# 理论力学/哈密顿原理

${\displaystyle S=\int _{t_{1}}^{t_{2}}Ldt}$

${\displaystyle \delta S=\Sigma _{i=1}^{N}\int _{t_{1}}^{t_{2}}dt({\frac {\partial L}{\partial q_{i}}}\delta q_{i}+{\frac {\partial L}{\partial {\dot {q_{i}}}}}\delta {\dot {q_{i}}})=0}$

${\displaystyle \Sigma _{i=1}^{N}\int _{t_{1}}^{t_{2}}({\frac {\partial L}{\partial q_{i}}}-{\frac {d}{dt}}({\frac {\partial L}{\partial {\dot {q_{i}}}}}))\delta q_{i}dt+{\frac {\partial L}{\partial {\dot {q_{i}}}}}\delta q_{i}|_{t_{1}}^{t_{2}}=0}$

${\displaystyle {\frac {\partial L}{\partial q_{i}}}-{\frac {d}{dt}}({\frac {\partial L}{\partial {\dot {q_{i}}}}})=0}$

${\displaystyle S'=S+\int _{t_{1}}^{t_{2}}{\frac {df}{dt}}dt=S+f|_{t_{1}}^{t2}}$

${\displaystyle L=\int d^{3}x{\mathcal {L}}(\phi ,\partial _{x}\phi ,...,\partial _{x}^{k}\phi ,\partial _{t}\phi ,....,\partial _{t}\partial _{x}^{k}\phi ,x,t)}$

${\displaystyle S=\int _{t_{1}}^{t_{2}}Ldt=\int {\mathcal {L}}dtd^{3}x}$

${\displaystyle S=\int {\mathcal {L}}(\phi ,\partial _{x_{i}}\phi ,\partial _{t}\phi ,x_{i},t)d^{3}xdt,i=1,2,3}$

${\displaystyle \delta S=0}$给出：

${\displaystyle \int ({\frac {\partial {\mathcal {L}}}{\partial \phi }}-\Sigma _{i}\partial _{x_{i}}({\frac {\partial {\mathcal {L}}}{\partial (\partial _{x_{i}}\phi )}})-\partial _{t}({\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\phi )}}))\delta \phi d^{3}xdt+\oint _{S}(\Sigma _{i}{\frac {\partial {\mathcal {L}}}{\partial (\partial _{x_{i}}\phi )}}+{\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\phi )}})\delta \phi dA=0}$

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \phi }}-\Sigma _{i}\partial _{x_{i}}({\frac {\partial {\mathcal {L}}}{\partial (\partial _{x_{i}}\phi )}})-\partial _{t}({\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\phi )}})=0}$

## 注释

1. 在后面我们会看到，这一假设实际上是狭义相对性原理的要求。最一般的情形下场运动方程的推导可以参考埃米·诺特的论文《Invariante Variationsprobleme》。