbf{X}}_{L} + u(\\mathfrak{u})\n\\left[\\mathbf{P}_{(1)}\\right]$, where\n$u(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$.\nAs the constraint \nalgorithm gives \n\\begin{equation}\n  \\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\gamma^{[1]} = v\\cdot\\Pi\n  \\cdot\\frac{\\partial \\gamma^{[1]}}{\\partial q} + u(\\mathfrak{u})\n  \\mathbf{U}^q_{(1)}\\gamma^{[1]},\n  \\label{stab}\n\\end{equation}\nwhether or not $u(\\mathfrak{u})$ (which in turn determines the\ndimensionality of\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}_L^{[n_f]}}$) is determined\nby the constraint condition also depends on the symmetries of $V(q)$. \n\nThere are three cases to consider.\n\n\\bigskip\n\\noindent{\\textit{The symmetric potential}}\n\\bigskip\n    \nFor $\\mathbf{P}_{(1)}$ to generate a generalized Lie symmetry of the\nEuler-Lagrange equations of motion, \n\\begin{equation}\n  0=\\frac{1}{\\vert q\\vert^2} \\widehat{q}\\cdot\\frac{\\partial \\>\\>\\>}{\\partial q} \\left(\\Pi_a^{\\>\\>b}(q)\\frac{\\partial\n    V}{\\partial \\widehat{q}^b}\\right),\n\\nonumber\\end{equation}\nand as such the potential must satisfy\n\\begin{equation}\n  \\frac{\\partial V}{\\partial\\widehat{q}^a} = \\frac{\\partial\n    V_{AS}(\\widehat{q}^a)}{\\partial \\widehat{q}^a},\n\\nonumber\\end{equation}\nwhere $V_{AS}$ is a function of $\\widehat{q}^a$ only. It follows that\n$\\mathbf{P}_{(1)}$ generates a generalized Lie symmetry iff $V(q^a) =\nV_{Sph}(\\vert q\\vert)+V_{AS}(\\widehat{q}^a)$, where $V_{Sph}$ is a function of\n$\\vert q\\vert$ only. For this potential, $\\mathcal{S}\\hbox{ym}$ is \none-dimensional, and is spanned by $\\mathbf{P}_{(1)}$. \n\nThe constraint condition Eq.~$(\\ref{stab})$ for this potential\nreduces to \n\\begin{equation}\n  0=u(\\mathfrak{u})\\frac{d^2V_{Sph}(q)}{d\\vert q\\vert^2},\n\\nonumber\\end{equation}\nwhich must be satisfied on $\\mathbb{P}_L^{[1]}$. There are two possibilities.\n\n\\bigskip\n\\textit{Case 1:} $\\frac{d^2V_{Sph}}{d\\vert q\\vert^2}=0$.\n\\bigskip\n\n\\noindent Then $V_{Sph}(\\vert q\\vert) = a\\vert q\\vert+b$, but since \n\\begin{equation}\n  \\gamma^{[1]}=\\frac{dV_{Sph}}{d\\vert q\\vert} =a,\n\\nonumber\\end{equation}\nthe condition $\\gamma^{[1]}=0$ requires $a=0$. It then follows that\n$\\gamma^{[1]} =0$ on $\\mathbb{P}_L$, and thus\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is one-dimensional; it also is\nspanned by $\\mathbf{P}_{(1)}$. The potential is then $V(q) =\nb+V_{AS}(\\widehat{q}^a)$, and the Lagrangian is   \ninvariant under the transformation $q^a\\to \\alpha q^a$,\nwhere $\\alpha$ is an arbitrary, nonvanishing function on\n$\\mathbb{P}_L$. This Lagrangian therefore has a local conformal symmetry.\nImportantly, the function $u(\\mathfrak{u})$ is\nnot determined, and thus the dynamics of the \nparticle is given only up to an arbitrary function. Then  \n$\\hbox{dim\n}(\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L})=1$ as well,\nand is also spanned by $\\mathbf{P}_{(1)}$.\n\n\\bigskip\n\\textit{Case 2:} $\\frac{d^2V_{Sph}}{d\\vert q\\vert^2}\\ne0$.\n\\bigskip\n\n\\noindent In this case $u(\\mathfrak{u})=0$, and the dynamics of the\nparticle is completely determined by its initial data;\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}=\\{\\overline{\\mathbf{X}}_{L}\\}$.\nThe first-order Lagrangian constraint $\\gamma^{[1]}$ does not vanish\nautomatically, but instead defines a surface\non $\\mathbf{P}_L$, and it follows that\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}=\\emptyset$. Indeed, the action's lack\nof a local gauge symmetry in this case can be seen explicitly.\n\n\nEquation $(\\ref{g-con})$ reduces to\n\\begin{equation}\n  0=\\widehat{q}\\cdot\\frac{\\mathbf{\\partial}V_{sph}}{\\mathbf{\\partial}q},\n\\nonumber\\end{equation}\nand for dynamics to be possible the set of solutions \n\\begin{equation}\n    \\left\\{R_i\\in\\mathbb{R} \\ \\Bigg\\vert\n    \\ \\frac{dV_{Sph}}{d\\vert q\\vert}\\Bigg\\vert_{R_i} =0\\right\\},\n\\nonumber\\end{equation}\nmust be non-empty. Dynamics are on the surfaces $\\vert q\\vert\n-R_i =0$ where the potential reduces to $V(q)=V_{Sph}(R_i) +\nV_{AS}(\\widehat{q}^a)$. This reduced potential has the same symmetry\nas the potential $V_{AS}(\\widehat{q}^a)$ in \\textit{Case 1}, and it is for\nthis reason that the Euler-Lagrange equations of motion have the same\ngeneralized Lie symmetry for the two cases. This is explicitly shown\nin the appendix.\n\nIn \\textit{Case 1} the action has a local conformal symmetry, while in\n\\textit{Case 2} it does not. (In \\cite{ADS2020} it was\nerroneously stated that in this case the action has a global rotational\nsymmetry.) The Lagrangian for the two cases do not have the same \ninvariances, resulting in one case dynamics that are determined\nonly up to \nan arbitrary $u(\\mathfrak{u})$, and in the other case to a\n$u(\\mathfrak{u})=0$ and dynamics\nthat are instead completely determined by the choice of initial data. \n\n\\bigskip\n\\noindent{\\textit{The asymmetric potential}}\n\\bigskip\n\nFor a general $V$ the second term in\nEq.~$(\\ref{e84})$ does not vanish, $\\mathbf{P}_{(1)}$ does not\ngenerate a symmetry of the equations of motion, and \n$\\mathcal{S}\\hbox{ym}=\\{\\emptyset\\}$. As before, $\\gamma^{[1]}$ does\nnot vanish, and thus $\\mathcal{S}\\hbox{ym}\\mathcal{L}=\\{\\emptyset\\}$\nas well. Furthermore, as Eq.$(\\ref{stab})$ results in\n\\begin{equation}\n  \\overline{\\mathbf{X}}_E =\\overline{\\mathbf{X}}_L-\\frac{v\\cdot\\Pi\n  \\cdot\\frac{\\partial \\gamma^{[1]}}{\\partial q}}\n  {q^2\\mathbf{U}^q_{(1)}\\gamma^{[1]}}[\\mathbf{P}_{(1)}],\n\\nonumber\\end{equation}\nthe dynamics of the particle is uniquely determined by its initial\ndata, and\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}=\\{\\overline{X}_{EL}\\}$\nonce again consists of a single point.\n\n\\subsection{A Lagrangian with Local\n  Conformal Symmetry}\n\nThe action,\n\\begin{equation}\n  S_{2} := \\int\\Bigg\\{\\frac{1}{2}m\n  \\left(\\frac{d\\widehat{q}_1}{dt}\\right)^2+\\frac{1}{2}m\n  \\left(\\frac{d\\widehat{q}_2}{dt}\\right)^2+\n  \\frac{\\lambda}{2}\\left[\\frac{q_1^a}{\\vert q_2\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{2a}}{\\vert q_1\\vert}\\right) -\n  \\frac{q_2^a}{\\vert q_1\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{1a}}{\\vert q_2\\vert}\\right)\n  \\right]\n  \\Bigg\\} dt,\n\\nonumber\\end{equation}\nwhere $a=1, \\dots, d$, $D=2d$, describes an interacting, two particle\nsystem that is invariant under the local conformal transformation\n$q_1^a \\to \\alpha(\\mathfrak{u}) q^a_1$ and $q_2^a \\to\n\\alpha(\\mathfrak{u}) q^a_2$. \n\nWith\n\\begin{widetext}\n\\begin{eqnarray}\n  \\mathbf{\\Omega}_M &=&  \\frac{m}{\\vert q_1\\vert^2} \\Pi_{ab}(q_1)\n  \\mathbf{d}q_1^a\\wedge \\mathbf{d} v_1^b + \\frac{m}{\\vert q_2\\vert^2}\n  \\Pi_{ab}(q_2) \\mathbf{d}q_2^a\\wedge \\mathbf{d} v_2^b, \\hbox{  and}\n  \\nonumber\n  \\\\\n  \\mathbf{\\Omega}_F &=& \\frac{m}{\\vert q_1\\vert^3}\n  \\left(\\widehat{q}_1\\cdot\\mathbf{d}\n  q_1\\right)\\wedge\\left(v_1\\cdot\\Pi(q_1)\\cdot \\mathbf{d} q_1\\right) + \n  \\frac{m}{\\vert q_2\\vert^3} \\left(\\widehat{q}_2\\cdot\\mathbf{d} \n  q_2\\right)\\wedge\\left(v_2\\cdot\\Pi(q_2)\\cdot \\mathbf{d} q_2\\right)- \n  \\nonumber\n  \\\\\n  &{}&\n  \\frac{\\lambda}{\\vert q_1\\vert \\vert\n    q_2\\vert}\\left[\\mathbf{d}q_1^a\\wedge\\left(\\Pi(q_2)\\cdot \n  \\mathbf{d} q_2\\right)_a+ \\left(\\Pi(q_1)\\cdot\n  \\mathbf{d} q_1\\right)_a\\wedge \\mathbf{d}q_2^a -\\left(\\Pi(q_1)\\cdot\n  \\mathbf{d} q_1\\right)^a\\wedge \\left(\\Pi(q_2)\\cdot\n  \\mathbf{d} q_2\\right)_a\\right]-\n  \\nonumber\n  \\\\\n  &{}& \\frac{\\lambda}{\\vert q_1\\vert^2}\n  \\left(\\widehat{q}_1\\cdot\\mathbf{d}q_1\\right)\\wedge\n  \\left(\\widehat{q}_2\\cdot\\Pi(q_1) \\cdot \\mathbf{d}q_1\\right) +\n  \\frac{\\lambda}{\\vert q_2\\vert^2} \n  \\left(\\widehat{q}_2\\cdot\\mathbf{d}q_2\\right)\\wedge\n  \\left(\\widehat{q}_1\\cdot\\Pi(q_2) \\cdot \\mathbf{d}q_2\\right),\n  \\nonumber\n\\end{eqnarray}\n\\end{widetext}\n$\\mathcal{C}$ and $\\mathcal{G}$ are two-dimensional, and\nare spanned by\n\\begin{equation}\n  \\mathbf{U}^q_{(1)} =\n  \\widehat{q}_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_1},\n    \\quad\n  \\mathbf{U}^q_{(2)} =\n  \\widehat{q}_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_2},\n  \\quad \\hbox{and}\\quad \n  \\mathbf{U}^v_{(1)} =\n  \\widehat{q}_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_1},\n    \\quad\n  \\mathbf{U}^v_{(2)} =\n  \\widehat{q}_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_2},\n\\nonumber\\end{equation}\nrespectively. The reduced $\\bar{F}=0$, and $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is spanned by $\\mathbf{U}^v_{(1)},\n\\mathbf{U}^v_{(2)}$, \n\\begin{eqnarray}\n  \\mathbf{P}_{(+)} &=&\n  q_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_1}+q_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_2}\n  +\n  v_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_1} \n  +\n  v_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_2},\n  \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n  \\mathbf{P}_{(-)} &=&\n  q_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_1}-q_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_2}\n  +\n  v_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_1} \n  -\n  v_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_2} \n  -\n  \\nonumber\n  \\\\\n  &{}&\n  2\\frac{\\lambda}{m}\n  \\left[\\frac{\\vert q_1\\vert}{\\vert\n      q_2\\vert}q_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} \n      v_1}\n    +\\frac{\\vert q_2\\vert}{\\vert q_1\\vert}q_1\\cdot\n    \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_2}\\right].\n  \\nonumber\n\\end{eqnarray}\nAs such, $\\hbox{dim }\\overline{(\\hbox{ker }\\mathbf{\\Omega}_L)}\/\\mathcal{G}=2$. \n\nThe energy is\n\\begin{equation}\n  E=\\frac{1}{2}\\frac{m}{\\vert q_1\\vert^2}v_1\\cdot\\Pi(q_1)\\cdot v_1 +\n  \\frac{1}{2}\\frac{m}{\\vert q_2\\vert^2}v_2\\cdot\\Pi(q_2)\\cdot v_2.\n\\nonumber\n\\end{equation}\nWe find that $\\gamma^{[1]}_{(+)} =0$ while\n\\begin{equation}\n  \\gamma^{[1]}_{(-)} =-\\frac{2\\lambda}{\\vert q_1\\vert \\vert q_2\\vert}\\left(\n  q_2\\cdot \\Pi(q_1)\\cdot v_1 +\n  q_1\\cdot \\Pi(q_2) \\cdot v_2\\right),\n\\nonumber\\end{equation}\ngiving,\n\\begin{eqnarray}\n  \\mathbf{\\beta}[\\mathbf{X}_{EL}] &=& \\frac{1}{2}\\gamma^{[1]}_{(-)}\n  \\left(\\frac{\\mathbf{\\Theta}_q^{(1)}}{\\vert q_1\\vert} -\n  \\frac{\\mathbf{\\Theta}_q^{(2)}}{\\vert q_2\\vert}\\right).\n  \\nonumber\n\\end{eqnarray}\nThen $\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is\none-dimensional and spanned by $\\mathbf{P}_{(+)}$. \nAs expected, $\\mathfrak{L}_{\\mathbf{P}_{(+)}}\n\\mathbf{\\beta} = 0$. Because\n\\begin{equation}\n  \\mathfrak{L}_{\\mathbf{P}_{(-)}} \\mathbf{\\beta} =\n  -\\frac{4\\lambda}{m}\\left[1-(\\widehat{q}_1\\cdot\\widehat{q}_2)^2\\right]\\left(\\frac{\\mathbf{\\Theta}_q^{(1)}}{\\vert q_1\\vert} -\n  \\frac{\\mathbf{\\Theta}_q^{(2)}}{\\vert q_2\\vert}\\right),\n  \\nonumber\n\\end{equation}\n$\\mathcal{S}\\hbox{ym}$ is also one-dimensional, and\nis also spanned by $\\mathbf{P}_{(+)}$. \n\nA general SOELVF is \n\\begin{equation}\n  \\overline{\\mathbf{X}}_{EL} = \\overline{\\mathbf{X}}_{L} -\n  \\frac{m}{8\\lambda^2}\\frac{\\overline{\\mathbf{X}}_{L}\\gamma^{[1]}_{(-)}}{ \n    \\left[1-(\\widehat{q}_1\\cdot\\widehat{q}_2)\\right]}\\left[\\mathbf{P}_{(-)}\\right] +\n  u^{(+)}(\\mathfrak{u}) \\left[\\mathbf{P}_{(+)}\\right],\n  \\label{103}\n\\end{equation}\nwhere $u^{(+)}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$, and from \\cite{ADS2020},\n\\begin{eqnarray}\n\\overline{\\mathbf{X}}_{L} &=&\nv_1\\cdot\\Pi(q_1)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n  q_1} + v_2\\cdot\\Pi(q_2)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n  q_2} +\n\\nonumber\n\\\\\n&{}&\n\\left(\\frac{\\widehat{q}_1\\cdot v_1}{\\vert q_1\\vert}\\right)\nv_1\\cdot\\Pi(q_1)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_1} +\n\\left(\\frac{\\widehat{q}_2\\cdot v_2}{\\vert q_2\\vert }\\right)\nv_2\\cdot\\Pi(q_2)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_2} \n+\n\\nonumber\n\\\\\n&{}&\n\\frac{\\lambda}{m} \\left(\\frac{\\vert q_1\\vert}{\\vert q_2\\vert}\n  v_2\\cdot\\Pi(q_2)\\cdot\\Pi(q_1)\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\>}{\\mathbf{\\partial} v_1}-\\frac{\\vert q_2\\vert}{\\vert q_1\\vert}\n  v_1\\cdot\\Pi(q_1)\\cdot\\Pi(q_2)\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\>}{\\mathbf{\\partial} v_2} \\right),\n  \\nonumber\n\\end{eqnarray}\nafter the constraint algorithm is applied. Equation $(\\ref{103})$ is\na consequence of the identity\n$\\langle\\mathbf{d}\\gamma^{[1]}_{(+)}\\vert \\overline{\\mathbf{X}}_{L}\\rangle=0$ and \n\\begin{eqnarray}\n -\\frac{1}{2\\lambda} \\langle\\mathbf{d}\\gamma^{[1]}_{(-)}\\vert \\overline{\\mathbf{X}}_{L}\\rangle&=& -\n  2(\\widehat{q}_1\\cdot\\widehat{q}_2)\\frac{E}{m}+ \\frac{2}{\\vert\n    q_1\\vert \\vert q_2\\vert}\n  v_1\\cdot\\Pi(q_1)\\cdot\\Pi(q_2)\\cdot v_2\n  -\n  \\nonumber\n  \\\\\n  &{}&\n  \\frac{\\lambda}{m}(\\widehat{q}_1\\cdot\\widehat{q}_2)\\left[v_2\\cdot\\Pi(q_2)\\cdot\\widehat{q}_1\n    - v_1\\cdot\\Pi(q_1)\\cdot \\widehat{q}_2\\right].