基本形式
泛函定义： \[ \min_{y(x)} ~I(y)=\int_{x_1}^{x_2}F\Big(x,~y(x),~y'(x)\Big)\mathrm{d}x \] 泛函取极值的必要条件，即Euler-Lagrange方程： \[ \frac{\partial F}{\partial y}-\frac{\mathrm{d}}{\mathrm{d}x}\bigg(\frac{\partial F}{\partial y'}\bigg)=0 \]
具有多个因变量
泛函定义： \[ \min_{y(x)} ~I(y)=\int_{x_1}^{x_2}F\Big(x,~y_1,y_2,\cdots,y_n,~y'_1,y'_2,\cdots,y'_n\Big)\mathrm{d}x \] 泛函取极值的必要条件： \[ \frac{\partial F}{\partial y_i}-\frac{\mathrm{d}}{\mathrm{d}x}\bigg(\frac{\partial F}{\partial y'_i}\bigg)=0\qquad (i=1,2,3,\cdots,n) \]
具有高阶导数
泛函定义： \[ \min_{y(x)} ~I(y)=\int_{x_1}^{x_2}F\Big(x,~y,~y',~y'',\cdots,y^{(n)}\Big)\mathrm{d}x \] 泛函取极值的必要条件，即Euler-Poisson方程： \[ \frac{\partial F}{\partial y}-\frac{\mathrm{d}}{\mathrm{d}x}\bigg(\frac{\partial F}{\partial y'}\bigg)+\frac{\mathrm{d}^2}{\mathrm{d}x^2}\bigg(\frac{\partial F}{\partial y''}\bigg)-\cdots+(-1)^n\frac{\mathrm{d}^n}{\mathrm{d}x^n}\bigg(\frac{\partial F}{\partial y^{(n)}}\bigg)=0 \]