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https://ja.wikipedia.org/wiki/%E5%80%8B%E4%BA%BA%E6%83%85%E5%A0%B1%E3%81%AE%E4%BF%9D%E8%AD%B7%E3%81%AB%E9%96%A2%E3%81%99%E3%82%8B%E6%B3%95%E5%BE%8B
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6,649 |
BC
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BCãBcãbc
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https://ja.wikipedia.org/wiki/BC
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6,650 |
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JIS X 0208
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çŽ ç²åç©çåŠã§ã¯ãçŽ ç²åã®è³ªéã鿢ãšãã«ã®ãŒå€(eV)ãçšã㊠eV/c ãšããåäœè¡šèšããªããããããã¯ãã鿢ãšãã«ã®ãŒãå
éã§é€ç®ããå€ããšããäžçš®ã®èšæ³ã§ããããé€ç®ããå€ãã§ã¯ãªãããšã«æ³šæãäŸãšããŠãé»åã®éæ¢ãšãã«ã®ãŒã¯ 511 keV ã§ãé»åã®éæ¢è³ªé㯠511 keV/c ãšè¡šãããã
| false | false | false |
https://ja.wikipedia.org/wiki/%E8%B3%AA%E9%87%8F
|
6,685 |
éåé
|
éåé(ãè±: momentum)ãšã¯ãåççã«ã¯ç©äœã®éåã®ç¶æ
ã衚ãç©çéã§ã質éãšé床ã®ç©ãšããŠå®çŸ©ãããããã®æå³ã®éåéã¯åŸè¿°ããäžè¬åãããéåéãšåºå¥ããŠãéååŠçéåé(ãããã¯åçéåé)ãšåŒã°ããããŸããè§éåéãšããéåéãšã¯ç°ãªãéãšå¯Ÿæ¯ããäžã§ãç·åéåéãªã©ãšåŒã°ããããšãããã
æ¥åžžç掻ã«ãããŠãç©äœã®æã€éåéã¯ãåããŠããç©äœã®æ¢ãã«ãããšããŠäœæããããã€ãŸããéããŠéãç©äœã»ã©éåéã倧ããã鿢ãããã®ã«å€§ããªåç©ãå¿
èŠã«ãªãã
ã¢ã€ã¶ãã¯ã»ãã¥ãŒãã³ã¯éåéã®æéçå€åãšåã®é¢ä¿ãéåã®ç¬¬2æ³åãšããŠæç€ºããã
è§£æååŠã§ã¯ãäžè¿°ã®å®çŸ©ããé¢ããéåéã¯äžè¬å座æšãšãªã€ã©ãŒ=ã©ã°ã©ã³ãžã¥æ¹çšåŒãéããŠäžããããããã®éåéã¯äžè¬å座æšç³»ã«ãããäžè¬åé床ã®å¯Ÿå¿ç©ãšããŠãäžè¬åéåéãšåŒã°ããã
ç¹ã«ããã«ãã³åœ¢åŒã®è§£æååŠã«ãããŠã¯ãæ£æºæ¹çšåŒãéããŠäžããããæ£æºå€æ°ã®äžæ¹ã座æšãšåŒã³ä»æ¹ãéåéãšåŒã¶ããã®æå³ã®éåéã¯ãä»ãšåºå¥ããŠãæ£æºéåéãšåŒã°ããããŸããæ£æºéåéã¯ãæ£æºæ¹çšåŒã«ãããŠåº§æšã®å¯Ÿãšãªããšããæå³ã§ãå
±åœ¹éåéãšåŒã°ãããéåéã¯ãããã«ãã³åœ¢åŒã®ååŠã§ã¯ãé床ãããåºæ¬çãªéã§ãããããã«ãã³åœ¢åŒã§èšè¿°ãããéåžžã®éåååŠã«ãããŠãéèŠãªåœ¹å²ãæããã
å
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§)ãé»ç£å Žäžãéåããé»åã«å¯ŸããŠã¯ããŒã¬ã³ãåãåããããã®ããŒã¬ã³ãåã«å¯Ÿå¿ããäžè¬åãããããã³ã·ã£ã«ãšãã«ã®ãŒã«ã¯é»åã®é床ã®é
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±åœ¹éåéã¯ã©ã°ã©ã³ãžã¢ã³ã®ããã³ã·ã£ã«é
ã«äŸåãã圢ã«ãªãããã®ãšãå
±åœ¹éåéãšéååŠçéåéã¯äžèŽããªãããŸããé»ç£å Žäžã®é»åã®éåãèšè¿°ããå€å
žçããã«ããã¢ã³ã§ã¯ãå
