# 刘维尔定理 (哈密顿力学)

## 刘维尔方程

$\frac{d\rho}{dt}= \frac{\partial\rho}{\partial t} +\sum_{i=1}^d\left(\frac{\partial\rho}{\partial q^i}\dot{q}^i +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0.\,$

$\frac{\partial\rho}{\partial t}+\sum_{i=1}^d\left(\frac{\partial(\rho\dot{q}^i)}{\partial q^i}+\frac{\partial(\rho\dot{p}_i)}{\partial p_i}\right)=0.$

$(\rho, \rho\dot{q}^i,\rho\dot{p}_i)$ 是一个保守流。注意到此式与刘维尔方程的差是

$\rho\sum_{i=1}^d\left( \frac{\partial\dot{q}^i}{\partial q^i} +\frac{\partial\dot{p}_i}{\partial p_i}\right) =\rho\sum_{i=1}^d\left( \frac{\partial^2 H}{\partial q^i\,\partial p_i} -\frac{\partial^2 H}{\partial p_i \partial q^i}\right)=0,$

## 物理解释

$N=\int d^dq\,d^dp\,\rho(p,q).\,$

$\frac{\partial\rho}{\partial t}+\frac{\mathbf{p}}{m}\cdot\nabla_\mathbf{x}\rho+\mathbf{F}\cdot\nabla_\mathbf{p}\rho=0.\,$

## 其他表述

### 泊松括号

$\frac{\partial\rho}{\partial t}=-\{\,\rho,H\,\}\,$

$\hat{\mathbf{L}}=\sum_{i=1}^{d}\left[\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial q^{i}}\frac{\partial }{\partial p_{i}}\right],\,$

$\frac{\partial \rho }{\partial t}+{\hat{L}}\rho =0.\,$

### 量子力学

$\frac{\partial}{\partial t}\rho=\frac{1}{i \hbar}[H,\rho],\,$

$\frac{d}{dt}\langle A\rangle = \frac{1}{i \hbar}\langle [A,H] \rangle,\,$

## 参考文献

• В.И.阿诺尔德，著. 齐民友，译. 经典力学中的数学方法（第4版）. 北京：高等教育出版社，2006年1月.