\n\\nonumber\n\\end{eqnarray}\nWe see that $\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}$ is also\none-dimensional, and is also spanned by $\\mathbf{P}_{(+)}$. \n\n\\subsection{A Lagrangian with Local Conformal and\n  Time-reparametization Invariance}\n\nThe action\n\\begin{equation}\n  S_{3} := sm\\int\\left[s\\left(\\frac{d\\widehat{q}}{dt}\\right)^2\\right]^{1\/2} dt,\n\\nonumber\\end{equation}\nwhere $s=\\pm1$, is invariant under both the local conformal\ntransformations, $q^a\\to \\alpha(\\mathfrak{u}) q^a$, and the\nreparametization of time $t\\to \\tau(t)$ where $\\tau$ is a\nmonotonically increasing function of $t$. Then\n\\begin{equation}\n  \\mathbf{\\Omega}_L = \\frac{m}{\\vert q\\vert}\n  \\frac{P_{ab}(u)}{\\sqrt{sv\\cdot \\Pi(q)\\cdot\n      v}}\\mathbf{d}q^a\\wedge\\mathbf{d}v^b,\n\\nonumber\\end{equation}\nand $\\mathbf{\\Omega}_F=0$. Here, $a=1. \\dots, D$,\n\\begin{equation}\n  u_a=\\frac{\\Pi_{ab}(q)v^b}{\\sqrt{sv\\cdot\\Pi(q)\\cdot v}},\n\\nonumber\\end{equation}\nso that $u^2 = s$, while $P_{ab}(u) = \\Pi_{ab}(q) -su_a u_b$. As such,\n$\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u}) = $ ker \n$\\mathbf{\\Omega}_M(\\mathfrak{u})$. Both $\\mathcal{C}$ and\n$\\mathcal{G}$ are two-dimensional, and are spanned by\n\\begin{equation}\n  \\mathbf{U}^q_{(1)} = \n  \\widehat{q}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}, \\quad\n    \\mathbf{U}^q_{(2)} =\n  u\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}, \\quad\n  \\hbox{and}\\quad \n    \\mathbf{U}^v_{(1)} =\n  \\widehat{q}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v}, \\quad\n    \\mathbf{U}^v_{(2)} =\n    u\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v},\n\\nonumber\\end{equation}\nrespectively. It follows that $\\hbox{dim }(\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L}\/\\mathcal{G})=2$. \n\nBecause this system is fully constrained, $E=0$. As\n$\\mathbf{\\Omega}_F=0$ as well, there are no\nLagrangian constraints. It follows that $\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is two\ndimensional and spanned by $\\mathbf{U}_{(1)}^q$ and\n$\\mathbf{U}_{(2)}^q$. As $\\mathbf{\\beta}=0$ as well, $\\mathcal{S}\\hbox{ym}$\nis also two dimensional, and is also spanned by  \n$\\mathbf{U}^q_{(1)}$ and $\\mathbf{U}^q_{(2)}$.   \n\nWe found in \\cite{ADS2020} that $\\overline{\\mathbf{X}}_{L} = 0$. A\ngeneral SOELVF is then $\\overline{\\mathbf{X}}_{EL} = u^{1}(\\mathfrak{u}) \n\\left[\\mathbf{U}^q_{(1)}\\right] + u^{2}(\\mathfrak{u})\n\\left[\\mathbf{U}^q_{(2)}\\right]$, with\n$u^{n}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$ for $n=1,2$. It follows\nthat $\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}$ \nis also two-dimensional, and is spanned by $\\mathbf{U}^{q}_{(1)}$ and\n$\\mathbf{U}^q_{(2)}$ as well.  \n\n\\begin{table}\n  \\begin{tabular}{c|c|ccccc}\n    \\hline\n    Action & Potential & $\\>\\>\\>\\overline{\\hbox{ker }\\mathbf{\\Omega}_L}\/\\mathcal{G}\\>\\>\\>$ &\n    $\\>\\>\\>\\mathcal{S}\\hbox{ym}\\>\\>\\>$ & $\\>\\>\\>\\mathcal{S}\\hbox{ym}\\mathcal{L}\\>\\>\\>$\n    & $\\>\\>\\> I_{\\left[1\\right]  }\\>\\>\\>$ &\n    $\\>\\>\\>\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L} \\>\\>\\>$ \\\\\n    \\hline\n    {} & $V_{AS}(\\hat{q}^a)$ & 1 & 1 & 1 & 0 & 1 \\\\\n   \n    $S_{1}$ & $V_{sph}(\\vert q\\vert)+V_{AS}(\\hat{q}^a)$ & 1 & 1 & 0 & 1 & 0 \\\\\n   \n    {} & $V(q^a)$ & 1 & 0 & 0& 1& 0 \\\\\n    \\hline\n    $S_{2}$ & $\\frac{\\lambda}{2}\\left[\\frac{q_1^a}{\\vert q_2\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{2a}}{\\vert q_1\\vert}\\right) -\n  \\frac{q_2^a}{\\vert q_1\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{1a}}{\\vert q_2\\vert}\\right)\n  \\right]$ & 2 & 1 & 1& 1& 1 \\\\\n    \\hline\n    $S_{3}$ & $0$ & 2 & 2& 2& 0& 2\\\\\n  \\hline\n  \\end{tabular}\n  \\caption{A summary of the symmetries of the three examples\n    considered in this paper. With the \n    exception of the $I_{[1]}$ column, the numerical entries are the\n    dimensionality of the vector spaces listed along the first row. Notice\n    the case where the Euler-Lagrange equations of motion has a\n    generalized Lie symmetry while the action itself does not. In all\n    three examples, $\\hbox{dim\n    }(\\mathcal{S}\\hbox{ym}\\mathcal{L})=\\hbox{dim\n    }(\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}_L^{[n_f]}})$. \n  }\n  \\label{Table}\n\\end{table}\n\n\n\\section{Concluding Remarks\\label{&Conc}}\n\nThat each generalized Lie symmetry of the action\ncontributes one arbitrary function to the SOELVF for a dynamical\nsystem is known anecdotally, and is a result expected on physical\ngrounds. For almost regular Lagrangians, the\nappearance in physics of a generalized Lie symmetry is due to a local\ngauge symmetry of the \ndynamical system, and thus to the absence of a gauge\\textemdash the\nlength of vectors for local conformal invariance, or a measure\nfor time for time-reparametization invariance\\textemdash for some\ndynamical property of the system. As the generalized Lie symmetries of\nthe action for an almost regular Lagrangian would have\n$N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ of these gauge freedoms, it is\nreasonable that the absence of these gauges will result in an \nequal number of arbitrary functions in the SOELVF. An equal\nnumber of terms to fix these gauges would then be needed to determine the\ndynamics of the system uniquely. But while these expectations are\nreasonable, up to now they have been fulfilled only on a case-by-case\nbasis. This is in great part because the analysis of dynamical systems\nwith a local gauge symmetry has traditionally been done using\nconstrained Hamiltonian mechanics. Such analysis relies on the\ncanonical Hamiltonian, however, and the connection\nbetween the canonical Hamiltonian and the symmetries of the Lagrangian\nis indirect at best, in contrast to the Lagrangian approach followed\nhere. Moreover, the process of determining the total Hamiltonian\nfor the system is often prescriptive, with results that are specific to\nthe system at hand. By focusing on the Lagrangian and on the Lagrangian\nphase space, we have been able to show for all systems with an almost\nregular Lagrangian that has a constant rank Lagrangian two-form,\na direct link between local gauge symmetries and its dynamics. In\nparticular, it establishes a link between the number of gauge\nsymmetries of the action and the number of arbitrary functions that\nnaturally appear in the evolution of such dynamical systems.  \n\nAs $\\gamma_{\\mathbf{P}}^{[1]}=0$ for any choice of\n$\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$, the vectors in\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ do not contribute to the\nfirst-order constraint manifold $\\mathbb{P}_L^{[1]}$, and as such do\nnot contribute to the Lagrangian constraint algorithm at this order, or at\nany higher orders. It is for this reason that the\n$N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions $u^m_{[1]}$\nare not determined by the algorithm, and why these functions will\nstill contribute to $\\overline{\\mathbf{X}}_{EL}$ even after the algorithm has been\ncompleted. It also means that if second- and higher-order Lagrangian constraints\nare introduced, they are accidental and cannot be due to the local\ngauge symmetries of the action. Interestingly, we have yet to find a dynamical system\nwith a Lagrangian that is both almost-regular and has a Lagrangian\ntwo-form with constant rank where second- or higher-order Lagrangian\nconstraints are introduced. \n\nThis impact of generalized Lie symmetries on the dynamics of particles\nillustrates the inherent differences between the analysis of the\nsymmetries of regular Lagrangians and that of almost regular\nLagrangians. For regular Lagragians, the generator of the generalized\nLie symmetry (at times referred to as a global symmetry) gives rise to\na prolongation vector, and the action of this prolongation on the Lagrangian gives\nthe variation of the action, $\\delta S$, under this symmetry. When the\nEuler-Lagrange equations of motion are thenimposed, the conserved\nquantity for this symmetry along the path given by the solution of these equations of\nmotion is then obtained. While the generator of the generalized \nLie symmetry for the almost regular Lagrangian $\\mathbf{g}_L$ does\ngive a prolongation vector \\textbf{pr} $\\mathbf{g}_L$ Eq.~$(\\ref{prog})$, and\nwhile the action of \\textbf{pr} $\\mathbf{g}_L$ on $L$ does give\n$\\delta S$, imposing the Euler-Lagrange equations of motion on $\\delta\nS$ in Eq.~$(\\ref{Ae1})$ gives the vacuous statement $\\delta\nS=0$. Instead, the requirement that $\\delta S=0$ for all paths on\n$\\mathbb{Q}$ gives the conditions that the  \ngenerators of the symmetry must satisfy. This in turn shows that the\nexistence of these generators is due solely to the Lagrangian being\nsingular. These conditions then \naffect the dynamics of the system through\n$\\gamma_{\\mathbf{P}}^{[1]}=0$, and in doing so, sets a lower bound to\nthe dimensionality of \n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}_L^{[n_f]}}$.  \n\nWe have found it quite difficult to construct more than one example of a\ndynamical system that has an almost regular Lagrangian with both a\ngeneralized Lie symmetry and a \nLagrangian two-form with constant rank on $\\mathbb{P}_L$. We have,\non the other hand, found it quite easy to construct examples of dynamical\nsystems that have an almost regular Lagrangian with a generalized Lie\nsymmetry and a Lagrangian two-form whose rank varies across\n$\\mathbb{P}_L$. Indeed, it is the latter case that is the more\nprevalent one, and yet much of the results of this \npaper and a good portion of the results of our previous one \\cite{ADS2020} relies\non the condition that the rank of the Lagrangian two-form be constant on\n$\\mathbb{P}_L$. This is even more concerning when we realize that these\nmore prevalent systems are expected, by their nature, to have much\nricher dynamics and mathematical structures (indeed, we have found that\nsuch systems often require the introduction of second- or higher-order\nLagrangian constraints), and yet it is not known\nwhich of the results that have been shown to hold for systems with\nconstant rank Lagrangian two-forms will still hold when the rank varies\nacross $\\mathbb{P}_L$. Determining the generalized Lie symmetries of\nthese systems; showing that the passage from the Lagrangian to the\nHamiltonian phase space is possible; and finding the links between\nsymmetry and dynamics is a necessity for future research.  \n\n\\begin{acknowledgments}\n  This paper would not have been possible without the contributions by\n  John Garrison, who provided much of the underlying symmetry analysis\n  of the action used in \\textbf{Section \\ref{&A-Sym}}, and most\n  of the essential mathematics in \\textbf{Section\n    \\ref{&review}}. Publication made possible in \n  part by support from the Berkeley Research Impact Initiative (BRII)\n  sponsored by the UC Berkeley Library. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe simple loop conjecture, proved by Gabai in \\cite{Ga}, states that any non-injective homomorphism from a closed surface group to another closed surface group has an element represented by a simple closed curve in the kernel.  It has been conjectured that the result still holds if the target is replaced by the fundamental group of an orientable 3-manifold (see Kirby's problem list in \\cite{Ki}).  Although special cases have been proved (e.g. \\cite{Ha}, \\cite{RW}), the general hyperbolic case is still open. \n\nRecently, Cooper and Manning showed that if instead of a 3-manifold group the target group is $\\SL(2, \\mathbb{C})$, then the conjecture is false.  Precisely, they show: \n\n\\begin{theorem}[Cooper-Manning \\cite{CM}]  \\label{cmthm} Let $\\Sigma$ be a closed orientable surface of genus $g \\geq 4$.  Then there is a homomorphism $\\rho: \\pi_1(\\Sigma) \\to \\SL(2, \\mathbb{C})$ such that \n\\begin{enumerate} \n\\item $\\rho$ is not injective\n\\item If $\\rho(\\alpha) = \\pm I$, then $\\alpha$ is not represented by a simple closed curve\n\\item If $\\rho(\\alpha)$ has finite order, then $\\rho(\\alpha) = I$\n\\end{enumerate}\nThe third condition implies in particular that no \\emph{power} of a simple closed curve lies in the kernel.  \n\\end{theorem}\n\nInspired by this, we asked whether a similar result holds for $\\PSL(2,\\mathbb{R})$, this being an intermediate case between Gabai's result for surface groups and Cooper and Manning's for $\\SL(2,\\mathbb{C})$.  Cooper and Manning's proof uses a dimension count on the $\\SL(2,\\mathbb{C})$ character variety and a proof that a specific subvariety is irreducible and smooth on a dense subset, much of which does not carry over to the $\\PSL(2, \\mathbb{R})$ case.  In general, complex varieties and their real points can behave quite differently.   However, we show here with different methods that an analogous result does hold.  \n\nWhile this note was in progress, we learned of work of Louder and Calegari (independently in \\cite{Lo} and \\cite{Ca}) that can also be applied to answer our question in the affirmative. Louder shows the simple loop conjecture is false for representations into limit groups, and Calegari gives a practical way of verifying no simple closed curves lie in the kernel of a non-injective representation using stable commutator length and the Gromov norm.  \n\nThe difference here is that our construction is entirely elementary.  We use an explicit representation from DeBlois and Kent in \\cite{DK} and verify that this representation it is non injective and kills no simple closed curve by elementary means.   Our end result parallels that of Cooper and Manning but also include surfaces with boundary and all genera at least 1: \n\n\\begin{theorem}\\label{main}\nLet $\\Sigma$ a surface of negative Euler characteristic and of genus $g \\geq 1$\n, possibly with boundary.  \nThen there is a homomorphism $\\rho: \\pi_1(\\Sigma) \\to \\SL(2, \\mathbb{R})$ such that \n\\begin{enumerate} \n\\item $\\rho$ is not injective\n\\item If $\\rho(\\alpha) = \\pm I$, then $\\alpha$ is not represented by a simple closed curve\n\\item In fact, if $\\alpha$ is represented by a simple closed curve, then $\\rho(\\alpha^k) \\neq 1$ for any $k \\in \\mathbb{Z}$.  \n\\end{enumerate}\nMoreover, there are uncountably many non-conjugate representations satisfying 1. through 3.  \n\\end{theorem}\n\n\n\n\n\n\n\n\n \n\n\n\n\n\\section{Proof of theorem \\ref{main}} \n\nWe first present a construction of a (non-injective) representation from DeBlois and Kent in \\cite{DK}, and then show that no power of a simple closed curve lies in the kernel of this representation.  The full construction appears in \\cite{DK}, we describe it here for convenience. \n\nLet $\\Sigma$ be a surface of genus $g \\geq 1$ and negative Euler characteristic, possibly with boundary. Assume for the moment that $\\Sigma$ is not the once-puntured torus -- Theorem \\ref{main} for this case will follow easily later on.  \n\nLet $c \\subset \\Sigma$ be a simple closed curve separating $\\Sigma$ into a genus 1 subsurface with single boundary component $c$, and a genus ($g-1$) subsurface with one or more boundary components.  Let $\\Sigma_A$ denote the genus $(g-1)$ subsurface and $\\Sigma_B$ the genus 1 subsurface.  See Figure \\ref{setup} below.  Finally, we let $A= \\pi_1(\\Sigma_A)$ and $B = \\pi_1(\\Sigma_B)$, so that $\\pi_1(\\Sigma) = A \\ast_C B$, where $C$ is the $\\mathbb{Z}$-subgroup generated by the element $[c]$ represented by the curve $c$.  We assume that the basepoint for $\\pi_1(\\Sigma)$ lies on $c$.   \n\n  \\begin{figure*}[h]\n  \\centerline{\n    \\mbox{\\includegraphics[width=2.5in]{simpleloopsetup.pdf}}}\n \\caption{The setup: decomposition of $\\Sigma$ and generators $x$ and $y$ for $B$}\n  \\label{setup}\n  \\end{figure*}\n\nLet $x \\in B$ and $y \\in B$ be generators such that $B = \\langle x, y \\rangle$, and that $c$ represents the commutator $[x,y]$.   Fix $\\alpha$ and $\\beta$ in $\\mathbb{R} \\setminus \\{0,\\pm1\\}$, and following \\cite{DK} define $\\phi_B: B \\to \\SL(2,\\mathbb{R})$ by \n\n$$\\phi_B(x) = \n \\begin{pmatrix}\n  \\alpha & 0 \\\\\n  0 & \\alpha\n \\end{pmatrix}$$\n  $$\\phi_B(y) = \n  \\begin{pmatrix}\n  \\beta & 1 \\\\\n  0 & \\beta^{-1} \n \\end{pmatrix}$$  \n  We have then \n  $$\\phi_B([x, y]) =  \\begin{pmatrix}\n 1 & \\beta(\\alpha^2 - 1)  \\\\\n  0 & 1  \n \\end{pmatrix}$$\nso that $\\phi_B([x, y])$ is invariant under conjugation by the matrix $\\lambda_t := \\bigl( \\begin{smallmatrix} \n 1 & t  \\\\\n  0 & 1  \n\\end{smallmatrix} \\bigr)$.  \n  \nProjecting this representation to $\\PSL(2,\\mathbb{R})$ gives a representation which is upper triangular, hence solvable and therefore non-injective.  Abusing notation, let $\\phi_B$ denote the representation to $\\PSL(2,\\mathbb{R})$.\n \nNow let $\\phi_A : A \\to \\PSL(2, \\mathbb{R})$ be Fuchsian and such that the image of the boundary curve $c$ under $\\phi_A$ agrees with $\\phi_B([x,y])$.  Such a representation exists for the following reasons.  First, if $\\Sigma$ has negative Euler characteristic, genus $g>1$, and is not the once punctured torus, then $\\Sigma_A$ will have negative Euler characteristic as well and admit a hyperbolic structure.  Secondly, the Fuchsian representation coming from the hyperbolic structure will send the element $[c]$ representing the boundary curve to a parabolic, so after conjugation we may assume that it is equal to $\\phi_B([x,y])$, provided $\\phi_B([x,y])$ is parabolic, i.e. $\\beta(\\alpha^2 - 1) \\neq 0$.  \n \nFinally, combine $\\phi_A$ and $\\phi_B$ to get a one-parameter family of representations $\\phi_t$ of $\\pi_1(\\Sigma) = A \\ast_C B$ to $\\PSL(2,\\mathbb{R})$ as follows.  For $t \\in \\mathbb{R}$ and $g \\in A \\ast_C B$, let\n\n$$\\phi_t(g) = \\left\\{ \\begin{array}{rcl} \\phi_A(g) & \\mbox{if} & g \\in A \\\\ \\lambda_t \\circ \\phi_B(g) \\circ (\\lambda_t)^{-1} & \\mbox{if} & g \\in B\n\\end{array}\\right.$$\n\nThis representation is well defined because $\\phi_B([x, y]) = \\phi_A([x,y])$ and $\\phi_B([x, y])$ is invariant under conjugation by $\\lambda_t$.  \n\nOur next goal is to show that for appropriate choice of $\\alpha, \\beta$ and $t$, the representation $\\phi_t$  satisfies the criteria in Theorem \\ref{main}.  The main difficulty will be checking that no element representing a simple closed curve is of finite order.  To do so, we employ a stronger form of Lemma 2 from \\cite{DK}.  This is:\n\n\\begin{lemma} \\label{transcendental}\nSuppose $w \\in A \\ast_C B$ is a word of the form $w = a_1b_1a_2b_2 ... a_lb_l$ with $a_i \\in A$ and $b_i \\in B$ for $1 \\leq i \\leq l$. Assume that for each $i$, the matrix $\\phi_t(a_i)$ has a nonzero 2,1 entry and $\\phi_t(b_i)$ is hyperbolic. If $t$ is transcendental over the entry field of $\\phi_0(A \\ast_C B)$, then $\\phi_t(w)$ is not finite order.  \n\\end{lemma}\n\n\\noindent By \\emph{entry field} of a group $\\Gamma$ of matrices, we mean the field generated over $\\mathbb{Q}$ by the collection of all entries of matrices in $\\Gamma$.\n\n\\begin{remark} \nLemma 2 of \\cite{DK} is a proof that $\\phi_t(w)$ is not the \\emph{identity}, under the assumptions of Lemma \\ref{transcendental}.    We use some of their work in our proof.  \n\\end{remark}\n\n\\begin{proof}[Proof of Lemma \\ref{transcendental}]\nIn \\cite{DK}, DeBlois and Kent show by a straightforward induction that the entries of $\\phi_t(w)$ are polynomials in $t$, where the degree of the 2,2 entry is $l$, the degree of the 1,2 entry is at most $l$, and the other entries have degree at most $l-1$.  Now suppose that $\\phi_t(w)$ is finite order.  Then it is conjugate to a matrix of the form \n$ \\bigl( \\begin{smallmatrix} \n u & v  \\\\\n  -v & u  \n\\end{smallmatrix} \\bigr)$. \nwhere $u = \\cos(\\theta)$ and $v = \\sin(\\theta)$ for some rational angle $\\theta$.  In particular, it follows from the deMoivre formula for sine and cosine that $u$ and $v$ are algebraic.  \n\nNow suppose that the matrix conjugating $\\phi_t(w)$ to \n$ \\bigl( \\begin{smallmatrix} \n u & v  \\\\\n  -v & u  \n\\end{smallmatrix} \\bigr)$ has entries $a_{ij}$.  Then we have\n$$ \\phi_t(w) =  \\begin{pmatrix}\nu -  (a_{12}a_{22} -  a_{11}a_{21})v   &  (a_{12}^2  a_{11}^2) v \\\\\n -(a_{22}^2  a_{21}^2)v &   u + (a_{12}a_{22} +  a_{11}a_{21})v\n \\end{pmatrix}$$\nLooking at the 2,2 entry we see that $a_{12}a_{22} +  a_{11}a_{21}$ must be a polynomial in $t$ of degree $l$.  But this means that the 1,1 entry is also a polynomial in $t$ of degree $l$, contradicting Deblois and Kent's calculation.  This proves the lemma.\n\\end{proof} \n\n\nTo complete our construction, choose $t$ to be transcendental over the entry field of $\\phi_0(A\\ast_C B)$.  We want to show that no power of an element representing a simple closed curve lies in the kernel of $\\phi_t$.  To this end, consider any word $w$ in $A \\ast_C B$ that has a simple closed curve as a representative.  There are three cases to check.  First, if $w$ is a word in $A$ alone, then $\\phi_t(w)$ is not finite order, since $\\phi_t(A)$ is Fuchsian and therefore injective.  Secondly, if $w$ is a word in $B$, then an elementary geometric argument shows that $w$ can only be represented by a simple closed curve if it has one of the following forms: \n\\begin{enumerate} \n\\item $w=x ^{\\pm 1}$ or $w= y^{\\pm 1}$\n\\item $w = [x^{\\pm1},y^{\\pm1}]$\n\\item Up to replacing $x$ with $x^{-1}$, $y$ with $y^{-1}$ and interchanging $x$ and $y$, there is some $n \\in \\mathbb{Z}^+$ such that $w = x^{n_1}y x^{n_2} y ... x^{n_s}y$ where $n_i \\in \\{n, n+1\\}$.  \n\\end{enumerate}\nWe leave this as an exercise for the reader.  \nThis classification of words representing simple closed curves in $\\Sigma_B$ also follows from a much more general theorem in \\cite{BS}.\n\nBy construction, no word of type 1, 2 or 3 is finite order provided that $\\alpha^s \\beta^k \\neq 1$ for any integers $s$ and $k$ other than zero -- indeed, we only need to check words of type 3, and these necessarily have trace $\\alpha^s\\beta^k + \\alpha^{-s}\\beta^{-k}$ for some $s, k \\neq 0$.  Note that in particular, under the condition that $\\alpha^s \\beta^k \\neq 1$ for $s, k \\neq 0$, all type 3 words are hyperbolic.  We will use this fact again later on.  \n\nFor the remaining case where $w$ is a word with letters in both $A$ and $B$, we claim that it can be written in a form where Lemma \\ref{transcendental} applies.  To write it this way, use the following procedure: First take a simple representative $\\gamma$ for $w$ and apply an isotopy so that each crossing of $\\gamma$ with $c$ occurs in some small neighborhood of the basepoint $p$.  This gives us a well defined shortest path along $c$ to $p$ from each crossing.  After further isotopy, we may assume additionally that no segment of $\\gamma$ together with the shortest path along $c$ from its endpoints to $p$ bounds a disc, and that $\\gamma$ is transverse to $c$.  All this can be done without introducing any self-crossings in $\\gamma$.  Now $\\gamma$ is of the form $\\gamma_1 \\delta_1 \\gamma_2 \\delta_2 ... \\gamma_l \\delta_l$ where $\\gamma_i$ is a simple arc in $\\Sigma_A$ and $\\delta_i$ a simple arc in $\\Sigma_B$. Close each $\\gamma_i$ and $\\delta_i$ into a simple loop by connecting its endpoints to $p$ using the shortest path along $c$ and let $a_i \\in A$ (respectively $b_i \\in B$) be the corresponding element of the fundamental group.  See Figure \\ref{fig1}.  This gives us a word $a_1b_1a_2b_2 ... a_lb_l$ equivalent to $w$ after cyclic reduction, and each $a_i$ is represented by a simple closed curve in $\\Sigma_A$ and each $b_i$ by a simple closed curve in $\\Sigma_B$.  The elimination of discs bounded between segments of $\\gamma$ and short segments of $c$ ensures that each $a_i$ and $b_i$ is nontrivial.  \n\n  \\begin{figure*}\n  \\centerline{\n    \\mbox{\\includegraphics[width=3.3in]{simpleloop.pdf}}}\n \\caption{$a_1$ and $b_1$ in $w$, represented by $\\gamma_i$ and $\\delta_i$ joined to $p$}\n  \\label{fig1}\n  \\end{figure*}\n\nWe can also show that each $a_i$ either has a non-zero 2,1 entry or is represented by the curve $c$ or its inverse.  This is because $\\phi_A$ is Fuchsian, so the only elements fixing infinity -- that is, with 2,1 entry equal to zero -- are powers of $c$, and no powers of $c$ other than $c^{\\pm1}$ have a simple closed curve representative.  Similarly, the classification of words representing simple closed curves in $\\Sigma_B$ shows that each $b_i$ is either hyperbolic or represented by $c$ or $c^{-1}$.  We claim that we may now rewrite $w$ to eliminate all appearances of $c$, keeping each $a_i$ with a non-zero 2,1 entry and each $b_i$ hyperbolic.  After doing so, we will have $w$ in a form where we can apply Lemma \\ref{transcendental}.  \n\nTo rewrite $w$ in the desired form, first note that all $\\gamma_i$ such that $a_i$ is represented by $c$ may be homotoped (simultaneously) into $\\Sigma_b$ without introducing any self intersections of $\\gamma$.  Thus, we can replace each such $\\delta_{i-1} \\gamma_i \\delta_i$ with a simple loop $\\delta'_i$ in $\\Sigma_B$ alone, and rewrite $w = a_1b_1... a_{i-1} b'_{i} a_{i+1} ... a_lb_l$.  Reindex so that $w = a_1b_1a_2b_2 ... a_kb_k$ for $k < l$, and reindex the corresponding $\\delta_i$ and $\\gamma_i$ as well.  \nNow repeat the procedure on this new word with each $b_i$: homotope all $\\delta_i$ such that $b_i$ is represented by $c$ over to $\\Sigma_A$ without introducing any self intersections of $\\gamma$, and then replace each such $\\gamma_i \\delta_i \\gamma_{i-1}$ with a simple loop $\\gamma_i'$ in $\\Sigma_B$ alone. Then rewrite $w$ so that, after reindexing, $w = a_1b_1a_2b_2 ... a_mb_m$ with $m<k$ and each $a_i$ and $b_i$ is a simple closed curve.  Repeat the process again with the $a_i$ of this new word.  The procedure ends when either no $a_i$ or $b_i$ is represented by $c$, or when $w$ is a word in $A$ or $B$ alone, represented by a simple loop in $\\Sigma_A$ or $\\Sigma_B$.  In the first case, Lemma \\ref{transcendental} applies to show that $\\phi_t(w)$ is not finite order.  In the second case, we have already shown that a word in $A$ or $B$ represented by a simple loop in $\\Sigma_A$ or $\\Sigma_B$ cannot be finite order.  \n\nIt remains only to remark that the representation $\\phi_t$ is non-injective and that, by choosing appropriate parameters, we can produce uncountably many nonconjugate representations.   Non-injectivity follows immediately since $\\phi_t(B)$ is solvable so the restriction of $\\phi_t$ to $B$ is non-injective.  Now for any fixed $\\alpha$ and $\\beta$  (satisfying the requirement that $\\alpha^s \\beta^k \\neq 1$ for all integers $s, k$), varying $t$ among transcendentals over the entry field of $\\phi_0(A \\ast_C B)$ produces uncountably many non-conjugate representations that are all non-injective, but have no power of a simple closed curve in the kernel.  This concludes the proof of Theorem \\ref{main}, assuming that the surface was not the punctured torus.  \n\nThe punctured torus case is now immediate: any representation of the form of $\\phi_B$ where $\\alpha^s \\beta^k \\neq 1$ for any integers $s$ and $k$ is non-injective and our work above shows that no element represented by a simple closed curve has finite order.  Fixing $\\alpha$ and varying $\\beta$ produces uncountably many non-conjugate representations.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis is a simple template for authors to write new MNRAS papers.