正则变换
$\quad\\$
正则变换定义
$\quad\\$
使变换后的动力学方程仍为哈密顿方程正则方程的广义坐标变换称为正则变换，作为同一个力学系统的动力学方程，变换前后的正则方程彼此等价，即二者的变分原理等价，拉格朗日函数$L(q,p,t)$与$L(Q,P,t)$相差某个函数$U$对时间的全导数，用数学形式可表示为
$$\left\lbrace\begin{array}&Q_i=Q_i(q,p,t)\\
P_i=P_i(q,p,t)\end{array}\right.\\
\quad\\
\delta\int_{t_1}^{t_2}[\sum\limits_{i=1}^{n}p_i\dot{q_i}-H(p,q,t)]dt=0\\
\quad\\
\delta\int_{t_1}^{t_2}[\sum\limits_{i=1}^{n}P_i\dot{Q_i}-K(P,Q,t)]dt=0\\
\quad\\
\Downarrow\\
\quad\\
\sum\limits_{i=1}^{n}p_idq_i-\sum\limits_{i=1}^{n}P_idQ_i+(K-H)dt=dU$$
$\quad\\$
生成函数
$\quad\\$
函数$U$决定了正则变换，称为正则变换的生成函数(母函数)，上式提示我们可以将$U$用$q,Q,t$表出，记为$U=U_1(q,Q,t)$，可以直接得出
$$\left\lbrace\begin{array}{l}\frac{\partial{U_1}}{\partial{q_i}}=p_i\\
\frac{\partial{U_1}}{\partial{Q_i}}=-P_i\\
\frac{\partial{U_1}}{\partial{t}}=K-H\end{array}\right.\\
\quad\\
\small 作以下三种勒让德变换可以得出另外三种生成函数\\
\quad\\
\left\lbrace\begin{array}{l}U_2=U_1-\sum\limits_{i=1}^{n}p_iq_i\\
U_3=U_1+\sum\limits_{i=1}^{n}P_iQ_i\\
U_4=U_1-\sum\limits_{i=1}^{n}p_iq_i+\sum\limits_{i=1}^{n}P_iQ_i\end{array}\right.\\
\quad\\
\small则其全微分可表示为\\
\quad\\
\left\lbrace\begin{array}{l}dU_2(p,Q,t)=-\sum\limits_{i=1}^{n}q_idp_i-\sum\limits_{i=1}^{n}P_idQ_i+(K-H)dt=dU\\
dU_3(q,P,t)=\sum\limits_{i=1}^{n}p_idq_i+\sum\limits_{i=1}^{n}Q_idP_i+(K-H)dt=dU\\
dU_4(p,P,t)=-\sum\limits_{i=1}^{n}q_idp_i+\sum\limits_{i=1}^{n}Q_idP_i+(K-H)dt=dU\end{array}\right.\\
\quad\\
\small结合U_1有\\
\quad\
\left\lbrace\begin{array}{l}\frac{\partial{U_1}}{\partial{q_i}}=+p_i\\
\frac{\partial{U_1}}{\partial{Q_i}}=-P_i\\
\frac{\partial{U_1}}{\partial{t}}=K-H\end{array}\right.\qquad\qquad\left\lbrace\begin{array}{l}\frac{\partial{U_2}}{\partial{p_i}}=-q_i\\
\frac{\partial{U_2}}{\partial{Q_i}}=-P_i\\
\frac{\partial{U_2}}{\partial{t}}=K-H\end{array}\right.\\
\quad\\
\;\;\;\;\;\left\lbrace\begin{array}{l}\frac{\partial{U_3}}{\partial{q_i}}=+p_i\\
\frac{\partial{U_3}}{\partial{P_i}}=+Q_i\\
\frac{\partial{U_3}}{\partial{t}}=K-H\end{array}\right.\qquad\qquad\left\lbrace\begin{array}{l}\frac{\partial{U_4}}{\partial{p_i}}=-q_i\\
\frac{\partial{U_4}}{\partial{P_i}}=+Q_i\\
\frac{\partial{U_4}}{\partial{t}}=K-H\end{array}\right.\\
$$
$\quad\\$
泊松括号的不变性
$\quad\\$
将正则变换前的正则坐标记为$q,p$，变换后的正则坐标记为$Q,P$，则两个力学量的泊松括号在正则变换前后具有不变性，即$[\varphi,\psi]_{qp}=[\varphi,\psi]_{QP}$
$$\small 取生成函数为U_3,将Q、p看成q、P的函数\\
\quad\\
\small 则任何一力学量f可表示为f(q,p(q,P))或f(Q(q,P),P,t)\\
