正则变换 2

             

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预备知识 泊松括号

   首先回顾拉格朗日方程

\begin{equation} \frac{\mathrm{d}}{\mathrm{d}{t}} \frac{\partial L}{\partial \dot q_i} = \frac{\partial L}{\partial q_i} \qquad L(q,\dot q, t) = T - V \end{equation}
正则动量为 $p_i = \partial L/\partial \dot q_i $,广义力为 $ \partial L/\partial q_i $, 拉氏方程就是广义力与正则动量的牛顿第二定律.对于任何广义坐标,拉格朗日方程的形式不变.

   勒让德变换后,得到哈密顿正则方程为

\begin{equation} \dot q_i = \frac{\partial H}{\partial p_i} \qquad \dot p_i = - \frac{\partial H}{\partial q_i} \end{equation}
其中 $H(p,q) = T + V$.我们通常简记 $q_1, \dots, q_N$ 为 $q$,简记 $\dot q_1, \dots, \dot q_N$ 为 $\dot q$.

1. 正则变换

   若广义坐标 $q$ 变换到另一套广义坐标 $q'$,假设变换不显含时间,每个 $q_k$ 都是新坐标的函数 $q_k(q')$,有

\begin{equation} \dot q_i = \sum_j \frac{\partial q_i}{\partial q'_j} \dot q'_j \end{equation}
所以
\begin{equation} \frac{\partial \dot q_k}{\partial \dot q'_i} = \frac{\partial q_k}{\partial q'_i} \end{equation}
另外可见广义速度的变换也和广义坐标有关:$\dot q_i = \dot q_i(q', \dot q')$.

   拉格朗日量是系统的状态量,所以 $L(q',\dot q', t) = L[q(q'),\dot q(q',\dot q'), t]$, 所以

\begin{equation} p'_i = \frac{\partial L}{\partial \dot q'_i} = \sum_k \frac{\partial L}{\partial \dot q_k} \frac{\partial \dot q_k}{\partial \dot q'_i} = \sum_k \frac{\partial q_k}{\partial q'_i} p_k \end{equation}
这就从坐标变换推出了动量变换.对于任何广义坐标以及对应的正则动量,哈密顿方程的形式不变(因为拉格朗日方程的形式不变,哈密顿方程是由拉格朗日方程推出来的),也有其他情况也不变.所有使正则方程成立的坐标叫做正则坐标(canonical coordinates).下面推导判断正则坐标的一般条件.

   对于不显含时的物理量 $\omega(q, p)$,有(式 3

\begin{equation} \dot \omega = \left\{\omega, H\right\} = \sum_i \left( \frac{\partial \omega}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial H}{\partial q_i} \frac{\partial \omega}{\partial p_i} \right) \end{equation}
现在若把 $H$ 看成是 $H[q'(q,p),p'(q,p)]$,
\begin{equation} \frac{\partial H}{\partial p_i} = \sum_k \frac{\partial H}{\partial q'_k} \frac{\partial q'_k}{\partial p_i} + \frac{\partial H}{\partial p'_k} \frac{\partial p'_k}{\partial p_i} \end{equation}
\begin{equation} \frac{\partial H}{\partial q_i} = \sum_k \frac{\partial H}{\partial q'_k} \frac{\partial q'_k}{\partial q_i} + \frac{\partial H}{\partial p'_k} \frac{\partial p'_k}{\partial q_i} \end{equation}
代入并对 $H$ 的偏微分合并同类项得
\begin{equation} \dot \omega = \left\{\omega, H\right\} = \sum_k \left[ \frac{\partial H}{\partial q'_k} \left\{\omega, q'_k\right\} + \frac{\partial H}{\partial p'_k} \left\{\omega, p'_k\right\} \right] \end{equation}
注意泊松括号是对 $q,p$ 进行偏微分,记为 $\{ {}\}_{q,p}$. 分别代入 $\omega = q'_i, p'_i$, 得到转换坐标后的哈密顿方程的一般形式.为了保持正则方程的形式,必须要求
\begin{equation} \left\{q'_i, q'_k\right\} _{q,p} = \left\{p'_i, p'_k\right\} _{q,p} = 0 \end{equation}
\begin{equation} \left\{q'_i, p'_k\right\} _{q,p} = \delta_{ik} \end{equation}
这就是判断正则变换的一般条件