\nSee \\texttt{mnras\\_sample.tex} for a more complex example, and \\texttt{mnras\\_guide.tex}\nfor a full user guide.\n\nAll papers should start with an Introduction section, which sets the work\nin context, cites relevant earlier studies in the field by \\citet{Others2013},\nand describes the problem the authors aim to solve \\citep[e.g.][]{Author2012}.\n\n\\section{Methods, Observations, Simulations etc.}\n\nNormally the next section describes the techniques the authors used.\nIt is frequently split into subsections, such as Section~\\ref{sec:maths} below.\n\n\\subsection{Maths}\n\\label{sec:maths}\n\nSimple mathematics can be inserted into the flow of the text e.g. $2\\times3=6$\nor $v=220$\\,km\\,s$^{-1}$, but more complicated expressions should be entered\nas a numbered equation:\n\n\\begin{equation}\n    x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n\t\\label{eq:quadratic}\n\\end{equation}\n\nRefer back to them as e.g. equation~(\\ref{eq:quadratic}).\n\n\\subsection{Figures and tables}\n\nFigures and tables should be placed at logical positions in the text. Don't\nworry about the exact layout, which will be handled by the publishers.\n\nFigures are referred to as e.g. Fig.~\\ref{fig:example_figure}, and tables as\ne.g. Table~\\ref{tab:example_table}.\n\n\\begin{figure}\n\n\n\n\t\\includegraphics[width=\\columnwidth]{example}\n    \\caption{This is an example figure. Captions appear below each figure.\n\tGive enough detail for the reader to understand what they're looking at,\n\tbut leave detailed discussion to the main body of the text.}\n    \\label{fig:example_figure}\n\\end{figure}\n\n\\begin{table}\n\t\\centering\n\t\\caption{This is an example table. Captions appear above each table.\n\tRemember to define the quantities, symbols and units used.}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{lccr}\n\t\t\\hline\n\t\tA & B & C & D\\\\\n\t\t\\hline\n\t\t1 & 2 & 3 & 4\\\\\n\t\t2 & 4 & 6 & 8\\\\\n\t\t3 & 5 & 7 & 9\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}\n\nThe last numbered section should briefly summarise what has been done, and describe\nthe final conclusions which the authors draw from their work.\n\n\\section*{Acknowledgements}\n\nThe Acknowledgements section is not numbered. Here you can thank helpful\ncolleagues, acknowledge funding agencies, telescopes and facilities used etc.\nTry to keep it short.\n\n\n\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\nThe first gravitational wave (GW) detection of a binary neutron star (BNS) merger, GW170817, by the Advanced\nLIGO and Virgo (aLIGO\/Virgo) facilities, was an eagerly awaited event \\citep{abbott17a}. Such a merger was predicted to produce a multi-wavelength electromagnetic afterglow radiating from radio to gamma-rays, and GW170817 did not disappoint \\citep{abbott17b,andreoni17}. The most conspicuous predicted counterpart was the prompt ejection of collimated, short-lived gamma-ray emitting jets, similar to the observed \nshort gamma-ray burst \\citep[SGRB, gamma-ray durations $<2$\\,s;][]{narayan92} phenomenon, which is one of the two main classes of gamma-ray burst \\citep[GRB;][]{norris84,dezalay92,kouveliotou93} detected by the \\textit{Neil Gehrels Swift Observatory} \\citep[hereafter {\\em Swift};][]{gehrels04} Burst Alert Telescope (BAT) and the \\textit{Fermi Gamma-ray Space Telescope} (hereafter {\\em Fermi}{}) Gamma-ray Burst Monitor \\citep[GBM;][]{meegan09}. \nThe other dominant population known as long GRBs (LGRBs) typically have durations $>2$\\,s and are attributed to core collapse supernovae \\citep[e.g.][]{galama98,bloom98}.\nIt was therefore the near-simultaneous detection of GW170817 and GRB 170817A \\citep[the latter of which was a SGRB detected by {\\em Fermi}{};][]{abbott17b} and the late-time radio and X-ray follow-up confirming the presence of an off-axis jet \\citep[e.g.][]{mooley18b,ghirlanda19,troja19gw} that strongly supported the link between BNS mergers and SGRBs.\n\nThe detection of the electromagnetic counterpart to a aLIGO\\slash Virgo-detected BNS merger is of great importance as it enables the localisation of the source, along with providing complementary information such as an independent distance measurement, insight into the central engine, the energy released, and the final merger remnant. However, the initial localisation of a GW event by aLIGO\/Virgo is tens to hundreds of square degrees, making it difficult to search for counterparts. \nWe therefore introduce a method designed to exploit the established link between GW-detected BNS mergers and SGRBs by using the Australia Telescope Compact Array (ATCA) rapid-response mode to trigger on {\\em Swift}{}-detected SGRBs. \n\nWhile the radio emission from SGRBs is usually short-lived \\citep[$\\lesssim2$\\,days;][]{fong15}, the ATCA rapid-response mode is capable of being on-source within $10$\\,minutes. \nBy rapidly responding to {\\em Swift}{} SGRB triggers, ATCA can become a new diagnostic tool for\nuncovering the range of radio behaviour shown by SGRBs to help interpret what to look for from GW events that have off-axis gamma-ray jets. \nAs targeted observations can usually reach deeper sensitivities than wide-field surveys, ATCA observations can provide\na template of the radio brightness and timing properties of BNS mergers, which will \nin-turn inform the follow-up strategies of the next era of aLIGO\/Virgo GW events by wide-field radio telescopes, such as Australian instruments like the Murchison Widefield Array \\citep[MWA;][]{tingay13} and the Australian Square Kilometre Array Pathfinder \\citep[ASKAP;][]{johnston08}.\n\nThe jet launched during an SGRB is expected to produce a radio afterglow as predicted by the fireball model \\citep{cavallo78,rees92}. \nIn this model, the relativistic ejecta interact with the circumstellar medium (CSM) producing a forward shock that accelerates electrons and generates synchrotron emission. Reverse shock synchrotron emission, produced by the shock that propagates back into the post-shock ejecta, may also be observed  \ndepending on the density of the CSM and the ejecta. \nThe broadband spectrum produced by the jet interactions in the GRB afterglow is described by the peak flux and 3 characteristic frequencies ($\\nu_m$, the minimum electron energy frequency; $\\nu_{sa}$, the synchrotron self-absorption frequency; and $\\nu_c$, the electron cooling frequency), which evolve over time \\citep{sari98,wijers99,granot99}. Only early-time radio observations are able to properly constrain 2 of these 3 frequencies ($\\nu_m$ and $\\nu_{sa}$), and also disentangle the reverse and forward shock components. \nBy combining ATCA observations with \nmulti-wavelength observations to perform SED modelling, these parameters can be derived, thus providing information about the blast wave kinetic energy, the CSM density, the magnetic field energy density and the power law electron energy distribution \\citep{sari98,wijers99,granot99}. \nLimits on the linear polarisation of the reverse shock can also provide information on the jet magnetic field structure \\citep{granot14}.\nEarly-time radio observations of SGRBs are also sensitive to temporal steepening from the jet-break \\citep{sari99},\nwhich constrains the jet opening-angle used to calculate the true energy released \\citep[and therefore merger BNS\/GW event rates, e.g.][]{fong14,fong15,deugartepostigo14}. \nEven early-time non-detections in the radio band can allow us to make predictions about when the forward-shock emission may peak, which can inform the cadence and duration of follow-up radio observations, potentially optimising the success of a late-time detection as we demonstrate in this paper.  \nIn addition, sensitive, multi-frequency, high-cadence radio observations may allow us to distinguish between more exotic emission models caused by the ejection of neutron star material or the propagation of shocks caused by the merger event, which may produce non- to ultra-relativistic omnidirectional radio emission \\citep[e.g.][]{nakar11,kyutoku14}.\nIt is therefore crucial to obtain early-time radio observations (within minutes to days) of a larger sample of SGRBs to better characterise the timescales and frequencies necessary for understanding the range of behaviours we might expect from GW radio counterparts. \n\nThere are also several BNS merger models that suggest \na short-lived, supramassive and highly magnetised neutron star (NS) or ``magnetar'', supported by rotation, can exist for a short time ($<10^{4}$\\,s) before finally forming a stable magnetar or further collapsing into a black hole \\citep[BH, e.g.][]{usov92,zhang01,falcke14,zhang14}.  \nEvidence for such merger products comes from the detection of a ``plateau phase'' in some SGRB X-ray light curves between $\\sim10^{2}-10^{4}$\\,s post-burst, where this departure from power-law decay indicates ongoing energy injection \\citep{rowlinson13}. \nSuch merger remnant scenarios may be sources of prompt, coherent radio emission \\citep[see][for a review]{rowlinson19}. However, no continuous monitoring of the radio behaviour has yet been performed at GHz frequencies during the plateau phase. Such detections or upper limits could constrain different central engine models as has been done at late-times \\citep[e.g.][]{fong16}. \n\nOnly eight SGRBs have published detections in the radio band to date: GRB 050724A, 051221A, 130603B, 140903A, 141212A, 150424A, 160821B and 200522A\n\\citep{berger05,soderberg06,fong14,fong15,fong17,troja16,zhang17,troja19,lamb19,fong21}.\nNote that this does not include GW170817 as it had a far more off-axis outflow \nthan standard cosmological SGRBs so the corresponding radio afterglow was detected much later when the ejecta had moved into our line-of-sight \\citep{mooley18b}. \nOut of a sample of $>70$ radio-observed SGRBs, only $\\sim10$\\% have been detected in the radio band at GHz frequencies \\citep{fong21}.\nThis low detection rate may be due to an observed fast rise in radio emission with a potentially short radio afterglow lifetime. \nFor example, 7 of the 8 radio-detected SGRBs were detected within 1\\,day post-burst, at least half of which faded below detectability within $\\sim2$ days (see Figure~\\ref{fig:lc}). \nGiven these short timescales, it is possible the radio emission is frequently dominated by the reverse-shock \\citep[as was the case for GRB 051221A;][]{soderberg06} since simulations of BNS mergers demonstrate forward shock radio emission may evolve over days to weeks \\citep{hotokezaka16} as is also the case for many LGRBs \\citep[e.g.][]{vanderhorst08,vanderhorst14}. \nIf we instead compare the radio-detected sample to those SGRBs that were initially observed at radio wavelengths $<1$\\,day post-burst, this gives a much higher radio detection rate of $\\sim30$\\% \\citep{fong15}. \nHowever, while the first four radio-detected SGRBs showed initial flux densities of $>0.1$ mJy\/beam at GHz frequencies, few of the other $<1$\\,day post-burst pre-2016 observations had sufficient sensitivity to detect a predicted peak flux density of $\\sim40\\mu$\\,Jy\/beam at 10\\,GHz for an SGRB at an average redshfit of $z=0.5$ with an expected CSM density of $n_{0} \\sim0.1$\\,cm$^{-3}$ \\citep{berger14}. In fact, the four most recent radio-detected SGRBs peak at $\\lesssim40\\mu$\\,Jy\/beam.\n\nThe small sample of radio detected SGRBs therefore provides limited knowledge of their radio afterglow brightnesses and timescales, and is insufficient for deriving the energy outputs and environmental properties of the population through multi-wavelength modelling. It is therefore vital to perform both rapid and sensitive radio follow-up observations of SGRBs to capture these short-lived and faint events. The key to achieving this is through the use of rapid-response (also known as triggering) systems, where a telescope has the ability to automatically respond to a transient alert, and either repoint at the event or update its observing schedule to begin observations when the source has risen above the horizon. Rapid-response radio telescopes have been in use since the 1990's \\citep[for example see][]{green95,dessenne96,bannister12,palaniswamy14,kaplan15} \nbut predominantly at low radio frequencies (100\\,MHz to 2.3\\,GHz), with the majority of experiments being designed to search for prompt, coherent radio emission. However, until recently, the only high frequency ($>5$\\,GHz) rapid-response program designed to target incoherent (synchrotron) radio emission from GRBs has been run on the Arcminute Microkelvin Imager (AMI) Large Array (LA), known as ALARRM (the AMI-LA Rapid Response Mode), which has been active since 2012 \\citep[][]{staley13,anderson18}. It was only through ALARRM that it was possible to be on-source fast enough to detect the rise and peak in the reverse-shock radio emission at 15\\,GHz from GRB 130427A within 1\\,day post-burst, which also represents one of the earliest radio detections of a GRB to date \\citep{anderson14}. In addition, the radio catalogue of AMI observations of 139 GRBs (12 were short GRBs but non-detections), the majority of which were automatically triggered on using the rapid-response mode within $1$\\,day post-burst, was the first representative sample of GRB radio properties that was unbiased by multi-wavelength selection criteria \\citep{anderson18}. This work revealed that possibly up to $\\sim44-56$\\% of {\\em Swift}{}-detected LGRBs have a radio counterpart (down to $\\sim0.1-0.15$\\,mJy\/beam), with the increase in detection rate from previous studies \\citep[$\\sim30$\\%;][]{chandra12} likely being due to the AMI rapid-response mode, which allows observations to begin while the reverse-shock is contributing to the radio afterglow. This program has motivated the installation of a rapid-response mode on the ATCA. \n\nHere we present the first triggered observation of a SGRB using the new ATCA rapid-response mode. \nATCA is an ideal instrument for performing triggered radio follow-up of {\\em Swift}{} SGRBs due to its high sensitivity and broadband receivers that provide simultaneous multi-frequency coverage. The ATCA response times (which can be as short as minutes) \nhave the potential to be \nmuch faster than the current median SGRB response of the Karl G. Jansky Very Large Array (VLA; $\\sim24.7$\\,hrs), which rely on manually scheduling target-of-opportunity observations \\citep{fong15}.