10.6.1 一阶和二阶变分

利用一个可比较函数, 借助于被积函数的泰勒展开 (参见第 806 页 10.3.2) 对欧拉微分方程的推导,在 $\epsilon$ 的线性项后即停止:

$$\begin{array}{}\text{(10.53)}& I\left(\epsilon \right)={\int }_a^{b}F\left(x,y_0+\epsilon \eta ,y_0^{\prime }+\epsilon {\eta }^{\prime }\right)\mathrm{d}x.\end{array}$$

也考虑二次项, 即有

$$I\left(\epsilon \right)-I\left(0\right)=\epsilon {\int }_a^b\left[\frac{\partial F}{\partial y}\left(x,y_0,y_0^{\prime }\right)\eta +\frac{\partial F}{\partial y^{\prime }}\left(x,y_0,y_0^{\prime }\right){\eta }^{\prime }\right]\mathrm{d}x$$$$+\frac{{\epsilon }^2}{2}{\int }_a^b[\frac{{\partial }^{2}F}{\partial y^2}\left(x,y_0,y_0^{\prime }\right){\eta }^2+2\frac{{\partial }^{2}F}{\partial y\partial y^{\prime }}\left(x,y_0,y_0^{\prime }\right)\eta {\eta }^{\prime }$$$$\begin{array}{}\text{(10.54)}& +\frac{{\partial }^{2}F}{\partial y^{\prime 2}}\left(x,y_0,y_0^{\prime }\right){\eta }^{\prime 2}+O\left(\epsilon \right)]\mathrm{d}x.\end{array}$$

(1)泛函 $I\left[y\right]$ 的变分 $\delta I$ 用表达式

$$\begin{array}{}\text{(10.55)}& \delta I={\int }_a^b\left[\frac{\partial F}{\partial y}\left(x,y_0,y_0^{\prime }\right)\eta +\frac{\partial F}{\partial y^{\prime }}\left(x,y_0,y_0^{\prime }\right){\eta }^{\prime }\right]\mathrm{d}x\end{array}$$

表示.

(2)泛函 $I\left[y\right]$ 的变分 ${\delta }^{2}I$ 用表达式

$$\begin{array}{}\text{(10.56)}& {\delta }^{2}I={\int }_a^b\left[\frac{{\partial }^{2}F}{\partial y^2}\left(x,y_0,y_0^{\prime }\right){\eta }^2+2\frac{{\partial }^{2}F}{\partial y\partial y^{\prime }}\left(x,y_0,y_0^{\prime }\right)\eta {\eta }^{\prime }+\frac{{\partial }^{2}F}{\partial y^{\prime 2}}\left(x,y_0,y_0^{\prime }\right){\eta }^{\prime 2}\right]\mathrm{d}x\end{array}$$

表示, 因此可以写为

$$\begin{array}{}\text{(10.57)}& I\left(\epsilon \right)-I\left(0\right)\approx \epsilon \delta I+\frac{{\epsilon }^2}{2}{\delta }^{2}I.\end{array}$$

由于这些变分,可以形成泛函 $I\left[y\right]$ 的最优性条件 (见 [10.6]).