±åœ¹éåéã®éšåããã¹ãŠå
±åœ¹éåéãããã¯ãã«ããã³ã·ã£ã«ã®å¯äžãåŒãããã®ã«çœ®ãæããã
éåéã¯ãéåã®ç¬¬2æ³åã«ãããŠããã®æéã«å¯Ÿããå€åã®å²åãåãšçããéãšããŠå°å
¥ãããã
ã€ãŸããéåé p ã¯ãã¥ãŒãã³ã®éåæ¹çšåŒã
d p d t = F ( t ) {\displaystyle {\frac {d{\boldsymbol {p}}}{dt}}={\boldsymbol {F}}(t)}
ãæºãããå F ã¯ãã¯ãã«éã§ãããéåéããŸããã¯ãã«éã§ããããŸããå®çŸ©ããæãããªããã«ãéåéã¯æå» t ã®é¢æ°ãšããŠè¡šãããéã§ããã
質ç¹ã®éåéã¯ã質ç¹ã®éåºŠã«æ¯äŸããã質ç¹ã®éåéã¯ã質ç¹ã®é床ã v ãšè¡šããæ¯äŸä¿æ°ã m ãšãããšã
p = m v {\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}}
ã§äžããããã
ããã§å°å
¥ãããæ¯äŸä¿æ° m ã¯æ
£æ§è³ªé (inertial mass) ãšåŒã°ã質ç¹ã®é床ã®å€åãé£ãã衚ãã
éåéã®å€åéã¯åç©ã§ããããéåã®éãæ
£æ§è³ªéãäžå®ã§ãããšããã°ãé床ã®å€åéã¯åç©ãæ
£æ§è³ªéã§å²ã£ããã®ãšãªããåŸã£ãŠãåã倧ããã®åç©ã«å¯ŸããŠã¯ãæ
£æ§è³ªéã倧ããã»ã©é床ã®å€åã¯å°ãããã®ãšãªãã
æå» t0 ãã t1 ã®éã®ç©äœã®éåéã®å€åéã
Î p ( t 0 ; t 1 ) := p ( t 1 ) â p ( t 0 ) = â« t 0 t 1 d p d t d t {\displaystyle \Delta {\boldsymbol {p}}(t_{0};t_{1}):={\boldsymbol {p}}(t_{1})-{\boldsymbol {p}}(t_{0})=\int _{t_{0}}^{t_{1}}{\frac {d{\boldsymbol {p}}}{dt}}\,dt}
ãšããã ãã®ç©äœãæå» t ã«å F(t) ãåããªããéåããŠãããšãããšãéåæ¹çšåŒããéåéã®æéå€åç dp/dt ã¯å F(t) ã«çãããããéåéã®å€åé Îp ã¯
Î p ( t 0 ; t 1 ) = â« t 0 t 1 F ( t ) d t =: I {\displaystyle \Delta {\boldsymbol {p}}(t_{0};t_{1})=\int _{t_{0}}^{t_{1}}{\boldsymbol {F}}(t)\,dt=:{\boldsymbol {I}}}
ãšãªãå F(t) ãæå» t0 ãã t1 ãŸã§ç©åãããã®ã«çããã ãã®åã®æéç©å I ã¯åç©(impulse)ãšåŒã°ããéåéã®å€åéã«çããã
æé Ît=t1ât0 ã§ç©äœãåããåã®æéçãªå¹³åã¯
F ave ( t 0 ; t 1 ) := 1 Î t â« t 0 t 1 F ( t ) d t {\displaystyle {\boldsymbol {F}}_{\text{ave}}(t_{0};t_{1}):={\frac {1}{\Delta t}}\int _{t_{0}}^{t_{1}}{\boldsymbol {F}}(t)\,dt}
ã§å®çŸ©ããããåã®æéå¹³å Fave ãçšããã°åç©ã¯
I = Î p ( t 0 ; t 1 ) = F ave ( t 0 ; t 1 ) Î t {\displaystyle {\boldsymbol {I}}=\Delta {\boldsymbol {p}}(t_{0};t_{1})={\boldsymbol {F}}_{\text{ave}}(t_{0};t_{1})\Delta t}
ãšãªãã ç¹ã«æé Ît ãå
åã«çããåãäžå®ã§ãããšèŠãªããå Žåã«ã¯ãåç©ã¯åã«åãšæéã®ç©
I = Î p = F Î t {\displaystyle {\boldsymbol {I}}=\Delta {\boldsymbol {p}}={\boldsymbol {F}}\,\Delta t}
ãšããŠè¡šãããšãã§ããã
ã€ãŸããç©äœã«äžå®ã®åãå ããŠãç©äœã®éåéã®å€åã倧ããããã«ã¯ãåãäœçšããæéãé·ãããã°ãããéã«ã倧ããªåãå ãããšããŠãããããããçæéã®ãã®ã§ããã°ãç©äœã«äžããåç©ã¯å°ãããªãã
