变分原理初步

设任意\(N\)维空间的路线用一组参数方程描述: \[x_k = x_k(\beta),\;k=1, 2, \cdots, N.\] 在确定两个端点的情况下, 任意选择一条路线计算积分, 并将这条路线称为预设路线 (未变路线). 记两端点对应的参数方程值为 \(\beta_1, \beta_2\), 则起点和终点坐标为: \[x_k^{(1)} = x_k(\beta_1),\;x_k^{(2)} = x_k(\beta_2)\]

变化路线

定义预设路线之后,考虑经过相同起点与终点的另外路线.

变化路线记为 \(x_k(\beta, \delta\alpha)\), 描述变换路线的参数方程为: \[x_k(\beta, \delta\alpha) = x_k(\beta) +\delta\alpha\eta_k(\beta).\] 其中变形函数 \(\eta_k(\beta)\) 可以是 \(\beta\) 的任意有限、可微函数, 同时满足端点条件, 即: \[\eta_k(\beta_1) = \eta_k(\beta_2) = 0,\;k=1,2,\cdots, N.\]

标度参数\(\delta\alpha\)表征预设路线于变化路线的差别,与\(k, \beta\)无关.

当 \(\delta\alpha\rightarrow 0\)时, 只要变形函数 \(\eta_k(\beta)\) 有限, 则变化路线与预设路线重合.

坐标的变分

变化路线的坐标与未变路线的中抽出的差别. \[\delta x_k(\beta) = x_k(\beta, \delta\alpha) - x_k(\beta) = \delta\alpha\eta_k(\beta)\]

记未变路线坐标对\(\beta\)的导数 \(\dot{x}_k(\beta) = \frac{\mathrm{d}x_k(\beta)}{\mathrm{d}\beta}\). (本节\(\dot{x}\)表示\(x\)对\(\beta\)求导), 可得: \[\dot{x}_k(\beta, \delta\alpha) = \dot{x}_k(\beta) + \delta\alpha\dot{\eta}_k(\beta).\] 坐标导数的变分有: \[\delta \dot{x}_k(\beta) = \dot{x}_k(\beta, \delta\alpha) - \dot{x}_k(\beta) =\delta\alpha\dot{\eta}_k(\beta) = \frac{\mathrm{d}}{\mathrm{d}\beta}[\delta x_k(\beta)]\] 即: \[\delta \frac{\mathrm{d}x_k(\beta)}{\mathrm{d}\beta} = \frac{\mathrm{d}}{\mathrm{d}\beta}[\delta x_k(\beta)].\] 微分、变分运算可交换次序.

由此可得: \[x_k(\beta, \delta\alpha) = x_k(\beta) + \delta x_k(\beta), \;\dot{x}_k(\beta, \delta\alpha) = \dot{x}_k(\beta) + \delta \dot{x}_k(\beta).\] 即:对每个给定\(\beta\)值,变化路线上的值等于未变路线上的值加相应的变化值.

函数变分

\[\delta f = f \left(x(\beta, \delta\alpha), \dot{x}(\beta, \delta\alpha)\right)- f \left( x(\beta), \dot{x}(\beta) \right).\]

视 \(\delta f = \delta f(\delta\alpha)\), 做 Taylor 一阶展开:

\begin{aligned} \delta f &= \left[ \frac{\partial f \left(x(\beta, \delta\alpha), \dot{x}(\beta, \delta\alpha)\right)} {\partial h} \right]_{h=0}\delta\alpha\\ &= \left[ \frac{\partial f(x, \dot{x})}{\partial x_k}\delta x_k(\beta) + \frac{\partial f(x, \dot{x})}{\partial \dot{x}_k}\partial \dot{x}_k(\beta) \right]. \end{aligned}

函数 \(f(x, \dot{x})\) 沿某路线的线积分为: \[I = \int_{\beta_1}^{\beta_2} f(x(\beta), \dot{x}(\beta))\mathrm{d}\beta\] 沿预设路线积分.

