泊松括号
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泊松括号定义
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$$力学量\varphi=\varphi(q,p,t)\\
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\frac{d\varphi}{dt}=\frac{\partial \varphi}{\partial t}+\sum\limits_{i=1}^{n}(\frac{\partial \varphi}{\partial q_i}\dot{q_i}+\frac{\partial \varphi}{\partial p_i}\dot{p_i})\\
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哈密顿正则方程\\
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\left\lbrace\begin{array}&\frac{\partial H}{\partial q_i}&=&-\dot{p_i}\\
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\frac{\partial H}{\partial p_i}&=&\dot{q_i}\end{array}\right.\\
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代入上式中\\
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\frac{d\varphi}{dt}=\frac{\partial \varphi}{\partial t}+\sum\limits_{i=1}^{n}(\frac{\partial \varphi}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial \varphi}{\partial p_i}\frac{\partial H}{\partial q_i})\\
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定义两个力学量\varphi=\varphi(q,p,t),\psi=\psi(q,p,t)的泊松括号如下\\
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[\varphi,\psi]=\sum\limits_{i=1}^{n}(\frac{\partial \varphi}{\partial q_i}\frac{\partial \psi}{\partial p_i}-\frac{\partial \varphi}{\partial p_i}\frac{\partial \psi}{\partial q_i})\\
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则有如下等式\\
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\frac{d\varphi}{dt}=\frac{\partial \varphi}{\partial t}+[\varphi,H]\\
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\dot{q_i}=[q_i,H]\quad\dot{p_i}=[p_i,H]\\
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[q_i,q_j]=0\quad[p_i,p_j]=0\quad[q_i,p_j]=\delta_{ij}
$$
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泊松括号性质
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$$[\varphi,C]=0\;(C是常数)\\
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[\varphi,\varphi]=0\\
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[\varphi,\psi]=-[\psi,\varphi]\;(反对称性)\\
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[\varphi,\sum\limits_{i=1}^{n}C_i\psi_i]=\sum\limits_{i=1}^{n}C_i[\varphi,\psi_i]\;(分配律)\\
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[\varphi,\psi_1\psi_2]=\psi_1[\varphi,\psi_2]+[\varphi,\psi_1]\psi_2\;(结合律)\\
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\frac{\partial}{\partial t}[\varphi,\psi]=[\frac{\partial \varphi}{\partial t},\psi]+[\varphi,\frac{\partial \psi}{\partial t}]\;(微商法则)\\
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[q_i,\varphi]=\frac{\varphi}{p_i}\\
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[p_i,\varphi]=-\frac{\varphi}{q_i}$$
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雅可比恒等式
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$$[\varphi,[\psi,\theta]]+[\theta,[\varphi,\psi]]+[\psi,[\theta,\varphi]]=0$$
$\nabla$证
$$ 由泊松括号的定义可知\\\\ \quad\\\\ 上式左边是\varphi,\psi,\theta二阶偏导数的齐次函数\\\\ \quad\\\\ 含\varphi的二阶偏导的项为[\theta,[\varphi,\psi]]+[\psi,[\theta,\varphi]]\\\\ \quad\\\\ 将泊松括号[X,Y]看作线性微分算符D_X=\sum\limits_{i=1}^n(\frac{\partial X}{\partial q_i}\frac{\partial}{\partial p_i}-\frac{\partial X}{\partial p_i}\frac{\partial}{\partial q_i})作用于Y的结果\\\\ \quad\\\\ \begin{aligned} \;[\theta,[\varphi,\psi]]+[\psi,[\theta,\varphi]]&=[\psi,[\theta,\varphi]]-[\theta,[\psi,\varphi]]\\\\ &=(D_{\psi}D_{\theta}-D_{\theta}D_{\psi})\varphi\end{aligned}\\\\ \quad\\\\ D_{\psi}D_{\theta}-D_{\theta}D_{\psi}中含二阶偏微分算符的项为\\\\ \quad\\\\ \sum\limits_{i=1}^n \sum\limits_{j=1}^n(\frac{\partial \psi}{\partial q_i}\frac{\partial \theta}{\partial q_j}\frac{\partial^2}{\partial p_i \partial p_j}+\frac{\partial \psi}{\partial p_i}\frac{\partial \theta}{\partial p_j}\frac{\partial^2}{\partial q_i \partial q_j}-\frac{\partial \psi}{\partial q_i}\frac{\partial \theta}{\partial p_j}\frac{\partial^2}{\partial p_i \partial q_j}-\frac{\partial \psi}{\partial p_i}\frac{\partial \theta}{\partial q_j}\frac{\partial^2}{\partial q_i \partial p_j})\\\\ \quad\\\\ -\sum\limits_{i=1}^n \sum\limits_{j=1}^n(\frac{\partial \theta}{\partial q_i}\frac{\partial \psi}{\partial q_j}\frac{\partial^2}{\partial p_i \partial p_j}+\frac{\partial \theta}{\partial p_i}\frac{\partial \psi}{\partial p_j}\frac{\partial^2}{\partial q_i \partial q_j}-\frac{\partial \theta}{\partial q_i}\frac{\partial \psi}{\partial p_j}\frac{\partial^2}{\partial p_i \partial q_j}-\frac{\partial \theta}{\partial p_i}\frac{\partial \psi}{\partial q_j}\frac{\partial^2}{\partial q_i \partial p_j})\\\\ \quad\\\\ \varphi为标量,偏导顺序可交换,故上式为零\\\\ \quad\\\\ 同理,含\psi和\theta的二阶偏导的项也为零\\\\ \quad\\\\ 故泊松括号的雅可比恒等式成立 $$
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泊松定理
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不随时间改变的力学量叫做运动积分，即若$\varphi$为运动积分，则有$\frac{d\varphi}{dt}=0$。泊松定理可叙述如下
$$若\varphi和\psi均为运动积分,则[\varphi,\psi]也为运动积分$$
$\nabla$证
$$因为\varphi和\psi均为运动积分,故有\\\\ \quad\\\\ \frac{d\varphi}{dt}=\frac{\partial \varphi}{\partial t}+[\varphi,H]=0\\\\ \quad\\\\ \frac{d\psi}{dt}=\frac{\partial \psi}{\partial t}+[\psi,H]=0\\\\ \quad\\\\ 雅可比恒等式\\\\ \quad\\\\ \begin{aligned}0&=[\varphi,[\psi,H]]+[\psi,[H,\varphi]]+[H,[\varphi,\psi]]\\\\ &=-[\varphi,\frac{\partial \psi}{\partial t}]+[\psi,\frac{\partial \varphi}{\partial t}]+[H,[\varphi,\psi]]\\\\ &=-(\frac{\partial [\varphi,\psi]}{\partial t}+[[\varphi,\psi],H])\\\\ &=-\frac{d[\varphi,\psi]}{dt}\end{aligned}\\\\ \quad\\\\ 故[\varphi,\psi]也为运动积分 $$
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参考与引用来源
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梁昆淼力学第四版