éåéã¯å æ³çãªéã§ãããç³»ã®å
šéåéã¯éšåã®éåéã®åã§è¡šãããã
質ç¹ç³»ã®å
šéåé P ã¯ãè³ªç¹ i = 1, 2, 3,... ã®éåé pi = mivi = midri/dt ãšããã°
P ( t ) = â i p i ( t ) = â i m i d r i d t = d d t ( â i m i r i ( t ) ) {\displaystyle {\boldsymbol {P}}(t)=\sum _{i}{\boldsymbol {p}}_{i}(t)=\sum _{i}m_{i}\,{\frac {d{\boldsymbol {r}}_{i}}{dt}}={\frac {d}{dt}}\left(\sum _{i}m_{i}\,{\boldsymbol {r}}_{i}(t)\right)}
ãšãªãã ããã§è³ªç¹ç³»ã®å
šè³ªé M ãšè³ªéäžå¿ rg ã
M = â i m i , r g ( t ) = 1 M â i m i r i ( t ) {\displaystyle M=\sum _{i}m_{i},~{\boldsymbol {r}}_{g}(t)={\frac {1}{M}}\sum _{i}m_{i}\,{\boldsymbol {r}}_{i}(t)}
ã«ããå°å
¥ããã°
P ( t ) = M d r g d t {\displaystyle {\boldsymbol {P}}(t)=M{\frac {d{\boldsymbol {r}}_{g}}{dt}}}
ãšãªãã å³ã¡ã質ç¹ç³»ã®å
šéåéã¯ã質éäžå¿ã«å
šè³ªéãéäžããŠãããšèãããšãã®éåéã«çããã
è³ªç¹ i ã®éåé pi ã®æéå€åã¯ãè³ªç¹ i ã«äœçšããå Fi ã«çãã
d p i d t = F i {\displaystyle {\frac {d{\boldsymbol {p}}_{i}}{dt}}={\boldsymbol {F}}_{i}}
ãæºããã ããã§è³ªç¹ i ã«äœçšããåã¯ã質ç¹ç³»ã®å€éšããäœçšããå€åãšãç³»ã«å«ãŸããä»ã®è³ªç¹ãšã®å
éšçžäºäœçšã«åããããã è³ªç¹ i ã«äœçšããå€åã fiãè³ªç¹ j ããè³ªç¹ i ã«äœçšããå
åã fij ãšããã°
F i = f i + â j f i j {\displaystyle {\boldsymbol {F}}_{i}={\boldsymbol {f}}_{i}+\sum _{j}{\boldsymbol {f}}_{ij}}
ãšè¡šãããã ãã ããè³ªç¹ i ããè³ªç¹ i èªèº«ã«äœçšããå㯠fii = 0 ãšããã å
šéåéã®æéå€åãèãããš
d P d t = â i d p i d t = â i f i + â i , j f i j {\displaystyle {\frac {d{\boldsymbol {P}}}{dt}}=\sum _{i}{\frac {d{\boldsymbol {p}}_{i}}{dt}}=\sum _{i}{\boldsymbol {f}}_{i}+\sum _{i,j}{\boldsymbol {f}}_{ij}}
ãšãªãã ããã§éåã®ç¬¬3æ³åãããè³ªç¹ j ããè³ªç¹ i ã«äœçšããå fij ãš è³ªç¹ i ããè³ªç¹ j ã«äœçšããå fji ã¯å€§ãããçãã笊å·ãéãªã®ã§
f i j = â f j i , f i j + f j i = 0 {\displaystyle {\boldsymbol {f}}_{ij}=-{\boldsymbol {f}}_{ji},~{\boldsymbol {f}}_{ij}+{\boldsymbol {f}}_{ji}=0}
ãæãç«ã¡ãå
åãå
šãŠè¶³ãåããããã®ã¯ 0 ãšãªãã åŸã£ãŠ
d P d t = M d 2 r g d t 2 = â i f i {\displaystyle {\frac {d{\boldsymbol {P}}}{dt}}=M{\frac {d^{2}{\boldsymbol {r}}_{g}}{dt^{2}}}=\sum _{i}{\boldsymbol {f}}_{i}}
ãšãªãã質ç¹ç³»ã®å
šéåéã®æéå€åã¯äœçšããå€åã®ç·åãšçããã ããã¯ãéåãªã©ã®åçŽãªå€åã®äžã§ã¯è³ªéäžå¿ã®éåãçžå¯Ÿäœçœ®ã®éåããåé¢ã§ããããšãæå³ããŠããã
