Euler-Lagrage方程

\frac{\partial{g}}{\partial{x}}(x(t),\dot{x}(t),t)-\frac{\mathrm{d}}{\mathrm{d}t}[\frac{\partial{g}}{\partial{\dot{x}}}(x(t),\dot{x}(t),t)]=0

当g二阶连续可微时，进一步上式的后项展开，得到：

\begin{array}{l}
\frac{\partial g}{\partial x_{i}}(x(t), \dot{x}(t), t)-\sum_{j=1}^{n} \frac{\partial^{2} g}{\partial \dot{x}_{i} \partial x_{j}}(x(t), \dot{x}(t), t) \dot{x}_{j}(t) \\
\quad-\sum_{j=1}^{n} \frac{\partial^{2} g}{\partial \dot{x}_{i} \partial \dot{x}_{j}}(x(t), \dot{x}(t), t) \ddot{x}_{j}(t) \\
\quad-\frac{\partial^{2} g}{\partial \dot{x}_{i} \partial t}(x(t), \dot{x}(t), t)=0, \quad i=1,2, \ldots, n \\
\end{array}

这是一个二阶常微分方程组, 看起来就很难求解, 在以下几种特殊情况中，可以化简为简洁形式进行求解.

特殊情况

情况一

若g不显含\dot{x}:
EL方程可化简为:

\frac{\partial{g}}{\partial{x}}(x(t),t)=0

此时EL退化为普通方程.

情况二

若g不显含x, EL方程的前项为零, 可化简为:

\frac{\mathrm{d}}{\mathrm{d}t}[\frac{\partial{g}}{\partial{\dot{x}}}(\dot{x}(t),t)]=0

即:

\frac{\partial{g}}{\partial{\dot{x}}}(\dot{x}(t),t)=c_1 \space\space\space (c_1\in \mathbb{R})

例: 两点之间线段最短

J(x)=\int_{t_0}^{t_f}\sqrt{1+\dot{x}^2(t)}\mathrm{d}t

情况三

若g不显含t(常见):
EL方程可化简为:

\frac{\mathrm{d}}{\mathrm{d}t}[g(x(t),\dot{x}(t))-\frac{\partial{g}}{\partial{\dot{x}}}(x(t),\dot{x}(t))\cdot \dot{x}(t)]=0

即:

g(x(t),\dot{x}(t))-\frac{\partial{g}}{\partial{\dot{x}}}(x(t),\dot{x}(t))\cdot \dot x(t)=c_1 \space\space\space (c_1\in \mathbb{R})

例: 最速降线问题求解

T(y)=\int^{1}_{0}\frac{\sqrt{1+(\mathrm{d}y/\mathrm{d}x)^2}}{\sqrt{2gy}}\mathrm{d}x