\quad\\
\small 用这两种形式分别对q_i求偏导数\\
\quad\\
\small 可得\\
\quad\\
\frac{\partial}{\partial q_i}+\sum\limits_{j=1}^{n}\frac{\partial p_j}{\partial q_i}\frac{\partial}{\partial p_j}
=\sum\limits_{j=1}^{n}\frac{\partial Q_j}{\partial q_i}\frac{\partial}{\partial Q_j}\\
\quad\\
\small即\\
\quad\\
\frac{\partial}{\partial q_i}=\sum\limits_{j=1}^{n}(-\frac{\partial p_j}{\partial q_i}\frac{\partial}{\partial p_j}+\frac{\partial Q_j}{\partial q_i}\frac{\partial}{\partial Q_j})\\
\quad\\
\begin{aligned}\;[\varphi,\psi]_ {qp}&=\sum\limits_{i=1}^{n}[\sum\limits_{j=1}^{n}(-\frac{\partial p_j}{\partial q_i}\frac{\partial \varphi}{\partial p_j}+\frac{\partial Q_j}{\partial q_i}\frac{\partial \varphi}{\partial Q_j})\frac{\partial \psi}{\partial p_i}
\\&\quad-\frac{\partial \varphi}{\partial p_i}\sum\limits_{j=1}^{n}(-\frac{\partial p_j}{\partial q_i}\frac{\partial \psi}{\partial p_j}+\frac{\partial Q_j}{\partial q_i}\frac{\partial \psi}{\partial Q_j})]\\
\quad\\
&=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{\partial p_j}{\partial q_i}(\frac{\partial \varphi}{\partial p_i}\frac{\partial \psi}{\partial p_j}-\frac{\partial \varphi}{\partial p_j}\frac{\partial \psi}{\partial p_i})
\\&\quad+\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{\partial Q_j}{\partial q_i}(\frac{\partial \varphi}{\partial Q_j}\frac{\partial \psi}{\partial p_i}-\frac{\partial \varphi}{\partial p_i}\frac{\partial \psi}{\partial Q_j})\\
\quad\\
&=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{\partial ^2U_3}{\partial q_i\partial q_j}(\frac{\partial \varphi}{\partial p_i}\frac{\partial \psi}{\partial p_j}-\frac{\partial \varphi}{\partial p_j}\frac{\partial \psi}{\partial p_i})
\\&\quad+\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{\partial ^2U_3}{\partial q_i\partial P_j}(\frac{\partial \varphi}{\partial Q_j}\frac{\partial \psi}{\partial p_i}-\frac{\partial \varphi}{\partial p_i}\frac{\partial \psi}{\partial Q_j})\\
\quad\\
&=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{\partial ^2U_3}{\partial q_i\partial P_j}(\frac{\partial \varphi}{\partial Q_j}\frac{\partial \psi}{\partial p_i}-\frac{\partial \varphi}{\partial p_i}\frac{\partial \psi}{\partial Q_j})\\
\end{aligned}\\
\quad\\
\small 同理,用这两种形式分别对P_i求导有\\
\quad\\
\frac{\partial}{\partial P_i}