質ç¹ç³»ã®éåã«ãããŠãç¹ã«äœçšããå€åãé£ãåã£ãŠããå Žåã¯
d P d t = d d t ( â i p i ( t ) ) = 0 {\displaystyle {\frac {d{\boldsymbol {P}}}{dt}}={\frac {d}{dt}}\left(\sum _{i}{\boldsymbol {p}}_{i}(t)\right)=0}
P ( t ) = â i p i ( t ) = const. {\displaystyle {\boldsymbol {P}}(t)=\sum _{i}{\boldsymbol {p}}_{i}(t)={\text{const.}}}
ãæãç«ã€ã ã€ãŸãããã®ç³»ã§ã¯ç³»ã®å
šéåéã¯æéçã«å€åããªããããã¯éåéä¿åã®æ³å (law of conservation of momentum) ãšåŒã°ãããéåéä¿åã®æ³åã¯ããã¥ãŒãã³ååŠã«ãããŠã¯äœçšåäœçšã®æ³åããå°ãããããéåéä¿ååèªäœã¯äœçšåäœçšã®æ³åããäžè¬çã«æãç«ã€æ³åã§ãããããšãã°ãé»ç£æ°åŠãªã©ã®å Žã®çè«ã§ã¯è¿æ¥äœçšè«ã®ç«å Žããšããé éäœçšè«çãªæ³åã§ããäœçšåäœçšã®æ³åããã®åºç€ã«ã¯çœ®ããªããããããªãããé»ç£æ°åŠã«ãããŠãéåéä¿åã®æ³åã¯æãç«ã¡ãããã«äŒŽãéåéã®å®çŸ©ãæ¡åŒµãããã
ç©çåŠã«ãããŠããã¯ãã«ã§è¡šãããç©çéãšããåç¹ã«å¯Ÿããäœçœ®ã®å€ç©ãã¢ãŒã¡ã³ããšãããéåéã®ã¢ãŒã¡ã³ãã¯ãè§éåé (angular momentum) ãšåŒã°ããæ¬¡ã®ããã«å®çŸ©ãããã
L := r à p . {\displaystyle {\boldsymbol {L}}:={\boldsymbol {r}}\times {\boldsymbol {p}}.}
å€å
žçãªè§éåéã®å€§ããã¯ãäœçœ®ãã¯ãã« r ã®å€§ãããšãéåé p ã® r ã«çŽäº€ããæåã®å€§ããã®ç©ãšããŠè¡šãããã2 ã€ã®ãã¯ãã« r, p ãèŒã£ãŠããå¹³é¢äžã®ã2 ã€ã®ãã¯ãã« r, p ã®éã®è§åºŠã Ξ ãšããã°ãè§éåéã®å€§ããã¯æ¬¡ã®ããã«è¡šãããã
| L | = | r | | p ⥠| = | r | | p | sin Ξ . {\displaystyle \left|{\boldsymbol {L}}\right|=\left|{\boldsymbol {r}}\right|\left|{\boldsymbol {p}}_{\perp }\right|=\left|{\boldsymbol {r}}\right|\left|{\boldsymbol {p}}\right|\sin \theta .}
è§£æååŠã«ãããŠã¯ãè§éåéã¯è§åºŠã«å¯Ÿå¿ããäžè¬åéåéãšããŠåŸãããã
è§éåéã¯ããã¥ãŒãã³ã®éåæ¹çšåŒãšåæ§ãªæ¹çšåŒã
d L d t = N {\displaystyle {\frac {d{\boldsymbol {L}}}{dt}}={\boldsymbol {N}}}
ãæºãããããã§ N â r à F ã¯ç©äœã«äœçšããåã®ã¢ãŒã¡ã³ãã§ããã
è§£æååŠã«ãããŠãäžè¬ååº§æš qi ã«å¯Ÿå¿ããäžè¬åéåé (generalized momentum) pi ã¯ãã®ç³»ã®ã©ã°ã©ã³ãžã¢ã³ L(q, ·q) ã®äžè¬åé床 ·qi ã«ããå埮åãšããŠå®çŸ©ãããã
p i := â L ( q , q Ì ) â q Ì i {\displaystyle p_{i}:={\frac {\partial L({\boldsymbol {q}},{\dot {\boldsymbol {q}}})}{\partial {\dot {q}}_{i}}}}
ããã§ãã©ã°ã©ã³ãžã¢ã³ L(q, ·q) ã¯ãéåãšãã«ã®ãŒ Kãããã³ã·ã£ã« U ã®å·®ãšããŠå®çŸ©ãããã
L = K â U . {\displaystyle L=K-U.}
ããã«ãã³åœ¢åŒã®ååŠã§ã¯ãäžè¬åé床ã®ä»£ããã«äžè¬åéåéãååŠå€æ°ãšããŠçšãããããããã«ããã¢ã³ H(q, p) ã¯ãã©ã°ã©ã³ãžã¢ã³ L(q, ·q) ã®ã«ãžã£ã³ãã«å€æãšããŠå®çŸ©ããããã«ãžã£ã³ãã«å€æ
