10.4.2 较一般的变分问题

这里考虑简单变分问题的两个推广.

1. $F=F\left(x,y,u\left(x,y\right),u_x,u_y,u_{xx},u_{xy},u_{yy}\right)$

泛函依赖于未知函数 $u\left(x,y\right)$ 的高阶导数. 如果出现的偏导数最高阶数为二,那么欧拉微分方程为

$$\frac{\partial F}{\partial u}-\frac{\partial }{\partial x}\left(\frac{\partial F}{\partial u_x}\right)-\frac{\partial }{\partial y}\left(\frac{\partial F}{\partial u_y}\right)+\frac{{\partial }^2}{\partial x^2}\left(\frac{\partial F}{\partial u_{xx}}\right)+\frac{{\partial }^2}{\partial x\partial y}\left(\frac{\partial F}{\partial u_{xy}}\right)+\frac{{\partial }^2}{\partial y^2}\left(\frac{\partial F}{\partial u_{yy}}\right)=0.$$

(10.50)

2. $F=F\left(x_1,x_2,\cdots ,x_n,u\left(x_1,\cdots ,x_n\right),u_{x_1},\cdots ,u_{x_n}\right)$

在有 $n$ 个自变量 $x_1,x_2,\cdots ,x_n$ 的情形,欧拉微分方程为

$$\begin{array}{}\text{(10.51)}& \frac{\partial F}{\partial u}-\sum _{k=1}^n\frac{\partial }{\partial x_k}\left(\frac{\partial F}{\partial u_{x_k}}\right)=0.\end{array}$$