+\sum\limits_{j=1}^n\frac{\partial Q_j}{\partial P_i}\frac{\partial}{\partial Q_j}
=\sum\limits_{j=1}^n\frac{\partial p_j}{\partial P_i}\frac{\partial}{\partial p_j}\\
\quad\\
\small即\\
\quad\\
\frac{\partial}{\partial P_i}
=\sum\limits_{j=1}^n(-\frac{\partial Q_j}{\partial P_i}\frac{\partial}{\partial Q_j}
+\frac{\partial p_j}{\partial P_i}\frac{\partial}{\partial p_j})\\
\quad\\
\begin{aligned}\;[\varphi,\psi]_ {QP}&=\sum\limits_{i=1}^n[\frac{\partial \varphi}{\partial Q_i}\sum\limits_{j=1}^n(-\frac{\partial Q_j}{\partial P_i}\frac{\partial \psi}{\partial Q_j}
+\frac{\partial p_j}{\partial P_i}\frac{\partial \psi}{\partial p_j})\\&\quad- \sum\limits_{j=1}^n(-\frac{\partial Q_j}{\partial P_i}\frac{\partial \varphi}{\partial Q_j}
+\frac{\partial p_j}{\partial P_i}\frac{\partial \varphi}{\partial p_j})\frac{\partial \psi}{\partial Q_i}]\\
\quad\\
&=\sum\limits_{i=1}^n\sum\limits_{j=1}^n
\frac{\partial Q_j}{\partial P_i}(\frac{\partial \varphi}{\partial Q_j}\frac{\partial \psi}{\partial Q_i}-\frac{\partial \varphi}{\partial Q_i}\frac{\partial \psi}{\partial Q_j})
\\&\quad+\sum\limits_{i=1}^n\sum\limits_{j=1}^n
\frac{\partial p_j}{\partial P_i}(\frac{\partial \varphi}{\partial Q_i}\frac{\partial \psi}{\partial p_j}-\frac{\partial \varphi}{\partial p_j}\frac{\partial \psi}{\partial Q_i})\\
\quad\\
&=\sum\limits_{i=1}^n\sum\limits_{j=1}^n
\frac{\partial^2 U_3}{\partial P_i\partial P_j}(\frac{\partial \varphi}{\partial Q_j}\frac{\partial \psi}{\partial Q_i}-\frac{\partial \varphi}{\partial Q_i}\frac{\partial \psi}{\partial Q_j})
\\&\quad+\sum\limits_{i=1}^n\sum\limits_{j=1}^n
\frac{\partial^2 U_3}{\partial P_i\partial q_j}(\frac{\partial \varphi}{\partial Q_i}\frac{\partial \psi}{\partial p_j}-\frac{\partial \varphi}{\partial p_j}\frac{\partial \psi}{\partial Q_i})\\
\quad\\
&=\sum\limits_{i=1}^n\sum\limits_{j=1}^n
\frac{\partial^2 U_3}{\partial P_i\partial q_j}(\frac{\partial \varphi}{\partial Q_i}\frac{\partial \psi}{\partial p_j}-\frac{\partial \varphi}{\partial p_j}\frac{\partial \psi}{\partial Q_i})\\
\end{aligned}\\
\quad\\
\small 故\\
\quad\\
[\varphi,\psi]_{qp}=[\varphi,\psi] _{QP}\\
\quad\\
\small由正则变换泊松括号的不变性可以得出如下正则变换的条件\\
\quad\\
[Q_i,Q_j]_{qp}=0\qquad [P_i,P_j] _{qp}=0\qquad[Q_i,P_j] _{qp}=\delta _{ij}
$$
$\quad\\$
泊松括号的辛形式
$\quad\\$
