# 哈密顿力学

（重定向自哈密頓函數

## 作为拉格朗日力学的重新表述

${\displaystyle \left\{\,q_{j}|j=1,\ldots ,N\,\right\}}$

${\displaystyle \left\{\,{\dot {q}}_{j}|j=1,\ldots ,N\,\right\}}$

${\displaystyle L(q_{j},{\dot {q}}_{j},t)}$

${\displaystyle p_{j}={\partial L \over \partial {\dot {q}}_{j}}}$

${\displaystyle H\left(q_{j},p_{j},t\right)=\sum _{i}{\dot {q}}_{i}p_{i}-L(q_{j},{\dot {q}}_{j},t)}$

${\displaystyle H}$ 的定义的每边各产生一个微分：

${\displaystyle {\begin{matrix}dH&=&\sum _{i}\left[\left({\partial H \over \partial q_{i}}\right)dq_{i}+\left({\partial H \over \partial p_{i}}\right)dp_{i}\right]+\left({\partial H \over \partial t}\right)dt\qquad \qquad \quad \quad \\\\&=&\sum _{i}\left[{\dot {q}}_{i}\,dp_{i}+p_{i}\,d{\dot {q}}_{i}-\left({\partial L \over \partial q_{i}}\right)dq_{i}-\left({\partial L \over \partial {\dot {q}}_{i}}\right)d{\dot {q}}_{i}\right]-\left({\partial L \over \partial t}\right)dt\end{matrix}}}$

${\displaystyle {\partial H \over \partial q_{j}}=-{\dot {p}}_{j},\qquad {\partial H \over \partial p_{j}}={\dot {q}}_{j},\qquad {\partial H \over \partial t}=-{\partial L \over \partial t}}$

## 数学表述

${\displaystyle {\frac {d}{dt}}f={\frac {\partial }{\partial t}}f+\{\,f,H\,\}.}$

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

### 黎曼流形

${\displaystyle H(q,p)={\frac {1}{2}}\langle p,p\rangle _{q}}$

### 亚黎曼流形

${\displaystyle H(x,y,z,p_{x},p_{y},p_{z})={\frac {1}{2}}\left(p_{x}^{2}+p_{y}^{2}\right)}$ .

${\displaystyle p_{z}}$ 没有在哈密顿量中被涉及到。

## 註釋

1. ^ 拉格朗日力学是经典力学的另一表述，由拉格朗日于1788年建立。