Euler-Lagrange 方程
$$\dfrac {\mathrm d}{\mathrm dt} \dfrac {\partial L}{\partial \dot q_i} - \dfrac {\partial L}{\partial q_i} = 0.$$正则动量
$$p_i = \dfrac{\partial L}{\partial \dot q_i},\quad \dot p_i = \dfrac {\partial L}{\partial q_i}.$$Hamiltonian，可以对于正则 Lagrangian 定义
$$H(q,p,t) = \sum_i p_i \dot q_i - L(q, \dot q, t)=\sum_i p_i \dot q_i - L(q, p, t).$$Hamilton 方程
$$\dot q_i = \dfrac{\partial H}{\partial p_i}, \quad \dot p_i = -\dfrac{\partial H}{\partial q_i},\quad \dfrac{\partial H}{\partial t} = -\dfrac{\partial L}{\partial t}.$$相空间作用量原理*
Poisson 括号
$$\{f, g\} = \sum_i \left( \dfrac{\partial f}{\partial q_i} \dfrac{\partial g}{\partial p_i} - \dfrac{\partial f}{\partial p_i} \dfrac{\partial g}{\partial q_i} \right).$$Poisson 括号与 Hamilton 方程
$$\dfrac {\mathrm d f}{\mathrm dt} = \{f, H\} + \dfrac{\partial f}{\partial t}.$$正则括号
$$\{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}.$$Jacobi 恒等式
$$\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0.$$
对于 $T^*V$ 上的任意一点，存在局部坐标 $(q_1, \dots, q_n, p_1, \dots, p_n)$，使得