\nIn Section 2, we describe the ATCA rapid-response system from the observer interaction (front-end) level and the observatory (back-end) level.\nIn Section 3, we describe the triggered ATCA observation and data reduction of GRB 181123B, and corresponding results. \nThis is followed by a comparison of our radio limits for GRB 181123B to the sample of radio-detected SGRBs and a discussion of the parameter space that the triggered ATCA observations are probing in Section 4. \nFinally, we perform modelling of the GRB 181123B afterglow \nand thus demonstrate the usefulness of obtaining early-time (within 1 day) radio observations of an SGRB (regardless of whether or not there is a detection) to place constraints on the GRB physics.  \n\n\\section{ATCA rapid-response mode}\n\nATCA is a six element, 22\\,m dish, East-West interferometer based in New South Wales in Australia. Its maximum baseline length is 6\\,km and it is capable of observing in multiple, broad frequency bands  \nwith full polarisation, and in a variety of array configurations. \nATCA is currently equipped with the Compact Array Broadband Backend \\citep[CABB;][]{wilson11}, which has a 2\\,GHz bandwidth that is capable of observing in two frequency bands simultaneously with tunable receivers that operate between 1.1-105\\,GHz. \n\nSince 2017 April 18, ATCA has been capable of rapidly responding to transient alerts. \nThe rapid-response mode can trigger using the 16\\,cm, 4\\,cm and 15\\,mm receivers, corresponding to a usable frequency range of $1.1-25$\\,GHz, and can observe in any CABB mode. \nIn the following, we describe both the observer front-end and the observatory back-end of this new triggering system.\n\n\\subsection{VOEvent parsing\/front-end}\\label{sec:front-end}\n\nThe front-end software we use to interface with the ATCA rapid-response system ({\\sc vo\\_atca})\\footnote{https:\/\/github.com\/mebell\/vo\\_atca} is designed to trigger on Virtual Observatory Events (VOEvent; \\citealt{seaman11}), which are the standard format for broadcasting machine readable astronomical alerts related to transient events. \nA VOEvent package contains all the required data (in {\\sc xml} format) that allow automated decisions to be made in real-time given certain keywords and parameters. \nVOEvents are brokered via the 4 Pi Sky VOEvent Broker \\citep{staley16pp} and the {\\sc comet} VOEvent client \\citep{swinbank14}. \nThese packages allow us to listen to multiple VOEvent streams, including those broadcast by {\\em Swift}{}. \nWe use the {\\sc Python} package {\\sc vovent-parse} \\citep{staley_voevent-parse_2014} as the main tool to read the VOEvents \nand to extract the required information to be assessed by the triggering algorithm. \n\nUpon receiving a {\\em Swift}{} VOEvent, the ATCA VOEvent parser uses the keyword {\\sc grb\\_identified = true} to initially identify a GRB packet. \nPackets containing {\\sc startrack\\_lost\\_lock=true} are ignored as it means that {\\em Swift}{} has lost its celestial position-lock so such an alert is unlikely to be from a real transient.  \nWhile the observatory back-end prevents the telescope from overriding for sources that are too far north (see Section~\\ref{sec:back-end}), we impose an additional declination cut-off for all SGRBs north of +15$^{\\circ}$ to \nensure the potential for $>8$\\,hr integrations for the triggered observations. \n\nOn passing these stages, the parser then assesses the duration of the trigger so that SGRB candidates can be identified. However, on the short timescales following the alert, and with growing uncertainty as the GRB burst duration increases, it is difficult to classify {\\em Swift}{} GRBs as short or long in an automated way.\nA rigorous classification of the GRB requires human inspection of the data, which is only published online on the Gamma-ray Coordinates Network Circulars (GCN)  Archive,\\footnote{https:\/\/gcn.gsfc.nasa.gov\/gcn3\\_archive.html} usually between 10\\,mins and 1\\,hr post-burst and therefore not via a VOEvent. \nTo account for this, we implemented a three-tiered system to flexibly respond to different GRB durations and therefore filter for those events more likely to be SGRBs. \nThe keyword {\\sc integ\\_time} (the length of time for the transient signal to reach a significant threshold) parameter is used as an estimator of the incoming GRB's true duration. \n\n\\begin{itemize}\n\\item GRBs with {\\sc integ\\_time}$<$0.257\\,s have a high probability of being SGRBs so the VOEvent parser will automatically submit these triggers to the observatory and alert team members via text and email of the override observation. \n\n\\item With durations $0.257$\\,s\\,$<${\\sc integ\\_time}\\,$<1.025$\\,s, we have implemented a \"wait-to-proceed\" algorithm as the probability of the GRB being a SGRB decreases with increasing {\\sc integ\\_time}. In this case, we issue email and text alerts so that team members can check the GCN Archive for adequate verification of the GRB classification. If the GRB is confirmed to be short, then the duty team member responds \"YES\" to the detection email, and this email reply is read by an algorithm (via the Google email Application Programming Interface\\footnote{https:\/\/developers.google.com\/gmail\/api}) \nthat then proceeds with submitting the trigger to ATCA, resulting in an override observation. This provides an easy interface to assess and submit triggers via a mobile phone, which can receive SMS alerts and allow responding to emails away from a computer. \n\n\\item If {\\sc integ\\_time}$>$1.025\\,s then we presume that the GRB is long and we do not proceed with submitting a trigger to override the telescope. \n\\end{itemize}\n\nAfter the parser (or duty team member) has successfully identified the event as an SGRB, \nour algorithm then searches the ATCA calibrator database for a nearby and suitable phase calibrator.  \nIt then automatically builds a schedule file (we use the ATCA scheduler software {\\sc cabb-schedule-api})\\footnote{https:\/\/github.com\/ste616\/cabb-schedule-api} for a 12-hour observation of the GRB in the requested frequency band (for GRB triggering we currently use the 4\\,cm receiver), which has interleaved \nphase calibrator observations every 20 minutes. \nNote that the total exposure time is also limited by how far the GRB is above the horizon at the time of the trigger. \nThe schedule file and override request is then submitted to the observatory where it is assessed for submission to the observing queue by the ATCA back-end.\n\n\\subsection{Observatory back-end}\\label{sec:back-end}\n\nTime on the ATCA is scheduled into two 6-month long semesters, and the order of observations in each semester is set months in advance. This is done to allow the project investigators, who are also responsible for conducting the observations, to plan their activities. A rapid-response system is not easily compatible with this mode of operation.\n\nNevertheless, demand for the telescope to quickly respond to events has been steadily rising. In 2016, roughly 10\\% of telescope time was given to  \nNAPA (Non A-priori Assignable) or ToO (Target of Opportunity) projects, while in 2019 this figure had risen to 19\\%. \nFor a NAPA project, a science case is given to the time assignment committee (TAC), which ranks its significance against the other projects for that semester. Provided the science is considered compelling, these projects are allowed to displace time from other projects during the semester, with the philosophy being that were we to know during the scheduling process when an event would happen, a compelling project would have been scheduled to observe it.\n\nRapid-response NAPAs operate in the same way. A scientific justification must be supplied to the TAC, who must agree that rapid response is warranted. The observatory then supplies an authentication Javascript Web Token (JWT) to the project, and assists the investigators to test their automatic triggering system.\n\nA web service is provided so that the trigger to start observations can be sent from any internet-connected device. A Python library ({\\sc atca-rapid-resonse-api}) is also available to make it easier to send requests to this service.\\footnote{https:\/\/github.com\/ste616\/atca-rapid-response-api} All requests must contain a valid schedule file, and must nominate the target coordinates and a nearby phase calibrator.\n\nUpon receipt of a trigger, the web service tries to schedule the observation as soon as possible. If the source is above the horizon and the user-nominated minimum useful observing time can be obtained before the source sets, the current and subsequent observations can be displaced  \nand the system can start the observations within 2 seconds of the trigger's arrival. Within that time, emails are sent to the projects that will be displaced, and to the triggering team, describing the new order of the scheduling. The schedule is also altered as necessary to add a scan on a flux density calibrator at an opportune time, and potentially to shorten the observations to fit the available time. At all times, the emphasis is to move the telescope to the target coordinates as quickly as possible.\n\nThe service can also provide an immediate rejection should no suitable time be found for the observation. For example, if no available time can be found up to 100 hours in the future (generally because the request was made during a time when the array is shutdown for maintenance or participating in VLBI observing), the observations are rejected and the proposal team are notified. \nWhile no explicit limit is set for the source declination, sources too far north may not be available for the user-nominated minimum useful observing time, and will thus be rejected.\n\nIf the web service can schedule the observations, a separate service then takes over, and takes control of the observing control software. Some more checks are made to see if the array can be used for observing, and will delay the start of the observations if the weather conditions are unsuitable. This service also monitors the observations for interruptions due to weather, equipment failure and human intervention. Rudimentary responses are pre-programmed for any such eventuality. The service stops once the observations have finished, the target sets, or the observations are cancelled, whichever comes first. Control of the telescope then goes to the investigators whose project was scheduled to be running at this end time.\n\nA more complete guide to the operation of the rapid-response system is provided in the ATCA Users Guide.\\footnote{https:\/\/www.narrabri.atnf.csiro.au\/observing\/users\\_guide\/html\/atug.html}\n\n\\subsection{Triggering performance} \\label{sec:trig_per}\n\nSince the commencement of the program, we have worked with the observatory to improve the success of SGRB triggered observations with ATCA, which involved extensive system and software debugging. Many SGRBs were missed due to the telescope being in uninterruptible modes such as maintenance, reconfiguration, participating in VLBI or operating in an incompatible correlator mode (the latter has since been resolved). \n\nOur original override strategy involved triggering on all {\\em Swift}{} GRBs with {\\sc integ\\_time}$<1.025$\\,s as SGRBs have been detected with {\\sc integ\\_time} up to $1.024$\\,s. However, as mentioned in Section~\\ref{sec:front-end}, the majority of events within this timescale are LGRBs. \n{\\em Swift}{} data requires a human in the loop to classify the event as long or short, which is usually based on the duration and the hardness of the event (note that SGRBs often produce higher energy prompt emission than LGRBs) and are only published on the GCN Archive up to an hour post-burst \\citep[also note that the distinction between events with durations between $1-2$\\,s can be tenuous and has led to discussions regarding intermediate GRB classes; e.g.][]{mukherjee98,horvath98,huja09,deugartepostigo11}. \nThis original strategy therefore resulted in several false ATCA triggers, most of which were identified and cancelled before the telescope was overridden as there was additional lead time before the event in question had risen above the horizon. However, \nthere were a few instances where some data was collected on LGRBs. \nRecent edits to the VOEvent parsing of event timescales using the keyword {\\sc integ\\_time}, which are described in Section~\\ref{sec:front-end} have resulted in a significant reduction in ATCA triggers of LGRB contaminants. \n\nWhen ATCA receives a trigger of an event that is above the horizon, the main limitation to the response time is the telescope slew speed. On receiving the VOEvent via the parsing code, it takes $2-3$\\,s for the observation to be queued and the subsequent maximum observing time calculated. Following a {\\em Swift}{} alert on the long GRB 190519A \\citep{ukwatta19}, ATCA was on target and observing the event in 2\\,min and 39\\,s. Other response times range between $3-6$\\,min post-burst, which make the ATCA rapid-response system competitive with other triggering facilities such as AMI \\citep[e.g.][]{anderson18} yet is also based in the Southern Hemisphere, has more collecting area, a larger number of frequency bands, polarisation capabilities, and (in some configurations) better angular resolution.