哈密顿力学¶

勒让德变换¶

$f (x)$ 是 $x$ 的函数，且 $u (x) = \frac{d f}{d x}$ .

构造

g (u) = x u - f (x)

$g$ 是 $u$ 的函数

取全微分，得

\begin{aligned} d g & = x d u + u d x - d f \\ = x d u + u d x - u d x \\ = x d u . \end{aligned}

故 $\frac{d g}{d u} = x$ , 即 $g (u)$ 是 $u$ 的函数。

称 $g (u)$ 为 $f (x)$ 的勒让德变换 (Legendre transformation).

函数 $f (x_{1}, x_{2}, \dots, x_{n}; α_{1}, α_{2}, \dots, α_{m})$ 是 $x_{i}$ 和 $α_{j}$ 的函数，且

y_{i} = \frac{\partial f}{\partial x_{i}}, (i = 1, 2, \dots, n) .

将变量 $x_{i}, α_{j}$ 变换为 $y_{i}, α_{j}$ ，构造

g = \sum_{i = 1}^{n} x_{i} y_{i} - f

$g$ 是 $y_{i}$ 的函数

取全微分，得

\begin{aligned} d g & = \sum_{i = 1}^{n} y_{i} d x_{i} + \sum_{i = 1}^{n} x_{i} d y_{i} - d f \\ = \sum_{i = 1}^{n} y_{i} d x_{i} + \sum_{i = 1}^{n} x_{i} d y_{i} - \sum_{i = 1}^{n} y_{i} d x_{i} \\ = \sum_{i = 1}^{n} x_{i} d y_{i} . \end{aligned}

故 $\frac{\partial g}{\partial y_{i}} = x_{i}$ , 即 $g$ 是 $y_{i}$ 的函数。

在物理学中，用于推导各种热力学势
- 热力学第一定律 $d U = d Q + d W$
- 热力学第二定律 $d Q = T d S$
$⟹ d U = T d S - P d V$
- $U (S, V)$ 是 $S$ 和 $V$ 的函数，且 $\frac{\partial U}{\partial S} = T, \frac{\partial U}{\partial V} = - P$
在数学中，用于求解 PDE
在弹性力学中
- 定义势能（应变能）
- 余能是势能的勒让德变换
- 势能和余能构成力学的数值分析方法（FEM）的基础
在理论力学中
- 推导哈密顿方程

设质点系由 $n$ 个质点组成，受到 $s$ 个完整约束，系统自由度为 $k = 3 n - s$ . 理想约束情况，保守系统的拉格朗日函数为

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} = 0, i = 1, 2, \dots, k .

p_{i} = \frac{\partial L}{\partial {\dot{q}}_{i}}, i = 1, 2, \dots, k .

\begin{aligned} H & = H (p_{1}, p_{2}, \dots, p_{k}; q_{1}, q_{2}, \dots, q_{k}; t) \\ = \sum_{i = 1}^{k} p_{i} {\dot{q}}_{i} - L, i = 1, 2, \dots, k . \end{aligned}

拉格朗日函数的勒让德变换为哈密顿函数 $H$ .

\begin{array}{r} det [\frac{\partial^{2} L}{\partial {\dot{q}}_{i} \partial {\dot{q}}_{j}}] \end{array}

广义速度表示的动能

\begin{aligned} T & = \frac{1}{2} \sum_{i = 1}^{k} \sum_{j = 1}^{k} a_{i j} {\dot{q}}_{i} {\dot{q}}_{j} + \sum_{i = 1}^{k} b_{i} {\dot{q}}_{i} + c \\ = T_{2} + T_{1} + T_{0} . \end{aligned}

其中 $T_{2}, T_{1}, T_{0}$ 分别表示二阶、一阶和常数项动能。

\begin{aligned} d H & = \sum_{i = 1}^{k} p_{i} d {\dot{q}}_{i} + \sum_{i = 1}^{k} {\dot{q}}_{i} d p_{i} - \sum_{i = 1}^{k} \frac{\partial L}{\partial q_{i}} d {\dot{q}}_{i} - \sum_{i = 1}^{k} \frac{\partial L}{\partial {\dot{q}}_{i}} d q_{i} \\ （利用广义动量定义消去两项） \\ = \sum_{i = 1}^{k} {\dot{q}}_{i} d p_{i} - \sum_{i = 1}^{k} \frac{\partial L}{\partial {\dot{q}}_{i}} d q_{i} . \end{aligned}

故得

\begin{aligned} \frac{\partial H}{\partial p_{i}} & = {\dot{q}}_{i}, \\ \frac{\partial H}{\partial q_{i}} & = - \frac{\partial L}{\partial q_{i}} . \end{aligned}

利用拉格朗日方程和广义动量定义，代换掉 $\frac{\partial L}{\partial q_{i}}$ :

\begin{aligned} \frac{\partial L}{\partial q_{i}} & = \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) \\ = \frac{d p_{i}}{d t} \\ = {\dot{p}}_{i} . \end{aligned}

最终得到

{\begin{cases} {\dot{q}}_{i} = \frac{\partial H}{\partial p_{i}} \\ {\dot{p}}_{i} = - \frac{\partial H}{\partial q_{i}} \end{cases}

其中 $i = 1, 2, \dots, k$ . 此即为哈密顿方程。

Remarks

对于非保守系统，主动力可分为有势力和非有势力两类，系统的拉格朗日方程为

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} = {\tilde{Q}}_{i}, i = 1, 2, \dots, k .

广义动量关于时间的全导数

{\dot{p}}_{i} = \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}}) = \frac{\partial L}{\partial q_{i}} + {\tilde{Q}}_{i} = - \frac{\partial H}{\partial q_{i}} + {\tilde{Q}}_{i} .

哈密顿方程

$$ \begin{aligned} H &= \left(\sum_{i=1}^k p_i \dot{q}i - L \right)}_i \to p_i

例题

某二自由度动力学系统的广义坐标为 $q_{1}, q_{2}$ , 拉格朗日量为

L = \frac{3}{2} {\dot{q}}_{1}^{2} + \frac{1}{2} {\dot{q}}_{2}^{2} - q_{1}^{2} - \frac{1}{2} q_{2}^{2} - q_{1} q_{2} .

求哈密顿函数 $H$ .

解

系统的广义动量

p_{1} = \frac{\partial L}{\partial {\dot{q}}_{1}} = 3 {\dot{q}}_{1}, p_{2} = \frac{\partial L}{\partial {\dot{q}}_{2}} = {\dot{q}}_{2} .

解得广义速度

{\dot{q}}_{1} = \frac{1}{3} p_{1}, {\dot{q}}_{2} = p_{2} .

代入哈密顿函数

\begin{aligned} H & = p_{1} {\dot{q}}_{1} + p_{2} {\dot{q}}_{2} - L \\ = \frac{1}{3} p_{1}^{2} + p_{2}^{2} - \frac{3}{2} {\dot{q}}_{1}^{2} - \frac{1}{2} {\dot{q}}_{2}^{2} + q_{1}^{2} + \frac{1}{2} q_{2}^{2} + q_{1} q_{2} \\ = - \frac{2}{3} p_{1}^{2} - \frac{1}{2} p_{2}^{2} + q_{1}^{2} + \frac{1}{2} q_{2}^{2} + q_{1} q_{2} . \end{aligned}