因为$q$与$p$可以联立变换，不妨将$q$与$p$视作一组坐标$\gamma=[q,p]^T$，称为相空间的辛坐标，正则变换后的$Q$与$P$也对应一组辛坐标$\Gamma=[Q,P]^T$，则不含时间的正则变换可以写为简单的形式$\Gamma=\Gamma(\gamma)$，以下来分析此类正则变换在辛形式下对应的性质
$$\small定义正则变换的雅可比矩阵为[以下一维向量均默认为列向量]\\
\quad\\
J=\frac{\partial\Gamma}{\partial\gamma}=\left[
\begin{array}&\left(\frac{\partial Q}{\partial q}\right)&
\left(\frac{\partial Q}{\partial p}\right)\\
\left(\frac{\partial P}{\partial q}\right)&
\left(\frac{\partial P}{\partial p}\right)\end{array}\right]\\
\quad\\
\small则\\
\quad\\
\dot{\Gamma}=J\dot{\gamma}\\
\quad\\
\small 定义2n维的辛矩阵\\
\quad\\
s=\left[\begin{array}&O&I_n\\
-I_n&O\end{array}\right]\\
\quad\\
\small则泊松括号在辛形式下可写为\\
\quad\\
[\varphi,\psi]_{qp}=\left(\frac{\partial \varphi}{\partial \gamma}\right)^Ts\left(\frac{\partial \psi}{\partial \gamma}\right)\\
\quad\\
\small将\Gamma代入有\\
\quad\\
[\Gamma_i,\Gamma_j]_{qp}=\left(\frac{\partial \Gamma_i}{\partial \gamma}\right)^Ts\left(\frac{\partial \Gamma_j}{\partial \gamma}\right)\\
\quad\\
\small 由正则变换的泊松括号条件即知\\
\quad\\
JsJ^T=s\\
\quad\\
\small另外\\
\quad\\
ss=-I_{2n}\\
\quad\\
JsJ^TsJ=-J\\
\quad\\
J^TsJ=s\\
\quad\\
\small此式称为拉格朗日括号条件$$
$\quad\\$
无限小正则变换
$\quad\\$
选取生成函数$U_3=\sum\limits_{i=1}^n+\varepsilon G(q,P,t)$，其中$\varepsilon$为一小量，由第三类生成函数的变换公式可得
$$\left\lbrace\begin{array}&Q_i=q_i+\varepsilon\frac{\partial G}{\partial p_i}\\
P_i=p_i-\varepsilon\frac{\partial G}{\partial q_i}\end{array}\right.\qquad即\qquad
\left\lbrace\begin{array}&dq_i=\varepsilon\frac{\partial G}{\partial p_i}\\
dp_i=-\varepsilon\frac{\partial G}{\partial q_i}\end{array}\right.$$
变化前后的正则变量仅相差一小量，故称为无限小正则变换，若令$G=H,\varepsilon=dt$，则上述变换正好给出哈密顿正则方程，亦即此无限小正则变换描述力学系统在$dt$时间里的演变，通过无穷多个这样的无限小正则变换复合，则可以在给定正则变量初始值的情况下得出任意时刻$t$的正则变量的值。另外对于任意力学量$\varphi(q,p,t)$有
$$d\varphi=\sum(\frac{\partial \varphi}{\partial q_i}dq_i+\frac{\partial \varphi}{\partial p_i}dp_i)+\frac{\partial \varphi}{\partial t}dt\\
\quad\\
\small 代入无限小正则变换得\\
\quad\\
\begin{aligned}d\varphi&=\varepsilon\sum(\frac{\partial \varphi}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial \varphi}{\partial p_i}\frac{\partial G}{\partial q_i})+\frac{\partial \varphi}{\partial t}dt\\
\quad\\
&=\varepsilon [\varphi,G]+\frac{\partial \varphi}{\partial t}dt\end{aligned}\\
\quad\\
\small 令\varphi=H,若H不显含时间则有\\
\quad\\
dH=\varepsilon [H,G]=\varepsilon(\frac{\partial G}{\partial t}-\frac{dG}{dt})
$$
从上式可以看出，若哈密顿函数$H$与无限小正则变换中的函数$G$不显含时间，且此变换使哈密顿函数保持不变，即$dH=0$，则$G$为力学系统的运动积分