Euler-Lagrange方程
$\quad\\$
单变量单函数泛函取驻值
$\quad\\$
$$I(f)=\int_{x_1}^{x_2}L(x,f,f’,f’’,\cdots,f^{(k)})dx\\
\quad\\
\begin{aligned}
\delta I(f)&=I(f+\delta f)-I(f)\\
\quad\\
&=\int_{x_1}^{x_2}(\frac{\partial L}{\partial f}\delta f+\frac{\partial L}{\partial f’}\delta f’
+\frac{\partial L}{\partial f’’}\delta f’’+\cdots+\frac{\partial L}{\partial f^{(k)}}\delta f^{(k)})dx\\
\quad\\
&\qquad\qquad\qquad\qquad令\delta f(x)=\epsilon \eta(x)\\
\quad\\
\end{aligned}
\quad\\
驻值条件\\
\quad\\
\left.\frac{\partial I(f+\epsilon \eta)}{\partial \epsilon}\right|_{\epsilon=0}=\left.\frac{I(f+\epsilon \eta)-I(f)}{\epsilon}\right|_{\epsilon\to 0}=0\\
\quad\\
=\int_{x_1}^{x_2}(\frac{\partial L}{\partial f}\delta \eta+\frac{\partial L}{\partial f’}\delta \eta’
+\frac{\partial L}{\partial f’’}\delta \eta’’+\cdots+\frac{\partial L}{\partial f^{(k)}}\delta \eta^{(k)})dx=0\\
\quad\\
即\\
\quad\\
\int_{x_1}^{x_2}(\frac{\partial L}{\partial f}\delta f+\frac{\partial L}{\partial f’}\delta f’
+\frac{\partial L}{\partial f’’}\delta f’’+\cdots+\frac{\partial L}{\partial f^{(k)}}\delta f^{(k)})dx=0\\
\quad\\
分部积分\\
\quad\\
\begin{aligned}\int_{x_1}^{x_2}\frac{\partial L}{\partial f^{(k)}}\delta f^{(k)}dx&=\int_{x_1}^{x_2}\frac{\partial L}{\partial f^{(k)}}d(\delta f^{(k-1)})\\ \quad\\&=\left.\frac{\partial L}{\partial f^{(k)}}\delta f^{(k-1)}\right|_{x_1}^{x_2}
-\int_{x_1}^{x_2}\frac{d}{dx}(\frac{\partial L}{\partial f^{(k)}})\delta f^{(k-1)}dx\\
\quad\\
\end{aligned}\\
边界条件\\
\quad\\
f的k-1阶及更低阶变分在端点处为0\\
\quad\\
\begin{aligned}\int_{x_1}^{x_2}\frac{\partial L}{\partial f^{(k)}}\delta f^{(k)}dx&=-\int_{x_1}^{x_2}\frac{d}{dx}(\frac{\partial L}{\partial f^{(k)}})\delta f^{(k-1)}dx\\ \quad\\
&=\int_{x_1}^{x_2}(-1)^k\frac{d^k}{dx^k}(\frac{\partial L}{\partial f^{(k)}})\delta f\;dx\end{aligned}\\
\quad\\
故有\\
\quad\\
\int_{x_1}^{x_2}[\frac{\partial L}{\partial f}-\frac{d}{dx}(\frac{\partial L}{\partial f’})
+\frac{d^2}{dx^2}(\frac{\partial L}{\partial f’’})+\cdots+(-1)^k\frac{d^k}{dx^k}(\frac{\partial L}{\partial f^{(k)}})]\delta f\;dx\\
\quad\\
=0\\
\quad\\
因为\delta f任意\\
\quad\\
故有\\
\quad\\
\frac{\partial L}{\partial f}-\frac{d}{dx}(\frac{\partial L}{\partial f’})
+\frac{d^2}{dx^2}(\frac{\partial L}{\partial f’’})+\cdots+(-1)^k\frac{d^k}{dx^k}(\frac{\partial L}{\partial f^{(k)}})=0
\quad\\
$$
$\quad\\$
单变量多函数泛函取驻值
$\quad\\$
$$
I(f)=\int_{x_1}^{x_2}L(x,f_1,f_1’,f_1’’,\cdots,f_1^{(k)},\cdots,f_i,f_i’,f_i’’,\cdots,f_i^{(k)},\cdots)dx\\
\quad\\
\frac{\partial L}{\partial f_i}-\frac{d}{dx}(\frac{\partial L}{\partial f_i’})
+\frac{d^2}{dx^2}(\frac{\partial L}{\partial f_i’’})+\cdots+(-1)^k\frac{d^k}{dx^k}(\frac{\partial L}{\partial f_i^{(k)}})=0\\
\quad\\
对所有i均成立
$$
$\quad\\$
多变量单函数一阶微分泛函取驻值
$\quad\\$
$$
I(f)=\int_{x_{k1}}^{x_{k2}}\cdots\int_{x_{11}}^{x_{12}}(x_1,x_2,\cdots,x_k,f,f_{x_1}’,f_{x_2}’,\cdots,f_{x_k}’)\\
\quad\\dx_1\;dx_2\;\cdots\;dx_k\\
\quad\\
\frac{\partial L}{\partial f}-\sum\limits_{i=1}^{k}\frac{\partial}{\partial x_i}(\frac{\partial L}{\partial(\frac{\partial f}{\partial x_i})})=0
$$
参考与引用来源
梁昆淼力学第四版