若沿变换路线积分, 则积分值依赖于标度参量 \(\delta\alpha\), 未变路线 \(x\), 变形函数 \(\eta\). 故有: \[I(\delta\alpha, [x], [\eta]) = \int_{\beta_1}^{\beta_2} f(x(\beta, \delta\alpha), \dot{x}(\beta, \delta\alpha))\mathrm{d}\beta. \] 易证, 预设路线 \(I = I(0, [x], [\eta])\).

线积分值的变分为:

\begin{aligned} \delta I &= \int_{\beta_1}^{\beta_2}\partial f\mathrm{d}\beta\\ &=\int_{\beta_1}^{\beta_2} \left[ \frac{\partial f(x, \dot{x})}{\partial x_k}\delta x^k(\beta) + \frac{\partial f(x, \dot{x})}{\partial \dot{x}_k}\partial \dot{x}_k(\beta) \right]\mathrm{d}\beta\\ &=-\int_{\beta_1}^{\beta_2} \left[ \frac{\mathrm{d}}{\mathrm{d}\beta} \left( \frac{\partial f(x, \dot{x})}{\partial \dot{x}_k} \right) - \frac{\partial f(x, \dot{x})}{\partial x_k} \right]\mathrm{d}\beta. \end{aligned}

拉格朗日形式.

在 \(N\) 维空间中有两个端点, 其坐标记为: \(x_k^{(1)}(\beta_1),\;x_k^{(2)}(\beta_2),\;k=1, 2, \cdots, N\) 沿过两个端点的一条由参数方程 \(x_k(\beta)\) 描述的路线, 其线积分: \[I = \int_{\beta_1}^{\beta_2}f(x(\beta), \dot{x}(\beta))\mathrm{d}\beta.\] \(I\) 取极值等价于积分路线是欧拉-拉格朗日微分方程的解, 即 \[\frac{\mathrm{d}}{\mathrm{d}\beta}\left( \frac{\partial f(x, \dot{x})}{\partial \dot{x}_k} \right)- \frac{\partial f(x, \dot{x})}{\partial x_k} = 0,\;k = 1, 2, \cdots, N.\]

光线总是沿时间取极值路线传播. \[\mathrm{d}T = \frac{\mathrm{d}s}{c/n}= n \mathrm{d}s / c\] 即 \(\int n\mathrm{d}s\) 取极值.

由欧拉-拉格朗日定理推得: \[\frac{\mathrm{d}}{\mathrm{d}s}(n\vec{\tau}) = \nabla n.\]

\[\frac{m}{2}\left( \frac{\mathrm{d}s}{\mathrm{d}t} \right)^2 = mgy \Rightarrow \mathrm{d}t = \frac{\mathrm{d}s}{\sqrt{2gy}}\] 即 \(\int \mathrm{d}s/\sqrt{2gy}\) 取极值.

解为摆线.

考虑到几何约束 \[G_a(x) = 0,\;a = 1, 2, \cdots, C.\] 无论是未变路线还是变化路线都必须满足约束条件: \[G_a(x(\beta)) = 0,\;G_a(x(\beta, \delta\alpha)) = 0.\] 则约束函数的变分也为零,即: \[\delta G_a = \frac{\partial G_a(x)}{\partial x_k}\delta x^k(\beta) = 0.\] 得到约束条件下的欧拉-拉格朗日定理: \[I = \int_{\beta_1}^{\beta_2}f(x(\beta), \dot{x}(\beta))\mathrm{d}\beta\] 上式取极值等价于: 存在 \(C\) 个函数 \(\lambda_a\) 满足: \[\frac{\mathrm{d}}{\mathrm{d}\beta}\left( \frac{\partial f(x, \dot{x})}{\delta \dot{x}_k} \right) - \frac{\partial f(x, \dot{x})}{\partial x_k} = \lambda_a\frac{\partial G_a(x)}{\partial x_k}, \;k = 1, 2, \cdots, N.\] 共求解 \(N + C\) 个未知量. 函数 \(\lambda_a\) 称为拉格朗日乘子.