H ( q , p ) := max q Ì â D { q Ì â
p â L ( q , q Ì ) } {\displaystyle H({\boldsymbol {q}},{\boldsymbol {p}}):=\max _{{\dot {\boldsymbol {q}}}\in D}\left\{{\dot {\boldsymbol {q}}}\cdot {\boldsymbol {p}}-L({\boldsymbol {q}},{\dot {\boldsymbol {q}}})\right\}}
ã®å³èŸºãæå€§åãã ·q ãèãããšãã«ãžã£ã³ãã«å€æãããé å D ã®äžã§ã©ã°ã©ã³ãžã¢ã³ãåžã§ãããã€å
åæ»ãããªãããã®ãã㪠·q ã¯ä»¥äžã®é¢ä¿ãæºããã
â â q Ì ( q Ì â
p â L ( q , q Ì ) ) = 0 . {\displaystyle {\frac {\partial }{\partial {\dot {\boldsymbol {q}}}}}\left({\dot {\boldsymbol {q}}}\cdot {\boldsymbol {p}}-L({\boldsymbol {q}},{\dot {\boldsymbol {q}}})\right)={\boldsymbol {0}}.}
ããã¯ããªãã¡ãããã«ããã¢ã³ã®å€æ° p ãäžè¬åéåéã«çããããšãæå³ããã
3 次å
ã®çŽäº€åº§æšç³» x = xËx + yËy + zËz ã«ãããŠã¯ãããã³ã·ã£ã«ãé床 ·x ã«äŸåããªããšãã«ã¯
L ( x , x Ì ) = m 2 ( x Ì 2 + y Ì 2 + z Ì 2 ) â U ( x ) {\displaystyle L({\boldsymbol {x}},{\dot {\boldsymbol {x}}})={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-U({\boldsymbol {x}})}
p α = â L â α Ì = m α Ì , α = x , y , z {\displaystyle p_{\alpha }={\frac {\partial L}{\partial {\dot {\alpha }}}}=m{\dot {\alpha }},\quad \alpha =x,y,z}
ã§ããããã®ãšãäžè¬åéåé p ã¯è³ªéãšé床ã®ç©ãšãªã£ãŠãããããã¯ãã¥ãŒãã³åœ¢åŒã®éåéã«äžèŽããã
äžè¬å座æšãšããŠäºæ¬¡å
æ¥µåº§æš x = (r, Ξ) ãéžã¶ãšãã©ã°ã©ã³ãžã¢ã³åã³ r, Ξ ã«å
±åœ¹ãªéåé pr, pΞ ã¯ãããã
L ( r , Ξ , r Ì , Ξ Ì ) = m 2 ( r Ì 2 + r 2 Ξ Ì 2 ) â U ( x ) {\displaystyle L(r,\theta ,{\dot {r}},{\dot {\theta }})={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2})-U({\boldsymbol {x}})}
p r = â L â r Ì = m r Ì {\displaystyle p_{r}={\frac {\partial L}{\partial {\dot {r}}}}=m{\dot {r}}}
p Ξ = â L â Ξ Ì = m r 2 Ξ Ì {\displaystyle p_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=mr^{2}{\dot {\theta }}}
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p i = â ( K ( v ) â U ( x , v ) ) â v i â â K ( v ) â v i = m v i . {\displaystyle p_{i}={\frac {\partial (K({\boldsymbol {v}})-U({\boldsymbol {x}},{\boldsymbol {v}}))}{\partial v_{i}}}\neq {\frac {\partial K({\boldsymbol {v}})}{\partial v_{i}}}=mv_{i}.}
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L ( x , v ) = m 2 v 2 â e Ï ( x ) + e v â
