相対論的力学

${\displaystyle x=X(\lambda )}$

${\displaystyle {\frac {d\lambda '}{d\lambda }}={\dot {f}}(\lambda )\geq 0}$

${\displaystyle {\frac {dt}{d\lambda }}={\frac {1}{c}}{\dot {X}}^{0}(\lambda )\geq 0}$

${\displaystyle {\dot {X}}(\lambda )={\frac {dX}{d\lambda }}\to {\frac {dX}{d\lambda '}}={\frac {dX}{d\lambda }}{\bigg /}{\frac {d\lambda '}{d\lambda }}}$

${\displaystyle U^{\mu }(\lambda )={\frac {c{\dot {X}}^{\mu }(\lambda )}{\sqrt {-{\dot {X}}^{\nu }{\dot {X}}_{\nu }}}}}$

${\displaystyle U^{0}(t)={\frac {c}{\sqrt {1-v^{2}/c^{2}}}},}$

${\displaystyle {\boldsymbol {U}}(t)={\frac {\boldsymbol {v}}{\sqrt {1-v^{2}/c^{2}}}}}$

${\displaystyle U^{\mu }(t)=(\gamma c,\gamma {\boldsymbol {v}})}$

${\displaystyle {\frac {d\tau }{d\lambda }}={\frac {1}{c}}{\sqrt {-{\dot {X}}^{\mu }{\dot {X}}_{\mu }}}}$

${\displaystyle U^{\mu }(\tau )={\frac {dX^{\mu }}{d\tau }}}$

${\displaystyle p^{\mu }(\lambda )=mU^{\mu }(\lambda )}$

${\displaystyle {\frac {dp^{\mu }}{d\tau }}=K^{\mu }(\tau )}$

${\displaystyle K^{\mu }(\lambda )={\frac {c\,{\dot {p}}^{\mu }(\lambda )}{\sqrt {-{\dot {X}}^{\mu }{\dot {X}}_{\mu }}}}}$

${\displaystyle K^{0}(t)={\frac {{\boldsymbol {f}}\cdot {\boldsymbol {v}}/c}{\sqrt {1-v^{2}/c^{2}}}},}$

${\displaystyle {\boldsymbol {K}}(t)={\frac {\boldsymbol {f}}{\sqrt {1-v^{2}/c^{2}}}}}$

${\displaystyle K^{\mu }(t)=(\gamma {\boldsymbol {f}}\cdot {\boldsymbol {v}}/c,\gamma {\boldsymbol {f}})}$

${\displaystyle S_{X}[X,e]=\int L_{X}({\dot {X}},e)\,d\lambda }$

${\displaystyle L_{X}({\dot {X}},e)={\frac {e}{2}}\left[e^{-2}{\dot {X}}^{\mu }{\dot {X}}_{\mu }(\lambda )-m^{2}c^{2}\right]}$

${\displaystyle e(\lambda )\,d\lambda \to e'(\lambda ')\,d\lambda '}$

${\displaystyle P_{\mu }(\lambda )={\frac {\partial L_{X}}{\partial {\dot {X}}^{\mu }}}=e^{-1}{\dot {X}}_{\mu }(\lambda )}$

${\displaystyle {\frac {\delta S_{X}[X,e]}{\delta X^{\mu }(\lambda )}}=-{\dot {P}}_{\mu }(\lambda )=0}$

${\displaystyle {\frac {\delta S_{X}[X,e]}{\delta e(\lambda )}}=-{\frac {1}{2}}\left[e^{-2}{\dot {X}}^{\mu }{\dot {X}}_{\mu }(\lambda )+m^{2}c^{2}\right]=-{\frac {1}{2}}\left[P^{2}(\lambda )+m^{2}c^{2}\right]=0}$

${\displaystyle e(\lambda )={\frac {1}{mc}}{\sqrt {-{\dot {X}}^{\mu }{\dot {X}}_{\mu }}}}$