由约束函数可得 \(C\) 个受限坐标与 \(f = N - C\) 个自由坐标的依赖关系: \[x_{f+j}^{(b)} = x_{f+j}^{(b)}\left( x^{(f)} \right), \;\dot{x}_{f+j}^{(b)} = \dot{x}_{f+j}^{(b)}\left( x^{(f)}, \dot{x}^{(f)} \right).\] 代入线积分中被积函数: \[I = \int_{\beta_1}^{\beta_2} f \left( x(\beta), \dot{x}(\beta) \right) = \int_{\beta_1}^{\beta_2}\tilde{f}\left( x^{(f)}(\beta), \dot{x}^{(f)}(\beta) \right).\] 显然,上述线积分取极值的充要条件为: \[\frac{\mathrm{d}}{\mathrm{d}\beta} \left( \frac{\partial \tilde{f}(x^{(f)}, \dot{x}^{(f)})}{\partial \dot{x}_k^{(f)}} \right) - \frac{\partial \tilde{f}\left( x^{(f)}, \dot{x}^{(f)} \right)}{\partial x_k^{(f)}} = 0.\] 其中 \(k = 1, 2, \cdots, f = N - C\).

选取坐标要反映体系的对称性.

坐标参数法中, 取 \(N = D - 1\), \(x_1 = t\) (这里把时间看作广义坐标!), \(x_2, x_3, \cdots, x_N\) 为 \(q_1, q_2, \cdots, q_D\). 并记 \(x_{[1]} = q\), \(x_{[1]}^{\prime} = \dot{q}\). 则作为充要条件的欧拉-拉格朗日方程可以改写为: \[\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\partial g(t, q, \dot{q})}{\partial \dot{q}_k} \right)- \frac{\partial g(t, q, \dot{q})}{\partial q_k} = 0.\] 其中 \(k = 1, 2, \cdots, D\).

与无约束体系拉格朗日方程作比较,得到

1. 拉格朗日方程的意义即下式取极值: \[I = \int_{t(1)}^{t(2)}L(q, \dot{q}, t)\mathrm{d}t = \int_{t(1)}^{t(2)}(T - U)\mathrm{d}t.\]
2. 哈密顿原理: 上式一阶变分 \(\delta I\) 取零的充要条件为: 积分路线 \(q_k(t)\) 为无约束且非保守力为零体系的拉格朗日方程之解. \[\int_{t(1)}^{t(2)}(T - U)\mathrm{d}t = \Delta t \min \left[ \left\langle T \right\rangle-\left\langle U \right\rangle \right]\] 意味着动能的时间平均值尽可能等于势能的时间平均值.

当约束力对任意虚位移不做虚功时,将方程改写为: \[\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\partial L(t, q, \dot{q})}{\partial \dot{q}_k} \right)- \frac{\partial L(t, q, \dot{q})}{\partial q_k} = \lambda_a \frac{\partial G^a(q, t)}{\partial q_k}.\] 其中 \( k = 1, 2,\cdots, D.\)

\[I =\int_{t(1)}^{t(2)}L(q, \dot{q}, t)\mathrm{d}t = \int_{t(1)}^{t(2)}\left( p_k\dot{q}^k - H(q, p, t) \right)\mathrm{d}t.\] 求线积分的一阶变分. 认为在时间 \(t(1)\) 到 \(t(2)\) 之间, \(\delta p_k\) 和 \(\delta q_k\) 可以独立变化, 得到: 一阶变分为零, 积分取极值. 等价于满足如下哈密顿方程: \[\dot{q}_k = \frac{\partial H(q, p, t)}{\partial p_k}, \;\dot{p}_k =-\frac{\partial H(q, p, t)}{\partial q_k}.\]