A ( x ) {\displaystyle L({\boldsymbol {x}},{\boldsymbol {v}})={\frac {m}{2}}{\boldsymbol {v}}^{2}-e\phi ({\boldsymbol {x}})+e{\boldsymbol {v}}\cdot {\boldsymbol {A}}({\boldsymbol {x}})}
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p = m v + e A {\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}+e{\boldsymbol {A}}}
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H ( x , p ) = v â
p â L ( x , v ) {\displaystyle H({\boldsymbol {x}},{\boldsymbol {p}})={\boldsymbol {v}}\cdot {\boldsymbol {p}}-L({\boldsymbol {x}},{\boldsymbol {v}})}
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H ( x , p ) = ( p â e A ( x ) ) 2 2 m + e Ï ( x ) {\displaystyle H({\boldsymbol {x}},{\boldsymbol {p}})={\frac {\left({\boldsymbol {p}}-e{\boldsymbol {A}}({\boldsymbol {x}})\right)^{2}}{2m}}+e\phi ({\boldsymbol {x}})}
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žè«çéåé p â {\displaystyle {\vec {p}}} ã¯ãæ³¢å颿° Ï ( t , x â ) = Ï ( t , x , y , z ) {\displaystyle \psi (t,{\vec {x}})=\psi (t,x,y,z)} ã«å¯Ÿããã
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6,686 |
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https://ja.wikipedia.org/wiki/%E5%BC%B1%E3%81%84%E7%9B%B8%E4%BA%92%E4%BD%9C%E7%94%A8
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y ( x ) = C e x . {\displaystyle y(x)=C\mathrm {e} ^{x}.}
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ææ°é¢æ°ã®æçšãªæ§è³ªãšããŠã埮åäœçšçŽ ãå¥ã®å®æ°ã颿°ã«çœ®ãæããããããšãæãããããä¿æ°ã宿°ã®ææ¬¡æ¹çšåŒ
ã®è§£ãšããŠææ°é¢æ°ã§æžãããã®ãæ¢ããšãf(x) = Cexp(λx) ãšçœ®ãæããŠã
ãšæžãããšãã§ããããã㯠λ ã«å¯Ÿãã n 次ã®ä»£æ°æ¹çšåŒã«ãªã£ãŠããã éæ ¹ããªããã°æ¹çšåŒã®è§£ã n åæ±ãŸãããšã«ãªããææ¬¡æ¹çšåŒã®äžè¬è§£ã¯ãããã®ç·åçµåãšããŠè¡šãããã
ãã®åœ¢ã®æ¹çšåŒã®äžè¬è§£ãæ±ããæ¹æ³ãšããŠã¯å®æ°å€åæ³ãããã
äžã€ã®æªç¥é¢æ°ã«å¯Ÿãããäžè¬ã®äžéç·ååžžåŸ®åæ¹çšåŒã¯ãæ¢ç¥é¢æ°ã P(x)ãQ(x) ãšããŠã次ã®ããã«æžãããã
d y ( x ) d x + P ( x ) y ( x ) = Q ( x ) . {\displaystyle {\frac {\mathrm {d} y(x)}{\mathrm {d} x}}+P(x)y(x)=Q(x).}
ãã®äžéç·ååžžåŸ®åæ¹çšåŒã¯ãäžè¬è§£ãæ±ç©æ³ã§è§£ããã ãŸããææ¬¡æ¹çšåŒ
d y ( x ) d x + P ( x ) y ( x ) = 0 {\displaystyle {\frac {\mathrm {d} y(x)}{\mathrm {d} x}}+P(x)y(x)=0}