\n\n\\subsection{Short GRB experimental design}\n\nThe majority of GRBs detected by {\\em Swift}-BAT are LGRBs, with SGRBs (in this case events with $T_{90} \\leq 2$\\,s including those found in ground analysis) only accounting for $\\sim7-8$\\% \n(this is based on event numbers between 2017 and 2019 using the {\\em Swift}{} GRB Look-up Table,\\footnote{https:\/\/swift.gsfc.nasa.gov\/archive\/grb\\_table\/} where $T_{90}$ is the time between 5 and 95\\% of the fluence being emitted). We therefore expect $\\sim5-10$ SGRBs to be detected by {\\em Swift}{} per year, and therefore predict $\\lesssim2$ will be observable with ATCA (below a Declination cut-off of +15\\,deg) during an observing semester. \n\nOur rapid-response observations are performed using the $4$\\,cm receiver, which has dual 2\\,GHz windows that are usually centered at 5.5 and 9\\,GHz, which is the most sensitive ATCA band. This choice is based on several factors: the full-width half-maximum of the primary beam encompasses the initial {\\em Swift}{}-BAT positional uncertainty of newly detected GRBs \\citep[1-4 arcmin;][]{barthelmy05}, it is largely immune to atmospheric instabilities, and is less disrupted by RFI than other ATCA bands.\nIn addition, as synchrotron emission from GRB reverse and forward shocks peaks earlier and brighter with increasing frequency, the 4\\,cm band (5.5\/9 GHz) is optimal for ensuring the source will be bright but not peaking before the telescope is on-target.\n\nAs mentioned in Section~\\ref{sec:intro}, the radio afterglows from SGRBs are usually detected within 1\\,day post-burst \\citep[e.g.][]{fong15}, which strongly motivates our need for the ATCA rapid-response mode. The triggered observations are designed to  observe between $2-12$\\,h (depending on how long the source is above the horizon following the trigger). \nAs previous SGRB radio studies have shown that the radio afterglow has already switched-on within $4-16$\\,h post-burst \\citep[e.g.][]{anderson18}, a $\\leq12$\\,hr observation allows us to track the rapid rise in emission with a sensitivity of $\\sim60 \\, \\mu$Jy ($3\\sigma$) on one hour timescales.\\footnote{https:\/\/www.narrabri.atnf.csiro.au\/myatca\/interactive\\_senscalc.html} This means that any delays of $\\leq1$\\,hr related to waiting for the GRB classification does not affect the rapid-response science goal (see Section~\\ref{sec:front-end}). \nA $\\leq12$\\,hr track also ensures some periods of simultaneous {\\em Swift}{} X-ray Telescope \\citep[XRT, observing band between $0.3-10$\\,keV;][]{lien18} observations, \nwhich is essential for modelling the spectral energy distribution (SED), and for exploring the radio properties associated with the plateau phase (e.g. see our modelling in Section~\\ref{sec:mod}). \n\nFollowing the triggered, rapid-response observation, we also request three $\\sim4$\\,hr follow-up observations in the $4$\\,cm band to occur between $1-3$, $4-6$, and $8-12$\\,days post-burst, which can reach a sensitivity of $30\\,\\mu$Jy ($3\\sigma$). \nWhile 3 of the previous radio-detected SGRBs faded below detectability within 2 days post-burst, the other 2 were detected up to 10\\,days post-burst (see Figure~\\ref{fig:lc}), thus motivating this more long-term monitoring of any triggered candidate. \n\n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[width=0.49\\textwidth]{figures\/SGRBs_flux_lc_v3.pdf}\\label{fig:flux}\n  \\includegraphics[width=0.49\\textwidth]{figures\/SGRBs_lum_lc_v3.pdf}\\label{fig:lum}\n  \\caption{Radio light curves of SGRB radio detections ($1\\sigma$ error bars) and $3\\sigma$ upper-limits observed at frequencies between 6 and 10\\,GHz. \n  Left: Radio flux density vs days post-burst and Right: $k$-corrected spectral luminosity vs days post-burst in the rest-frame. The 9\\,GHz upper-limit of GRB 181123B is depicted as a large white triangle. For those GRBs without a known redshift we assume $z=0.5$. The $3\\sigma$ upper limits of those SGRBs that were observed but not detected in the radio band are depicted as grey triangles.  \n  References for all radio flux densities and redshifts for radio-detected SGRBs: \\citet{berger05}, \\citet{fox05}, \\citet{prochaska05}, \\citet{soderberg06}, \\citet{cucchiara13}, \\citet{deugartepostigo14}, \\citet{cucchiara14}, \\citet{chornock14}, \\citet{fong14}, \\citet{fong15}, \\citet{troja19}, \\citet{lamb19}, \\citet{paterson20}, \\citet{fong21}. All radio upper limits shown in grey were taken from \\citet[][see references therein]{fong15}.}\n  \\label{fig:lc}\n  \\end{center}\n\\end{figure*}\n\n\\section{ATCA observations of GRB 181123B}\n\n{\\em Swift}{}-BAT detected the short GRB 181123B at 05:33:03 UT (trigger=873186), which was rapidly detected in the X-rays by the {\\em Swift}{}-XRT  \nand localised to the position $\\alpha \\mathrm{(J2000.0)} =  12^{\\mathrm{h}}17^{\\mathrm{m}}28\\overset{\\mathrm{s}}{.}05$ and $\\delta (\\mathrm{J2000.0}) = +14^{\\circ}35'52\\overset{''}{.}4~$ with a 90\\% confidence of $1\\overset{''}{.}8$ \\citep{osborne18}. Further optical and near-infrared follow-up detected a source coincident with the {\\em Swift}{}-XRT position \\citep{fong18,paterson18a,paterson18b}, \nresulting in the identification of the host galaxy at redshift $z=1.754$ and the detection of the optical afterglow to GRB 181123B \\citep[$i=25.1$\\,mag at $9.1$\\,h post-burst;][]{paterson20}. This makes GRB 181123B one of only three SGRBs at $z>1.5$ \\citep{paterson20}. \n\nOn receiving the VOEvent trigger, ATCA was automatically scheduled to begin observations on 2018 Nov 23 at 18:07:24.9 UT (12.6\\,h post-burst) for 8.3\\,h  \\citep{anderson18gcn}, when the GRB had risen above the horizon (minimum elevation of 12\\,deg). On this date, ATCA was in the 6B array configuration, and the triggered observations were taken in the 4\\,cm band, with the dual 2\\,GHz bandwidth windows centered at 5.5 and 9\\,GHz. The observation pointing was at the initial BAT position, which was $1.2$\\,arcmin offset from the final {\\em Swift}{}-XRT position of GRB 181123B. \nNote that we requested no follow-up ATCA observations due to the imminent reconfiguration and correlator reprogramming, with many subsequent programmes having priority.\n\nThe ATCA rapid-response observation was reduced and analysed with the radio reduction software {\\sc MIRIAD} \\citep{sault95} using standard techniques. Flux and bandpass calibration were conducted using PKS 1934-638 and phase calibration with PKS 1222+216. Several rounds of phase and amplitude self calibration were also applied \\citep[this was possible due to the nearby bright field source FIRST~J121731.7+143953;][]{helfand15}. In order to obtain the most robust flux density upper limits at the position of the GRB, we used {\\sc mfclean} to create a clean model of the sources in the field (manually drawing clean boxes) and subtracted this model from the visibilities. A primary beam correction was then applied due to the 1.2\\,arcmin offset between the pointing centre and the best known GRB position from the {\\em Swift}{}-XRT. GRB 181123B was not detected, and the final $3\\sigma$ upper-limits can be found in Table~\\ref{tab:obs}. \n\nAs we know the precise location of GRB 181123B to within the ATCA beam, we also report the peak force-fitted flux density at both 5.5 and 9\\,GHz in Table~\\ref{tab:obs}.\nThese were calculated using the task {\\sc imfit} to force-fit a Gaussian to the beam that was fixed at the {\\em Swift}{}-XRT position of the GRB (errors are the $1\\sigma$ rms).\nThe advantage of quoting the force-fitted flux density over an upper-limit is that such a measurement also accounts for the presence of nearby sources, as well as variations in the noise across the image.\nThe data were also divided into 3\\,h and 1\\,h timescales and then re-imaged to search for evidence of emission that may have switched on nearer the end of the observation; however none was detected. \n\n\\begin{center}\n\\begin{table}\n\\caption{ATCA observations of GRB 181123B at 5.5 and 9 GHz, which began on 2018 Nov 23 at 18:07:24.9 UT (12.6\\,h post-burst) for 8.3\\,h.}\n\\label{tab:obs}\n\\begin{tabular}{lcc}\n\\\\\n\\hline\nFrequency & $3\\sigma$ Upper-limit & Forced-fit flux density\\\\\n(GHz) & ($\\mu$Jy\/beam) & ($\\mu$Jy\/beam)\\\\\n\\hline\n 5.5 & 34 & $7 \\pm 12$ \\\\\n 9.0 & 32 & $15 \\pm 11$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n\n\\section{Discussion}\n\nIn this section, we first demonstrate that our radio flux density limits for GRB 181123B are consistent and competitive with previous studies of the radio-detected SGRB population. This is followed by afterglow modelling to demonstrate the importance of obtaining early-time radio observations (regardless of whether there is a detection) to better constrain the properties of the blast wave.\n\nIn Figure~\\ref{fig:lc}, we show the light curves of SGRBs observed in the radio band between 6 and 10\\,GHz. The 8 radio-detected SGRBs are colour-coded with $3\\sigma$ upper-limits represented by triangles. The $3\\sigma$ upper limits of those SGRBs observed but not detected in the radio band have been plotted as grey triangles. The ATCA 9\\,GHz $3\\sigma$ upper-limit of GRB 181123B is shown as a large white triangle. In the left panel of Figure~\\ref{fig:lc}, we have plotted the observed radio flux density vs days post-burst, whereas in the right panel we have plotted the spectral luminosity vs days post-burst in the rest frame, assuming a redshift of $z=0.5$ \\citep{berger14} for those events with no known redshift.  \nWhen converting the flux ($F$) to luminosity ($L$), a $k$-correction was also applied such that $L=4 \\pi F d_{L}^{2} (1+z)^{\\alpha - \\beta -1}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$, where $d_{L}$ is the luminosity distance for the redshift $z$ \\citep[assuming $\\Lambda$CDM cosmology with $H_{0}=68$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$ and $\\Omega_m=0.3$;][]{planck16}, and $\\alpha$ and $\\beta$ are the temporal and spectral indices defined as $F \\propto t^{\\alpha}\\,\\nu^{\\beta}$ \\citep{bloom01}. We assume $\\alpha=0$ and $\\beta=1\/3$, which are appropriate for an optically thin, post-jet-break light curve \\citep[see][]{chandra12}.\n\n\nFrom Figure~\\ref{fig:lc} we can see that the ATCA flux limit for GRB 181123B is extremely competitive and consistent with the most constraining lower-limits. \nUsing formalism by \\citet{granot02}, \\citet{berger14} showed that if we assume fiducial parameters for SGRBs, along with typical microphysical parameters for LGRBs, the expected peak flux density at a redshift of $z=0.5$ is $F_{\\nu}\\sim40\\mu$\\,Jy at $\\sim10$\\,GHz for an ambient medium density of $n_{0}=0.1$\\,cm$^{-3}$. \nOur $3\\sigma$ sensitivity at 9\\,GHz was 32$\\mu$\\,Jy, and therefore sensitive enough to detect emission from a GRB with the above properties, however, it is important to note that some GRB microphysical and macrophysical parameters like the kinetic energy and the CSM density can vary by several orders of magnitude \\citep{granot14}.\n\nThe luminosity light curves in Figure~\\ref{fig:lc} show the $3\\sigma$ upper-limit for GRB 181123B at $\\sim1$\\,day post-burst (in the rest frame). \nGiven the very high redshift of GRB 181123B, even these sensitive ATCA observations would not have detected the radio counterpart to the seven SGRBs detected at early times (within a day post-burst in the rest frame) if they were placed at $z=1.754$.\nWe therefore cannot draw any further comparisons between the physical properties of the radio-detected GRB sample and GRB 181123B based on luminosity alone and require more detailed multi-wavelength light curve modelling (see Section~\\ref{sec:mod}).\n\n\\subsection{Modelling constraints} \\label{sec:mod}\n\nIn this section, we model the afterglow of GRB 181123B in order to explore how early-time ($<1$\\,day) radio observations of SGRBs can help to constrain the dynamical and microphysical parameters of such blast waves in the context of the fireball model. \nUsing the redshift derived from the identification of the host galaxy of GRB 181123B \\citep[$z=1.754$;][]{paterson20}, \nwe model the force-fitted flux density values at the {\\em Swift}{}-XRT position of the GRB from our ATCA observations together with the {\\em Swift}-XRT light curve \\citep{evans09,evans10}. \nWe have chosen to use the force-fitted flux measurements plus errors in our modelling as it allows us to assign a likelihood to a predicted model flux for a set of model parameters, which is not possible with an upper limit \n\\citep[for some examples of where radio force-fitted flux measurements are quoted and used in afterglow modelling see][]{galamawijers98,kulkarni99,vanderhorst11,vanderhorst15}. \n\nFor this modelling, we have chosen to only consider the forward-shock component to minimise complexity, particularly as we are dealing with a small number of data points. \nAs previously mentioned, \nthe reverse-shock could be dominant at early times ($\\lesssim1$\\,day) in the radio band as has been observed for some SGRBs \\citep[e.g.][]{soderberg06,lamb19,troja19}. \nGiven that the reverse-shock evolves to lower frequencies more rapidly than the forward-shock and we have no radio detection, our modelling depends primarily on the X-ray detections, which are always dominated by the forward-shock, thus motivating our model choice. \nOur afterglow fitting also does not rule out a reverse shock contribution.  \nWe therefore assume a spherical, relativistic, blast wave interacting with the circumburst medium and generating synchrotron emission. Since SGRBs are known to occur in homogeneous, low density environments \\citep[median densities of $n_{0}\\approx (3-15) \\times 10^{-3}$\\,cm$^{-3}$ with $\\approx80-95\\%$ of events being situated in environments of $n_{0}<1$\\,cm$^{-3}$;][]{fong15}, \nwe assume a constant density circumburst medium. \n\nWe use the \\texttt{boxfit} code to model the afterglow emission \\citep{vanEerten2012}. \\texttt{boxfit} makes use of pre-calculated hydrodynamics data to calculate the dynamics of the blast wave, and solves radiative transfer equations on the go. \nSince in this work we assume a spherical blast wave, we fix the opening angle ($\\theta_0$) to $\\pi \/ 2$. \nWe then use the C++ implementation of the \\texttt{MultiNest} nested sampling algorithm, which is a Bayesian inference tool, to determine the posterior distributions of the free parameters \\citep{feroz09}.  The free parameters of our model are defined as:\n\n\\begin{itemize}\n  \\item $E_{K, \\mathrm{iso}}$: Isotropic equivalent kinetic energy in units of erg.\n  \\item $n_0$: Circumburst medium number density in units of $\\mathrm{cm}^{-3}$.\n  \\item $p$: Power-law index of the accelerated electron distribution, such that $N(\\gamma) \\propto \\gamma^{-p}$, with some minimum Lorentz factor $\\gamma_{m}$ \\citep{wijers99}.\n  \\item $\\epsilon_B$: Fraction of thermal energy in the magnetic fields.\n  \\item $\\epsilon_e$: Fraction of thermal energy in the electrons.\n\\end{itemize}\n\nIn order to demonstrate how the inclusion of early-time radio data helps to further constrain \nthe dynamical and microphysical parameters (when combined with {\\em Swift}{}-XRT observations and regardless of whether or not there is a radio detection), we model the afterglow of GRB 181123B with and without the ATCA force-fitted fluxes and compare the posterior distributions of the free parameters. \nIn both fits, we use the same prior for the free parameters (Table~\\ref{tab:prior}) and the resulting best fit values in Table~\\ref{tab:post_param} are set with the lowest chi-squared value in the posterior. \n\nLight curves for the posterior predictive distribution when the ATCA force-fitted flux density values are included in the modelling, together with the best fit, can be found in Figure \\ref{fig:lc_post}. \nGiven that the modelling of the X-ray detections of GRB 181123B alone suggests an energetic solution, the inclusion of radio information aids to pull down the overall fit so that at both 5.5 and 9\\,GHz, the best fit light curves are clustered around the ATCA force-fitted flux densities. \nWhile the resulting model is consistent with the {\\em Swift}{} Ultraviolet\/Optical Telescope \\citep[UVOT;][]{roming05} upper-limits \\citep{oates18}, it over-predicts the Galactic extinction corrected $i$-band flux reported by \\citep{paterson20} by a factor of $\\sim3$ or 1.2 magnitudes. \nAt this high redshift, an $i$-band detection indicates the afterglow emission was produced at ultraviolet wavelengths in the rest frame, and would therefore be quite prone to extinction by dust.\nGiven our model does not consider extinction, intrinsic or otherwise, this over-prediction may therefore not be unreasonable. \nHowever, our $i$-band prediction is much higher than the host optical extinction calculated by \\citet{paterson20} from photometric observations ($A_{V}$=0.23) or calculated from their observed excess hydrogen column density ($N_{H}$; derived from X-ray afterglow spectral modelling), which is known to scale to optical extinction \\citep{guver09}, predicting $A_{V}$=0.38.\nThere are also other potential sources of optical and infrared emission from SGRBs such as a kilonova from r-process radiative decay \\citep[e.g.][]{metzger10}, which our model does not include. However, such emission usually does not dominate over the afterglow until $>1$\\,d post-burst \\citep[e.g.][]{tanvir13}. \n\nAs can be seen in Table~\\ref{tab:post_param}, the inclusion of the ATCA force-fitted fluxes in our modelling allows for much better constraints to be placed on $\\epsilon_e$ (see also Figure~\\ref{fig:marge_comp}, which shows a comparison between the marginal distributions of the parameters for both cases - modelling with and without the ATCA data). \nThe rest of the parameters are consistent between both modelling experiments but the $E_{K,\\mathrm{iso}}$ is on the brighter end of known SGRBs \\citep{fong15}. \nOur findings are consistent with those by \\cite{beniamini17}, who have shown that the flux density and time of the GRB radio light curve peak can be used to particularly constrain $\\epsilon_e$. We also note that our constraint on $\\epsilon_e$ is also consistent (within the 95\\% credible interval) with the distribution of $\\epsilon_e$ ($0.13-0.15$) found through the analysis of 36 GRB radio afterglows performed by \\citet{beniamini17}.\nThe predicted radio peak also suggests that at later times ($\\gtrsim3-4$\\,days post-burst), the forward shock radio emission from GRB 181123B may have been detectable at 5.5 and 9\\,GHz with $\\geq4$\\,hr ATCA integrations (see Figure~\\ref{fig:lc_post}). \nTherefore, the inclusion of early-time radio data in GRB afterglow modelling (regardless of whether or not there is a detection), \ntogether with an X-ray light curve, allows us to predict the forward shock peak radio flux density, \nthus constraining the fraction of shock energy in the relativistic electrons  ($\\epsilon_e$). \n\\citet{paterson20} also derived these same afterglow parameters for GRB 181123B but assumed fixed values of $\\epsilon_e=0.1$ and $\\epsilon_B=0.1$ or 0.01. While our parameters are far less constrained, our values for $E_{K,iso}$ and $n_{0}$ (as well as $\\epsilon_e$ and $\\epsilon_B$)  are consistent with \\citet{paterson20} within the 95\\% credible intervals. However, our value range for $p$ was higher and did not overlap with the range derived by \\citet{paterson20}. Note that our value range for $p$ is more consistent with those calculated for radio-detected SGRBs (see Section~\\ref{sec:comp}).\n\n\\begin{table}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Assumed priors for the free parameters for all modelling efforts.}\n    \\begin{tabular}{l|r}\n    \\hline\n    \\multicolumn{1}{l}{Parameter range} & \\multicolumn{1}{r}{Prior distribution} \\\\\n    \\hline\n    $10^{49}<E_{K, \\mathrm{iso}}~\\mathrm{(erg)}<10^{54}$ & log-uniform\\\\\n    $10^{-4}<n_{0}~\\mathrm{(cm^{-3})}<10$ &log-uniform\\\\\n    $2.0<p<3.5$& uniform\\\\\n    $10^{-7}<\\epsilon_B<0.50$& log-uniform\\\\\n    $10^{-4}<\\epsilon_e<0.50$& log-uniform\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:prior}\n\\end{table}\n\n\\begin{figure}\n  \\includegraphics[width=.45\\textwidth]{figures\/lc_post-z.pdf}\n  \\caption{Fit result for the afterglow light curves of GRB 181123B at ATCA observing frequencies 5.5\\,GHz (top panel) and 9.0\\,GHz (middle panel), and with the {\\em Swift}{}-XRT ($0.3-10$\\,keV; bottom panel) when the ATCA data are included (in this case the force-fitted flux densities; red data point and error bar plotted on the 5.5 and 9.0\\,GHz light curves). The plotted values in the {\\em Swift}{}-XRT light curve (also red data points) were downloaded via the {\\em Swift}{} Burst Analyser \\citep{evans10}. For each of the three frequency bands, 200 light curves are drawn by sampling the inferred posterior distribution of the parameters. The solid line represents the best fit model. The horizontal dashed lines show the 3-$\\sigma$ detection limit for various ATCA integration times.\n  }\n  \\label{fig:lc_post}\n\\end{figure}\n\n\\begin{table*}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Fit results for GRB 181123B for when the ATCA force-fitted data are excluded, included or the ATCA force-fitted mean is lowered by an order of magnitude. The errors on the parameters represent the 95\\% credible interval.\n    It can be seen that when the ATCA data are included, $\\epsilon_e$ is better constrained. \n    }\n    \\begin{tabular}{l|r|r|r}\n    \\hline\n    Parameter name &  \\multicolumn{1}{c}{ATCA data excluded} & \\multicolumn{1}{c}{ATCA data included} &  \\multicolumn{1}{c}{Lower ATCA forced fit} \\\\\n    \\hline\n    {\\boldmath$\\log_{10}E_{K, \\mathrm{iso}}$} & $52.4^{+1.4}_{-1.6}        $ & $52.0^{+1.5}_{-1.2}        $ & $51.5^{+1.1}_{-0.85}       $\\\\\n    \n    {\\boldmath$\\log_{10}n_{0}$} & $-0.4^{+1.4}_{-1.5}        $ & $-0.5^{+1.4}_{-1.4}        $ & $-1.1^{+1.4}_{-1.3}        $\\\\\n    \n    {\\boldmath$p$} & $2.92^{+0.42}_{-0.37}      $ & $2.90^{+0.42}_{-0.38}      $ & $2.97^{+0.38}_{-0.39}      $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_B$} & $-2.9^{+2.5}_{-3.2}        $ & $-3.0^{+2.7}_{-3.5}        $ & $-2.5^{+2.1}_{-2.3}        $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_e$} & $-1.13^{+0.82}_{-1.2}      $ & $-0.75^{+0.39}_{-0.40}     $ & $-0.60^{+0.30}_{-0.35}     $\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:post_param}\n\\end{table*}\n\n\\subsubsection{Robustness of the results for more complicated models} \\label{sec:robust}\n\nThe aim of the modelling analysis presented in Section~\\ref{sec:mod} is to demonstrate how early-time radio data (even a non-detection) can help to constrain physical parameters in the framework of the fireball model.\nHowever, the presence of a plateau feature in many GRB X-ray light curves indicates that energy injection and more complex emission mechanisms are at play beyond a simple forward and reverse shock. \nOne of the main interpretations of X-ray plateaus observed from SGRBs \nis likely an energy injection signature from a potentially short-lived, supramassive, highly magnetised, rapidly rotating neutron star remnant, often referred to as a `magnetar' \\citep[e.g.][]{zhang01,yu07,rowlinson13}. In fact, \\citet{gompertz15} has performed broad-band modelling of SGRBs that includes energy injection from the spin-down of such a magnetar. Alternatively, \\cite{leventis14} were able to demonstrate that X-ray plateaus could be explained by the combined emission from the reverse and forward shock, provided that the blast wave is in the thick shell regime, and such a reverse shock would also lead to an additional emission component in radio and optical wavelengths.\nIn the case of GRB 181123B, the X-ray light curve of the afterglow shows evidence of a plateau phase at early times that \\citet{rowlinson20pp} have interpreted as energy injection from an unstable magnetar that collapsed a few hundred seconds following its formation. \nWhile many models have been proposed in the literature to describe the X-ray light curve behaviour of GRBs, our limited radio data-set means that an exhaustive analysis of these complex models is beyond the scope of this paper. \nNonetheless, it is worth exploring whether the constraints we have derived from our simple forward shock model are still meaningful if we introduce additional free parameters. \n\nIn the following we investigate how the inclusion of the ATCA data affects the posterior of modelling efforts that also include energy injection. We therefore \nincorporate energy injection into \\texttt{boxfit}, \nwhich is modelled by varying the isotropic equivalent kinetic energy ($E_{K, \\mathrm{iso}}$) as a power-law in time. In this case, $E_{K, \\mathrm{iso}}$ is described as:\n\n\\begin{equation}\n    E = \n    \\begin{cases} \n      E_0 & t \\leq t_{\\mathrm{inj}} \\\\\n      E_0 (t \/ t_{\\mathrm{inj}})^\\alpha &  t_{\\mathrm{inj}} < t \\leq t_{\\mathrm{inj}} + dt_{\\mathrm{inj}} \\\\\n      E_0 (1 + dt_{\\mathrm{inj}} \/ t_{\\mathrm{inj}})^\\alpha & t_{\\mathrm{inj}} + dt_{\\mathrm{inj}} < t \n   \\end{cases}\n\\end{equation}\n\n\\noindent where the three additional parameters are defined as:\n\n\\begin{itemize}\n  \\item $t_{\\mathrm{inj}}$: Start time of the energy injection in seconds (s).\n  \\item $dt_{\\mathrm{inj}}$: Duration of energy injection in seconds (s).\n  \\item $\\alpha$: Power-law index of the energy injection.\n\\end{itemize}\n\n\\noindent The assumed prior distributions for the energy injection parameters can be seen in Table~\\ref{tab:prior_ei}. We use the same priors as before for all other burst parameters (Table~\\ref{tab:prior}).\n\nThe resulting best-fit parameters for the afterglow modelling of GRB 181123B that includes energy injection for the cases when the ATCA force-fitted flux densities are included and excluded can be found in Table~\\ref{tab:post_param_ei}, with the resulting light curves for the posterior predictive distribution, together with the best fit model plotted in Figure \\ref{fig:lc_post_ei}. \nDue to the limited multi-wavelength coverage of the afterglow, we are not able to place tight constraints on the energy injection parameters. \nHowever, note that $\\epsilon_e$ continues to be well constrained when ATCA data are included, even with a more complex model with additional parameters (see Figure~\\ref{fig:marge_comp_ei}, which shows the marginal distributions of the parameters for the energy injection cases when ATCA data are included and excluded). \nIn addition, Figure~\\ref{fig:marge_comp_classic_ei} compares the obtained marginalized distribution for the GRB parameters (omitting the energy injection parameters) for the two modelling cases that include the ATCA data: with and without including energy injection. It can be seen that the resulting parameter values common between both fits are consistent within the 95\\% credible intervals despite different model complexities.