   可以证明,用任何正则坐标作为泊松括号的角标,其值都不变:

\begin{equation} \left\{u, v\right\} _{q,p} = \sum_i \left( \frac{\partial u}{\partial q_i} \frac{\partial v}{\partial p_i} - \frac{\partial v}{\partial q_i} \frac{\partial u}{\partial p_i} \right) \end{equation}
其中
\begin{equation} \frac{\partial u}{\partial q_i} \frac{\partial v}{\partial p_i} = \sum_j \left( \frac{\partial u}{\partial q'_j} \frac{\partial q'_j}{\partial q_i} + \frac{\partial u}{\partial p'_j} \frac{\partial p'_j}{\partial q_i} \right) \sum_k \left( \frac{\partial v}{\partial q'_k} \frac{\partial q'_k}{\partial p_i} + \frac{\partial v}{\partial p'_k} \frac{\partial p'_k}{\partial p_i} \right) \end{equation}
\begin{equation} \frac{\partial v}{\partial q_i} \frac{\partial u}{\partial p_i} = \sum_k \left( \frac{\partial v}{\partial q'_k} \frac{\partial q'_k}{\partial q_i} + \frac{\partial v}{\partial p'_k} \frac{\partial p'_k}{\partial q_i} \right) \sum_j \left( \frac{\partial u}{\partial q'_j} \frac{\partial q'_j}{\partial p_i} + \frac{\partial u}{\partial p'_j} \frac{\partial p'_j}{\partial p_i} \right) \end{equation}

   现在我们要得到 $ \left\{u, v\right\} _{q',p'} = \sum_i \left( \frac{\partial u}{\partial q'_i} \frac{\partial v}{\partial p'_i} - \frac{\partial v}{\partial q'_i} \frac{\partial u}{\partial p'_i} \right) $, 可以把上两式代入式 12 后对 $ \frac{\partial u}{\partial q'} \frac{\partial v}{\partial p'} $ 和 $ \frac{\partial v}{\partial q'_i} \frac{\partial u}{\partial p'_i} $ 合并同类项,得

\begin{equation} \begin{aligned} \left\{u, v\right\} _{q,p} & = \sum_{jk} \frac{\partial u}{\partial q'_j} \frac{\partial v}{\partial p'_k} \sum_i \left( \frac{\partial q'_j}{\partial q_i} \frac{\partial p'_k}{\partial p_i} - \frac{\partial p'_k}{\partial q_i} \frac{\partial q'_j}{\partial p_i} \right) \\ & -\sum_{jk} \frac{\partial v}{\partial q'_k} \frac{\partial u}{\partial p'_j} \sum_i \left( \frac{\partial q'_k}{\partial q_i} \frac{\partial p'_j}{\partial p_i} - \frac{\partial p'_j}{\partial q_i} \frac{\partial q'_k}{\partial p_i} \right) \\ &= \sum_{jk} \frac{\partial u}{\partial q'_j} \frac{\partial v}{\partial p'_k} \left\{q'_j, p'_k\right\} _{q,p} - \sum_{jk} \frac{\partial v}{\partial q'_k} \frac{\partial u}{\partial p'_j} \left\{q'_k, p'_j\right\} _{q,p} \end{aligned} \end{equation}
代入正则坐标条件(式 11 ),得
\begin{equation} \begin{aligned} \left\{u, v\right\} _{q,p} & = \sum_{jk} \frac{\partial u}{\partial q'_j} \frac{\partial v}{\partial p'_k} \delta_{jk} - \sum_{jk} \frac{\partial v}{\partial q'_k} \frac{\partial u}{\partial p'_j} \delta_{jk} = \sum_j \left( \frac{\partial u}{\partial q'_j} \frac{\partial v}{\partial p'_j} - \frac{\partial v}{\partial q'_j} \frac{\partial u}{\partial p'_j} \right) \\ &= \left\{u, v\right\} _{q',p'} \end{aligned} \end{equation}
证毕.

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