ã®äžè¬è§£ã¯ãç©å宿°ã A â 0 ãšããŠã
y ( x ) = A exp ( â â« P ( x â² ) d x â² ) {\displaystyle y(x)=A\exp \left(-\int P(x')\,\mathrm {d} x'\right)}
ãšãªããäžéç·ååžžåŸ®åæ¹çšåŒã®äžè¬è§£ã¯ãææ¬¡æ¹çšåŒã®è§£ãå©çšã A ã x ã®é¢æ°ãšã¿ãªã宿°å€åæ³ã«ãã£ãŠæ±ããããã
y ( x ) = { â« Q ( x â² ) exp ( â« P ( x â²â² ) d x â²â² ) d x â² + C } exp ( â â« P ( x â² ) d x â² ) . {\displaystyle y(x)=\left\{\int Q(x')\exp \left(\int P(x'')\,\mathrm {d} x''\right)\mathrm {d} x'+C\right\}\exp \left(-\int P(x')\,\mathrm {d} x'\right).}
ããã§ C â 0 ã¯ç©å宿°ã§ããã
äºéç·ååžžåŸ®åæ¹çšåŒã®äžè¬åœ¢ã¯ãæ¢ç¥é¢æ°ã P(x), Q(x), R(x) ãšããŠã次ã®ããã«æžãããã
d 2 y ( x ) d x 2 + P ( x ) d y ( x ) d x + Q ( x ) y = R ( x ) . {\displaystyle {\frac {\mathrm {d} ^{2}y(x)}{\mathrm {d} x^{2}}}+P(x){\frac {\mathrm {d} y(x)}{\mathrm {d} x}}+Q(x)y=R(x).}
ãã®äºéç·ååžžåŸ®åæ¹çšåŒã¯ããã®ãŸãŸã®åœ¢ã§ã¯æ±ç©æ³ãçšããŠäžè¬è§£ã衚瀺ããããšã¯ã§ããªãã ãããå³èŸºã 0 ãšããææ¬¡æ¹çšåŒã®ç¹æ®è§£ãšããŠãy = y1 ãååšããã°ã
d 2 y 1 ( x ) d x 2 + P ( x ) d y 1 ( x ) d x + Q ( x ) y 1 ( x ) = 0 {\displaystyle {\frac {\mathrm {d} ^{2}y_{1}(x)}{\mathrm {d} x^{2}}}+P(x){\frac {\mathrm {d} y_{1}(x)}{\mathrm {d} x}}+Q(x)y_{1}(x)=0}
ãæãç«ã€ã®ã§ãz ãªãæªç¥é¢æ°ãå°å
¥ããŠã
y ( x ) = y 1 ( x ) z ( x ) {\displaystyle y(x)=y_{1}(x)z(x)}
ãšããã°ãäºéç·ååžžåŸ®åæ¹çšåŒããz ã«é¢ããåžžåŸ®åæ¹çšåŒã
y 1 d 2 z ( x ) d x 2 + ( 2 y 1 â² ( x ) + P ( x ) y 1 ( x ) ) d z ( x ) d x = R ( x ) , y 1 â² ( x ) = d y 1 ( x ) d x , {\displaystyle {\begin{aligned}&y_{1}{\frac {\mathrm {d} ^{2}z(x)}{\mathrm {d} x^{2}}}+{\Bigl (}2y'_{1}(x)+P(x)y_{1}(x){\Bigr )}{\frac {\mathrm {d} z(x)}{\mathrm {d} x}}=R(x),\\&y'_{1}(x)={\frac {\mathrm {d} y_{1}(x)}{\mathrm {d} x}},\end{aligned}}}
ã«å€æãããããã®åžžåŸ®åæ¹çšåŒã¯ãå°é¢æ° dz/dx ã«é¢ããŠäžéç·ååžžåŸ®åæ¹çšåŒãªã®ã§ãæ±ç©æ³ã§è§£ããããã®äžè¬è§£ã
z = Ï ( C 1 , C 2 , x ) {\displaystyle z=\psi (C_{1},C_{2},x)~~}
ãšãããšãäºéç·ååžžåŸ®åæ¹çšåŒã®äžè¬è§£ã¯ã
y = y 1 Ï ( C 1 , C 2 , x ) {\displaystyle y=y_{1}\psi (C_{1},C_{2},x)~~}
ã§äžããããããªããC1, C2 ã¯ç©å宿°ã§ããã x ã®æ¢ç¥é¢æ°ãå«ãäºéç·ååžžåŸ®åæ¹çšåŒã§ãæ±ç©æ³ã§è§£ããåŸ®åæ¹çšåŒã¯å°ãªããã æ¬¡ã®åŸ®åæ¹çšåŒãªã©ãç¥ãããŠããã
| false | false | false |
