哈密顿原理
$\quad\\$
位形空间的哈密顿原理
$\quad\\$
力学系统从时刻$t_1$到时刻$t_2$的一切可能运动之中，使哈密顿作用量$S=\int_{t_1}^{t_2}L(q,\dot{q},t)dt$取极值的运动为实际发生的运动，即
$$\delta\int_{t_1}^{t_2}L(q,\dot{q},t)\;dt=0$$
$$\frac{\partial L}{\partial q_i}-\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_i}})=0\\
\quad\\
\small经变化可得另一种形式\\
\quad\\
\frac{d L}{d t}=\frac{\partial L}{\partial t}+\sum\limits_{i=1}^n\frac{\partial L}{\partial q_i}\frac{d q_i}{d t}+
\sum\limits_{i=1}^n\frac{\partial L}{\partial \dot{q_i}}\frac{d \dot{q_i}}{d t}\\
\quad\\
\begin{aligned}\frac{\partial L}{\partial t}&=\frac{d L}{d t}-\sum\limits_{i=1}^n\frac{\partial L}{\partial q_i}\frac{d q_i}{d t}-\sum\limits_{i=1}^n\frac{\partial L}{\partial \dot{q_i}}\frac{d \dot{q_i}}{d t}\\
\quad\\
&=\frac{d}{d t}(L-\sum\limits_{i=1}^n\dot{q_i}\frac{\partial L}{\partial \dot{q_i}})-\sum\limits_{i=1}^n\dot{q_i}[\frac{\partial L}{\partial q_i}-\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_i}})]\\
\quad\\
&=\frac{d}{d t}(L-\sum\limits_{i=1}^n\dot{q_i}\frac{\partial L}{\partial \dot{q_i}})\end{aligned}\\
\quad\\
\small即\\
\quad\\
\frac{\partial L}{\partial t}+\frac{d}{d t}(\sum\limits_{i=1}^n\dot{q_i}\frac{\partial L}{\partial \dot{q_i}}-L)=0\\
$$
最后一个式子$\frac{\partial L}{\partial t}+\frac{d H}{d t}=0$，也被称为广义功-能定理
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相空间的哈密顿原理
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$n$维位形空间中的变分原理可以直接通过代换$p_i=\frac{\partial L}{\partial \dot{q_i}},H=\sum\limits_{i=1}^n p_i \dot{q_i}-L$得到$2n$维相空间中的哈密顿变分原理
$$\delta\int_{t_1}^{t_2}[\sum\limits_{i=1}^n p_i \dot{q_i}-H(q,p,t)]\;dt=0$$
通过变化可以得到哈密顿正则方程。
$$\begin{aligned}0&=\delta\int_{t_1}^{t_2}\sum\limits_{i=1}^n[\dot{q_i}\delta p_i+p_i\delta \dot{q_i}-\frac{\partial H}{\partial q_i}\delta q_i-\frac{\partial H}{\partial p_i}\delta p_i]\;dt\\
\quad\\
&=\int_{t_1}^{t_2}\sum\limits_{i=1}^n[\dot{q_i}\delta p_i+\frac{d}{dt}(p_i \delta q_i)-\dot{p_i}\delta q_i-\frac{\partial H}{\partial q_i}\delta q_i-\frac{\partial H}{\partial p_i}\delta p_i]\;dt\\
\quad\\
&\small[力学系统的始末状态给定，则端点处p和q变分为零]\\
\quad\\
&=\int_{t_1}^{t_2}\sum\limits_{i=1}^n[(\dot{q_i}-\frac{\partial H}{\partial p_i})\delta p_i-(\dot{p_i}+\frac{\partial H}{\partial q_i})\delta q_i]\;dt\end{aligned}\\
\quad\\
\small由于p和q的变分是任意的,故有哈密顿正则方程\\
\quad\\
\left\lbrace\begin{array}&\frac{\partial H}{\partial q_i}&=&-\dot{p_i}\\
\frac{\partial H}{\partial p_i}&=&\dot{q_i}\end{array}\right.$$
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位形世界的哈密顿原理
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在位形世界中，将时间$t$看作与广义坐标$q$地位平等的变量，于是$t$相当于第$n+1$个广义坐标，这$n+1$个广义坐标构成了推广的位形世界，另外选取一个参数$\tau$，则可以把运动规律$q_i=q_i(t)$表示成参数形式$q_i=q_i(\tau)$和$t=t(\tau)$，并记$\frac{d q_i}{d\tau}=q_i’$和$\frac{d t}{d\tau}=t’$，由于$\dot{q_i}=\frac{d q_i}{d t}=\frac{\frac{d q_i}{d \tau}}{\frac{d t}{d \tau}}=\frac{q_i’}{t’}$，故哈密顿作用量可表示为
$$S=\int L(q,t,\frac{q’}{t’})t’\;d\tau$$
记$t=q_0,q_{[0]}$为排除$t$之后的广义坐标，位形世界的拉格朗日函数
$\mathcal{L}=L(q_{[0]},t,\frac{q_{[0]}’}{t’})t’=\mathcal{L}(q_{[0]},t,q_{[0]}’,t’)=\mathcal{L}(q,q’)$，则哈密顿变分原理成为
$$\delta \int\mathcal{L}(q,q’)\;d\tau=0$$
同样可以得到Euler-Lagrange方程
$$\frac{d}{d\tau}(\frac{\partial \mathcal{L}}{\partial q_i’})-\frac{\partial \mathcal{L}}{\partial q_i}=0\quad \forall i=0,1,\cdots,n\\
\quad\\
\begin{aligned}\small对于q_i \in q_{[0]},\frac{\partial \mathcal{L}}{\partial q_i’}
&=\frac{\partial [L(q_{[0]},t,\frac{q_{[0]}’}{t’})t’]}{\partial q_i’}\\
\quad\\
&=\frac{\partial L(q_{[0]},\dot{q_{[0]}},t)}{\partial \dot{q_i}}\frac{\partial \dot{q_i}}{\partial q_i’}t’\\
\quad\\
&=\frac{\partial L(q_{[0]},\dot{q_{[0]}},t)}{\partial \dot{q_i}}\\
\quad\\ \end{aligned}\\
\small仍为位形空间的广义动量\\
\quad\\
\begin{aligned}对于q_0=t,\frac{\partial \mathcal{L}}{\partial q_0’}
&=\frac{\partial [L(q_{[0]},t,\frac{q_{[0]}’}{t’})t’]}{\partial t’}\\
\quad\\
&=L+\sum\limits_{i=1}^n\frac{\partial L(q_{[0]},\dot{q_{[0]}},t)}{\partial \dot{q_i}}\frac{\partial \dot{q_i}}{\partial t’}t’\\
\quad\\
&=L-\sum\limits_{i=1}^n\frac{\partial L(q_{[0]},\dot{q_{[0]}},t)}{\partial \dot{q_i}}\dot{q_i}\\
\quad\\
&=-H\\ \end{aligned}\\
\quad\\
\small于是位形世界中与t对应的广义动量为-H$$
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参考与引用来源
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梁昆淼力学第四版