无约束最小作用量推导欧拉拉格朗日方程随笔

我看到可以以Latex格式描述公式，我写个随笔尝试一下.

假设一个物理系统在\(position:A \to B\, , t:t_A \to t_B\)上有作用量：

\[S=\int_{t_A}^{t_B} L(q,\dot{q},t)\,dt \]

这里\(L(q,\dot{q})\)是拉格朗日函数，\(q,\dot{q}\)是广义坐标及广义速度；为了方便这里只取了一个维度的\(q\)，也就是\(q(t_A)=q_A,q(t_B)=q_B\).最小作用量原理描述真实路径\(\Gamma\)必须使\(S\)取极小值.

\[\min S\ \to \delta S=0 \]

取一个小的\(\delta q(t)\)，也有\(\delta \dot{q}(t)\)；由于所有路径必须满足\(A \to B\)，所以这个\(\delta q(t)\)必然不能改变\(t_A , t_B\)时刻的位置：

\[\delta q(t_A) = \delta q(t_B) = 0 \tag{1} \]

因此：

\[\begin{aligned} \delta S &= \int_{t_A}^{t_B} L(q+\delta q, \dot{q} + \delta \dot{q},t) \, dt-\int_{t_A}^{t_B} L(q,\dot{q},t)\,dt \\ &= \int_{t_A}^{t_B} L(q+\delta q, \dot{q} + \delta \dot{q},t)-L(q,\dot{q},t) \, dt \\ \end{aligned} \]

全微分得：

\[L(q+\delta q, \dot{q} + \delta \dot{q},t)-L(q,\dot{q},t)= \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot{q}}\delta \dot{q}+o(\delta q,\delta \dot{q})\]

多元函数取极小的必要条件为一阶偏导数为\(0\).所以取变分：

\[\begin{aligned} \delta S &= \int_{t_A}^{t_B} \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot{q}}\delta \dot{q}\,dt \\ &= \int_{t_A}^{t_B} \frac{\partial L}{\partial q}\delta q \, dt +\int_{t_A}^{t_B} \frac{\partial L}{\partial \dot{q}} \, d\delta q \\ &= \int_{t_A}^{t_B} \frac{\partial L}{\partial q}\delta q \, dt - \int_{t_A}^{t_B} \delta q \, d\frac{\partial L}{\partial \dot{q}} + \left. \frac{\partial L}{\partial \dot{q}}\delta q \right|_{t_A}^{t_B} \end{aligned} \tag{2} \]

良性函数有界\(|\frac{\partial L}{\partial \dot{q}}| < M,M \in \mathbb{R^+}\)；根据式\((1)\)有：

\[\left. \frac{\partial L}{\partial \dot{q}}\delta q \right|_{t_A}^{t_B} = \left. \frac{\partial L}{\partial \dot{q}}\right|_{t_B}\delta q(t_B) - \left. \frac{\partial L}{\partial \dot{q}}\right|_{t_A}\delta q(t_A) = 0 \]

代入式\((2)\)，变为：

\[\begin{aligned} \delta S &= \int_{t_A}^{t_B} \frac{\partial L}{\partial q}\delta q \, dt - \int_{t_A}^{t_B} \delta q \, d\frac{\partial L}{\partial \dot{q}} \\ &= \int_{t_A}^{t_B} \delta q (\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}) \, dt \\ \end{aligned} \tag{3} \]

极值对任意充分小的\(\delta q\)都有成立：

\[S + \delta S \geq S \]

式\((3)中\)由于\(\delta q\)的任意性，可知\(\delta S=0\)的必要条件是：

\[\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0 \tag{4} \]

式\((4)\)就是欧拉拉格朗日方程\((Euler\)-\(Lagrange \,\, equation)\)的形式.
如果系统有多维度\((q_1,q_2,...)\)，作用量：

\[S=\int_{t_A}^{t_B} L(t,q_1,\dot{q_1},q_2,\dot{q_2},...)\,dt \]

那么由于极值条件的独立性，式\((4)\)变成方程组：

\[\left\{ \begin{aligned} &\frac{\partial L}{\partial q_1} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q_1}} = 0 \\ &\frac{\partial L}{\partial q_2} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q_2}} = 0 \\ &... \end{aligned} \right. \]

这就是完整的欧拉拉格朗日方程组形式.