\nHowever, while the model is consistent with the {\\em Swift}{}-UVOT upper-limits, it over-predicts the $i$-band detection \\citep{paterson20}, which demonstrates the limitations of our modelling.\n\nThe modelling of GRB 181123B shows that early-time radio observations, \nregardless of whether they are detections or non-detections,\nare able to constrain the fraction of thermal energy in the accelerated electrons, $\\epsilon_e$, beyond what is possible with just the {\\em Swift}-XRT X-ray light curve data. \nOur modelling also predicted that observations at later times (1-10 days) may have resulted in a detection of the forward shock, which would further constrain the GRB parameters.\n\n\\begin{table}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Assumed priors for the energy injection parameters for all modelling efforts.}\n    \\begin{tabular}{l|r}\n    \\hline\n    \\multicolumn{1}{l}{Parameter range} & \\multicolumn{1}{r}{Prior distribution} \\\\\n    \\hline\n    $8.64<t_{\\mathrm{inj}}~\\mathrm{(s)}<4.32 \\times 10^5$ & log-uniform\\\\\n    \n    $8.64<dt_{\\mathrm{inj}}~\\mathrm{(s)}<4.32 \\times 10^5$ & log-uniform\\\\\n    \n    $0.0<\\alpha<2.0$& uniform \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:prior_ei}\n\\end{table}\n\n\\begin{table}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Fit results for GRB 181123B for when the ATCA force-fitted data are excluded and included, with \n    energy injection also included in our modelling. The errors on the parameters represent the 95\\% credible interval. It can be seen that when the ATCA data are included, $\\epsilon_e$ is better constrained. \n    }\n    \\begin{tabular}{l|r|r}\n    \\hline\n    Parameter name &  \\multicolumn{1}{c}{ATCA data excluded} & \\multicolumn{1}{c}{ATCA data included} \\\\\n    \\hline\n    {\\boldmath$\\log_{10}E_{K, \\mathrm{iso}}$} & $51.8^{+1.9}_{-1.3}        $   & $51.4^{+1.7}_{-0.91}       $\\\\\n    \n    {\\boldmath$\\log_{10}n_{0}$}    & $-0.7^{+1.6}_{-1.7}        $ & $-0.9^{+1.7}_{-1.8}        $ \\\\\n    \n    {\\boldmath$p              $} &$3.08^{+0.42}_{-0.47}      $ & $3.16^{+0.35}_{-0.50}      $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_{B}$} &$-1.9^{+1.6}_{-3.5}        $ & $-2.1^{+1.9}_{-4.0}        $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_{e}$} &$-1.01^{+0.70}_{-1.2}      $ & $-0.66^{+0.35}_{-0.39}     $\\\\\n    \n    {\\boldmath$\\log_{10}t_{\\mathrm{inj}}$ (days)} &$-1.23^{+1.7}_{-0.91}      $ & $-1.32^{+1.8}_{-0.84}      $\\\\\n    \n    {\\boldmath$\\log_{10}dt_{\\mathrm{inj}}$ (days)} & $-1.2^{+1.8}_{-2.3}        $ & $-1.1^{+1.8}_{-2.3}        $\\\\\n    \n    {\\boldmath$\\alpha         $} &\n    $0.86^{+0.94}_{-0.77}      $ & $0.88^{+0.92}_{-0.76}      $\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:post_param_ei}\n\\end{table}\n\n\\begin{figure}\n  \\includegraphics[width=.45\\textwidth]{figures\/lc_post-ei-z.pdf}\n  \\caption{As for Figure~\\ref{fig:lc_post} but with energy injection incorporated into our modelling. \n  } \n  \\label{fig:lc_post_ei}\n\\end{figure}\n\nTo test how the dependencies between the parameters in our model are affected by \nthe mean of the forced fitted values, we ran an additional fit using \\texttt{boxfit} (without energy injection) where we lowered the mean of the ATCA force-fitted flux density values by an order of magnitude. Table~\\ref{tab:post_param} shows the resulting inferred parameters with the best fit light curves shown in Figure~\\ref{fig:lc_post_mean}. \nThe main effect of lowering the mean value of the force-fitted ATCA flux densities on the parameters was to decrease the mean value of the circumburst medium density ($n_0$) and increase the fraction of thermal energy in the  magnetic fields ($\\epsilon_B$) by a similar amount (factor of $\\sim3-4$) when compared to the results from fitting the original force-fitted ATCA flux densities.\nHowever, note that all parameters presented in Table~\\ref{tab:post_param} that were derived from including ATCA force-fitted flux densities in our  modelling still agree within the 95\\% credible intervals of the modelling performed without the ATCA information.\nThe reduction in the ATCA force-fitted mean flux densities also indicate that the predicted peak forward shock emission in the radio band would be delayed and also drop below the ATCA $3\\sigma$ detection limits (see Figure~\\ref{fig:lc_post_mean}) when compared to the fit performed with the measured force-fitted flux densities shown in Figure~\\ref{fig:lc_post}.\n\n\\begin{figure}\n  \\includegraphics[width=.45\\textwidth]{figures\/lc_post-lower-z.pdf}\n  \\caption{As for Figure~\\ref{fig:lc_post} but with the ATCA forced-fitted flux densities lowered by an order of magnitude. \n  }\n  \\label{fig:lc_post_mean}\n\\end{figure}\n\n\nOverall, our analysis demonstrates the importance of quoting force-fitted flux density values for radio transients over just reporting $3\\sigma$ upper-limits, which is traditional in most fields of astrophysics. If only upper-limits are reported then it is not always possible to incorporate this information into some modelling analyses (e.g. like our GRB afterglow modelling), which means we are throwing away important data that could further constrain the physics of an event or source. \nFinally, the inclusion of early-time radio force-fitted flux densities allows us to make predictions about the time and brightness of the forward shock peak in the radio band, which can inform late-time radio follow-up strategies. \n\n\\subsection{Comparisons of GRB 181123B to radio-detected SGRBs}\\label{sec:comp}\n\nWe now compare our parameter constraints on the micro- and macro-physical properties of GRB 181123B resulting from our modelling to those obtained for other radio-detected GRBs. For this comparison, we only focus on the parameters derived from our forward-shock modelling using \\texttt{boxfit} with the inclusion of the ATCA data (see Table~\\ref{tab:post_param}). \nSix of the eight radio-detected SGRBs have constraints on the same parameters (e.g. Table~\\ref{tab:prior}) through afterglow modelling \n\\citep[GRB 050724A, 051221A, 130603B, 140903A, 160821B, GRB 200522A;][]{berger05,soderberg06,fong14,pandey19,troja16,zhang17,troja19,lamb19,fong21}. \nThe advantage of these modelling efforts was that these events had extensive multi-wavelength data (radio, infrared, optical, X-ray) so the afterglow analysis \nled to stringent constraints on these parameters and in many cases, an estimate of \nthe gamma-ray jet opening angle, which has important implications for rate calculations. \nWhile the parameters we derived for GRB 181123B were far less constrained, our 95\\% credible intervals agree with those values derived for the six  SGRBs mentioned above, but the upper limits in our prediction for both $n_0$ and $p$ are much higher overall. \nIn fact, our derived accelerated electron distribution is steeper than that usually expected for GRBs \\textbf{($p=2.90^{+0.42}_{-0.38}$)} but not unreasonably so.\nOverall, as previously mentioned, the inclusion of ATCA force-fitted flux densities within 1 day post-burst have allowed us to place \nreasonable constraints on $\\epsilon_{e}$. \n\n\\section{Conclusions}\n\nIn this paper, we introduce the ATCA rapid-response observing mode by presenting results from the first successful SGRB trigger on GRB 181123B. This new mode of operations allows the telescope to automatically and rapidly respond to transient alerts broadcast via VOEvents, causing the active observing programme to be interrupted to allow for time-critical observations of transient phenomena. Successful triggers on LGRBs (see Section~\\ref{sec:trig_per}) have demonstrated that if the source is above the horizon, the ATCA can be on target and observing the event as fast as 3 minutes post-burst, allowing us to probe this early-time radio regime over a wide range of frequencies ($1.1-25$\\,GHz) with full polarisation, and in a variety of array configurations.\n\nThe ATCA rapid-response observations of GRB 181123B began 12.6\\,hr post-burst, as soon as the target had risen above the horizon, collecting 8.3\\,h of data at 5.5 and 9\\,GHz. While no radio emission was detected from GRB 181123B, we quote force-fitted flux densities, which enabled more constraining GRB afterglow modelling to be performed then would usually be possible with just X-ray data from {\\em Swift}{}-XRT. The addition of early-time radio data in the modelling allowed us to obtain more stringent constraints on the fraction of thermal energy in the electrons behind the shock wave ($\\epsilon_e$), which in turn allowed us to predict the peak in the forward shock radio afterglow emission around $\\sim10$\\,days post-burst.  \nThis modelling indicates that $\\gtrsim3-4$\\,days post-burst, the radio afterglow of GRB 181123B may have been detectable with a $\\geq4$\\,hr ATCA integration. \n\nOur results demonstrate the importance of including early-time radio observations in afterglow modelling efforts for constraining GRB blast wave properties and predicting the late-time peak in the radio forward shock, regardless of whether or not there is a detection, provided that force-fitted flux densities are quoted rather than upper-limits in the case of a non-detection. \n\nThis project also demonstrates the importance of implementing rapid-response observing systems on radio telescopes to probe a new parameter space in transient science. \nEarly-time radio observations of SGRBs can allow us to distinguish between different sources of early-time synchrotron afterglow emission \\citep[e.g.][]{kyutoku14} and even detect prompt and persistent coherent signals predicted to be produced during the compact merger and from the merger remnant \\citep[e.g.][]{rowlinsonanderson19}.\nDetections from rapid-response radio observations will provide crucial insight into the radio brightness and timescales we might expect from aLIGO\/Virgo-detected merging BNSs or NS-BH systems, which will aid in our search for electromagnetic counterparts in the large GW positional error regions.\nOther science cases include \nLGRBs, which may show bright, early-time radio emission from the reverse shock \\citep[e.g.][]{anderson14} and flare stars, which have shown simultaneous high-energy and radio flaring behaviour \\citep[e.g.][]{fender15}. \n\n\nFinally, our efforts running rapid-response programs on ATCA act as an excellent test for transient observing strategies with the SKA. \nIt is only through the utilisation of a rapid-response system that we can exploit the SKA to study early-time BNS and BH-NS merger physics down to sub-micro-Jansky levels. \n\n\\section*{acknowledgements}\nWe thank the referee for their careful reading of the manuscript and recommendations.\nGEA is the recipient of an Australian Research Council Discovery Early Career Researcher Award (project number DE180100346) and JCAM-J is the recipient of Australian Research Council Future Fellowship (project number FT140101082) funded by the Australian Government. \n\nThe Australia Telescope Compact Array (ATCA) is part of the Australia Telescope National Facility, which is funded by the Australian Government for operation as a National Facility managed by CSIRO. \nThis work made use of data supplied by the UK {\\it Swift} Science Data Centre at the University of Leicester and the {\\it Swift} satellite. {\\it Swift}, launched in November 2004, is a NASA mission in partnership with the Italian Space Agency and the UK Space Agency. {\\it Swift} is managed by NASA Goddard. Penn State University controls science and flight operations from the Mission Operations Center in University Park, Pennsylvania. Los Alamos National Laboratory provides gamma-ray imaging analysis.\n\nThe ATCA rapid-response front-end software makes use of {\\sc comet} \\citep{swinbank14} and\n{\\sc voevent-parse} \\citep{staley16pp} to process the incoming VOEvents.\nBoth the front-end ({\\sc vo\\_atca})\\footnote{https:\/\/github.com\/mebell\/vo\\_atca} and back-end ({\\sc atca-rapid-resonse-api})\\footnote{https:\/\/github.com\/ste616\/atca-rapid-response-api} software for ATCA rapid-response triggering rely on the {\\sc Astropy}, a community-developed core Python package for Astronomy \\citep{TheAstropyCollaboration2013,TheAstropyCollaboration2018}, {\\sc numpy} \\citep{vanderWalt_numpy_2011} and {\\sc scipy} \\citep{Jones_scipy_2001} python modules. This research also makes use of {\\sc matplotlib} \\citep{hunter07}. \nThis research has made use of NASA's Astrophysics Data System. \nThis research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory.\nThis research has made use of the VizieR catalogue access tool \\citep{ochsenbein00} and the SIMBAD database \\citep{wenger00}, operated at CDS, Strasbourg, France.\n\n\\section*{Data Availability Statement}\nThe unprocessed ATCA visibility datasets are public and can be accessed via the ATCA online archive (\\href{https:\/\/atoa.atnf.csiro.au\/}{https:\/\/atoa.atnf.csiro.au\/}) under project code C3204 and PI G Anderson.\nThe reduced images of these observations have been uploaded to Zenodo \\citep{anderson21zen}.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}