贡献者： addis
首先回顾拉格朗日方程为
\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 2 ）
\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。