哈密顿力学

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。

注释

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。

参见

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