${\displaystyle L_{X}({\dot {X}})=-mc{\sqrt {-{\dot {X}}^{\mu }{\dot {X}}_{\mu }}}=-mc^{2}{\frac {d\tau }{d\lambda }}}$

${\displaystyle S_{X}[X]=-mc\int {\sqrt {-{\dot {X}}^{\mu }{\dot {X}}_{\mu }}}d\lambda =-mc^{2}\int d\tau }$

${\displaystyle S_{X}[X,e]={\frac {1}{2}}\int \sum _{i\in I}\left[e_{i}^{-2}{\dot {X}}_{i}^{\mu }{\dot {X}}_{i\mu }-m_{i}^{2}c^{2}\right]e_{i}\,d\lambda }$

${\displaystyle {\begin{aligned}S_{X}[X]&=-\sum _{i\in I}m_{i}c\int {\sqrt {-{\dot {X}}_{i}^{\mu }{\dot {X}}_{i\mu }}}d\lambda \\&=-\sum _{i\in I}m_{i}c^{2}\int d\tau _{i}\\\end{aligned}}}$

${\displaystyle L_{X}(X,{\dot {X}},e)={\frac {e}{2}}\left[e^{-2}g_{\mu \nu }(X){\dot {X}}^{\mu }{\dot {X}}^{\nu }(\lambda )-m^{2}c^{2}\right]}$

${\displaystyle P_{\mu }(\lambda )={\frac {\partial L_{X}}{\partial {\dot {X}}^{\mu }}}=g_{\mu \nu }(X)\,p^{\nu }(\lambda )}$

${\displaystyle {\begin{aligned}{\frac {\delta S[X,e]}{\delta X^{\mu }(\lambda )}}&={\frac {e^{-1}}{2}}\partial _{\mu }g_{\rho \nu }(X){\dot {X}}^{\rho }{\dot {X}}^{\nu }(\lambda )-{\dot {P}}_{\mu }(\lambda )\\&=-e^{-1}\varGamma _{\mu \rho \nu }(X){\dot {X}}^{\rho }{\dot {X}}^{\nu }(\lambda )-g_{\mu \nu }(X)\,{\dot {p}}^{\nu }(\lambda )\\&=0\\\end{aligned}}}$

${\displaystyle {\dot {p}}^{\mu }(\lambda )+e^{-1}\varGamma _{\rho \nu }^{\mu }(X){\dot {X}}^{\rho }{\dot {X}}^{\nu }(\lambda )=0}$

${\displaystyle K^{\mu }(\lambda )=-m\varGamma _{\rho \nu }^{\mu }(X)U^{\rho }U^{\nu }(\lambda )}$

${\displaystyle \varGamma _{\rho \nu }^{\mu }(x)={\frac {1}{2}}g^{\mu \sigma }\left[\partial _{\rho }g_{\nu \sigma }(x)+\partial _{\nu }g_{\rho \sigma }(x)-\partial _{\sigma }g_{\rho \nu }(x)\right]}$

${\displaystyle S_{\text{int}}[X]=\int L_{\text{int}}(X,{\dot {X}})\,d\lambda }$

${\displaystyle L_{\text{int}}(X,{\dot {X}})=qA_{\mu }(X)\,{\dot {X}}^{\mu }(\lambda )}$

${\displaystyle P_{\mu }(\lambda )={\frac {\partial }{\partial {\dot {X}}^{\mu }}}(L_{X}+L_{\text{int}})=p_{\mu }(\lambda )+qA_{\mu }(X)}$

${\displaystyle {\begin{aligned}{\frac {\delta S[X]}{\delta X^{\mu }(\lambda )}}&=q\partial _{\mu }A_{\nu }(X){\dot {X}}^{\nu }(\lambda )-{\dot {P}}_{\mu }(\lambda )\\&=qF_{\mu \nu }(X){\dot {X}}^{\nu }(\lambda )-{\dot {p}}_{\mu }(\lambda )\\&=0\\\end{aligned}}}$

${\displaystyle {\dot {p}}_{\mu }(\lambda )=qF_{\mu \nu }(X){\dot {X}}^{\nu }(\lambda )}$