https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%BC%8F
|
6,697 |
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ããŠãããè±: Fermat's Last Theorem)ãšã¯ã3 以äžã®èªç¶æ° n ã«ã€ããŠãx + y = z ãšãªãèªç¶æ°ã®çµ (x, y, z) ã¯ååšããªãããšããå®çã§ããã
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n = 4 ã®å Žåããã§ã«ããŒã«ãã£ãŠèšŒæãããåŸã¯ãæ®ãã®èšŒæã¯ n ãå¥çŽ æ°ã®å Žåã®ã¿ãèããã°ããããšã«ãªãããªããªããn ã奿°ã®å Žåã¯ãn = pq...r ã®ããã«å¥çŽ æ°ã®ç©ã§è¡šãããšãã§ããŠãå¥çŽ æ° p ã®ãšãã«æãç«ãŠã°ã(x) + (y) = (z) ãã n = pq...r ã®ãšããæãç«ã€ããšã瀺ããããããã«ãn ãå¶æ°ã®å Žåã¯ã4ã§å²ã£ãäœãã0ãŸãã¯2ãšãªãã®ã§ãäœãã0ããªãã¡ n = 4m ã®å Žå㯠(x) + (y) = (z) ããæãç«ã¡ãäœãã2ããªãã¡ n = 4m+2 ã®å Žå㯠n = 2(2m+1) ãã n ã奿°ã®å æ° 2m+1 ãæã€ããšã«ãªã 2m+1 ãçŽ å æ°åè§£ãããšãã®å¥çŽ æ°ã«ã€ããŠæãç«ã€ããã§ããã
ã¬ãªã³ãã«ãã»ãªã€ã©ãŒã¯1753幎ã«ã¯ãªã¹ãã£ã¢ã³ã»ãŽãŒã«ãããããžå®ãŠãæžç°¡ã®äžã§ n = 3 ã®å Žåã®èšŒææ³ã«ã€ããŠèšåãã1760幎ã«çŽåççã§å®å
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·äœçã«ã¯ a+bââ3 ã®åœ¢ã®æ°ãŸã§å æ°åè§£ãè¡ã£ããã®ã§ãçŸä»£ã®èšèã§èšãã°ãæŽæ°ç° Z [ â 3 ] {\displaystyle \mathbb {Z} [{\sqrt {-3}}]} ã§å æ°åè§£ãè¡ããã®ã§ãã£ããããã®æŽæ°ç°ã§ã¯çŽ å æ°åè§£ã®äžææ§ãæç«ããªã(äžæåè§£ç°ã§ã¯ãªã)ãšããäžåããã£ãã®ã§ãã®ã¡ã« ââ3 ã®ä»£ããã« 1ã®åå§3乿 ¹ ζ 3 = ( â 1 ± â 3 ) / 2 {\displaystyle \zeta _{3}=(-1\pm {\sqrt {-3}})/2} ãä»å ããæŽæ°ç° Z [ ζ 3 ] {\displaystyle \mathbb {Z} [\zeta _{3}]} (ããã¯ååäœ Q [ ζ 3 ] {\displaystyle \mathbb {Q} [\zeta _{3}]} ã®æŽæ°ç°ã§ããããçŽ å æ°åè§£ã®äžææ§ãæãç«ã€)ã䜿ãããšã§ä¿®æ£ãããã
1823幎ã«åœæã»ãšãã©ããªãã£ãå¥³æ§æ°åŠè
ã§ãã£ããœãã£ã»ãžã§ã«ãã³ã¯ããã§ã«ããŒäºæ³ãå¥çŽ æ° p ã«å¯ŸããŠã x + y = z ã«ãããŠã
ãšãã2ã€ã®ã±ãŒã¹ã«åé¡ããp ãš 2p+1 ãå
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ãœãã£ã»ãžã§ã«ãã³ã®å®ç â p ã 2p+1 ãçŽ æ°ã§ãããããªå¥çŽ æ°ãšãã.ãã®ãšããã§ã«ããŒã®å€§å®çã®ç¬¬äžã®å Žåã¯ææ° p ã«å¯ŸããŠæ£ãã.
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6,703 |
NaCl (ææ§ãåé¿)
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JIS X 0213
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ISO-2022-JP
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https://ja.wikipedia.org/wiki/ISO-2022-JP
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