# 刘维尔定理 (哈密顿力学)

## 刘维尔方程

${\displaystyle {\frac {d\rho }{dt}}={\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{d}\left({\frac {\partial \rho }{\partial q^{i}}}{\dot {q}}^{i}+{\frac {\partial \rho }{\partial p_{i}}}{\dot {p}}_{i}\right)=0.\,}$

${\displaystyle {\frac {\partial \rho }{\partial t}}+\sum _{i=1}^{d}\left({\frac {\partial (\rho {\dot {q}}^{i})}{\partial q^{i}}}+{\frac {\partial (\rho {\dot {p}}_{i})}{\partial p_{i}}}\right)=0.}$

${\displaystyle (\rho ,\rho {\dot {q}}^{i},\rho {\dot {p}}_{i})}$ 是一个守恒流。注意到此式与刘维尔方程的差是

${\displaystyle \rho \sum _{i=1}^{d}\left({\frac {\partial {\dot {q}}^{i}}{\partial q^{i}}}+{\frac {\partial {\dot {p}}_{i}}{\partial p_{i}}}\right)=\rho \sum _{i=1}^{d}\left({\frac {\partial ^{2}H}{\partial q^{i}\,\partial p_{i}}}-{\frac {\partial ^{2}H}{\partial p_{i}\partial q^{i}}}\right)=0,}$

## 物理解释

${\displaystyle N=\int d^{d}q\,d^{d}p\,\rho (p,q).\,}$

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

## 其他表述

### 泊松括号

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

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

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

### 量子力学

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

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

### 刘维尔定理之破坏

2005年，有论文[1]发现当x与p不是辛形式的时候，尤其演化中存在几何相位，流体密度将可能被压缩。

## 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.
1. ^ 存档副本 (PDF). [2020-10-23]. （原始内容存档 (PDF)于2020-10-26）.