哈密顿力学

作为拉格朗日力学的重新表述

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

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

$L(q_j, \dot{q}_j, t)$

$p_j = {\partial L \over \partial \dot{q}_j}$

$H\left(q_j,p_j,t\right) = \sum_i \dot{q}_i p_i - L(q_j,\dot{q}_j,t)$

H的定义的每边各产生一个微分：

$\begin{matrix} dH &=& \sum_i \left[ \left({\partial H \over \partial q_i}\right) dq_i + \left({\partial H \over \partial p_i}\right) dp_i \right] + \left({\partial H \over \partial t}\right) dt\qquad\qquad\quad\quad \\ \\ &=& \sum_i \left[ \dot{q}_i\, dp_i + p_i\, d\dot{q}_i - \left({\partial L \over \partial q_i}\right) dq_i - \left({\partial L \over \partial \dot{q}_i}\right) d\dot{q}_i \right] - \left({\partial L \over \partial t}\right) dt \end{matrix}$

${\partial H \over \partial q_j} = - \dot{p}_j, \qquad {\partial H \over \partial p_j} = \dot{q}_j, \qquad {\partial H \over \partial t } = - {\partial L \over \partial t}$

数学表述

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

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

黎曼流形

$H(q,p)= \frac{1}{2} \langle p,p\rangle_q$

亚黎曼流形

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

$p_z$没有在哈密顿量中被涉及到。