${\displaystyle K^{\mu }(\lambda )=qF^{\mu }{}_{\nu }(X)U^{\nu }(\lambda )}$

${\displaystyle F_{\mu \nu }(x)=\partial _{\mu }A_{\nu }(x)-\partial _{\nu }A_{\mu }(x)}$

${\displaystyle {\dot {p}}^{\mu }(\lambda )+e^{-1}\varGamma _{\rho \nu }^{\mu }(X){\dot {X}}^{\rho }{\dot {X}}^{\nu }(\lambda )=qF^{\mu }{}_{\nu }(X){\dot {X}}^{\nu }(\lambda )}$

${\displaystyle F^{\mu }{}_{\nu }(x)=g^{\mu \rho }(x)\,F_{\rho \nu }(x)}$

${\displaystyle S[X,e]=\int L(X,{\dot {X}},e)\,d\lambda }$

${\displaystyle L(X,{\dot {X}},e)=L_{X}({\dot {X}},e)-emV(X)={\frac {e}{2}}\left[e^{-2}{\dot {X}}^{\mu }{\dot {X}}_{\mu }(\lambda )-m^{2}c^{2}-2mV(X)\right]}$

${\displaystyle M={\frac {1}{c}}{\sqrt {m^{2}c^{2}+2mV(X)}}\simeq m+{\frac {1}{c^{2}}}V(X)}$

${\displaystyle P_{\mu }(\lambda )={\frac {\partial L}{\partial {\dot {X}}^{\mu }}}=e^{-1}{\dot {X}}_{\mu }(\lambda )}$

${\displaystyle {\frac {\delta S[X,e]}{\delta e(\lambda )}}=-{\frac {1}{2}}\left[e^{-2}{\dot {X}}^{\mu }{\dot {X}}_{\mu }(\lambda )+m^{2}c^{2}+2mV(X)\right]=-{\frac {1}{2}}\left[P^{2}(\lambda )+M^{2}c^{2}\right]=0}$

${\displaystyle P_{\mu }(\lambda )=M(X)U_{\mu }}$

${\displaystyle {\frac {\delta S[X,e]}{\delta X^{\mu }(\lambda )}}=-eMc^{2}\partial _{\mu }M-{\dot {P}}_{\mu }(\lambda )=0}$

${\displaystyle {\frac {dP_{\mu }}{d\tau }}=-c^{2}\partial _{\mu }M\simeq -\partial _{\mu }V}$

${\displaystyle H_{X}(P,e)=P_{\mu }{\dot {X}}^{\mu }(\lambda )-L_{X}={\frac {e}{2}}\left[P^{\mu }P_{\mu }(\lambda )+m^{2}c^{2}\right]}$

${\displaystyle H_{X}(X,P,e)={\frac {e}{2}}\left[g_{\mu \nu }(X)P^{\mu }P^{\nu }(\lambda )+m^{2}c^{2}\right]}$

${\displaystyle H(X,P,e)={\frac {e}{2}}\left[(P^{\mu }-qA^{\mu })(P_{\mu }-qA_{\mu })+m^{2}c^{2}\right]}$

${\displaystyle H(X,P,e)={\frac {e}{2}}\left[P^{\mu }P_{\mu }(\lambda )+m^{2}c^{2}+2mV\right]}$

${\displaystyle H(X,P,e)=e\cdot \chi (X,P)}$

${\displaystyle P_{e}(\lambda )={\frac {\partial L_{X}}{\partial {\dot {e}}}}=0}$

${\displaystyle H_{\text{tot}}=e\chi +b\phi }$

${\displaystyle {\dot {\phi }}=\{H_{\text{tot}},P_{e}\}=\{e,P_{e}\}\cdot \chi +\{b,P_{e}\}\cdot \phi \approx -\chi }$

${\displaystyle {\dot {\chi }}=\{H_{\text{tot}},\chi \}=\{b,\chi \}\cdot \phi \approx 0}$