驻点

${\displaystyle \left.{\frac {dy}{dx}}\right|_{p}=0\,}$

靜態平衡系統

${\displaystyle \delta W=\sum _{i}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}=0\,}$

${\displaystyle \delta W=\sum _{i}F_{i}\delta q_{i}=0\,}$

${\displaystyle F_{i}=-{\frac {\partial V}{\partial q_{i}}}\,}$

${\displaystyle \delta W=\sum _{i}-{\frac {\partial V}{\partial q_{i}}}\delta q_{i}=-\delta V=0\,}$

歐拉-拉格朗日方程式

${\displaystyle \mathbf {y} (x)=(y_{1}(x),\ y_{2}(x),\ \ldots ,y_{N}(x))\,\!}$
${\displaystyle {\dot {\mathbf {y} }}(x)=({\dot {y}}_{1}(x),\ {\dot {y}}_{2}(x),\ \ldots ,\ {\dot {y}}_{N}(x))\,\!}$
${\displaystyle f(\mathbf {y} ,\ {\dot {\mathbf {y} }},\ x)=f(y_{1}(x),\ y_{2}(x),\ \ldots ,\ y_{N}(x),\ {\dot {y}}_{1}(x),\ {\dot {y}}_{2}(x),\ \ldots ,\ {\dot {y}}_{N}(x),\ x)\,\!}$

${\displaystyle \mathbf {y} (x)\in (C^{1}[a,\ b])^{N}\,\!}$ 使泛函${\displaystyle J(\mathbf {y} )=\int _{a}^{b}f(\mathbf {y} ,\ {\dot {\mathbf {y} }},\ x)dx\,\!}$ 取得局部平穩值，則在區間${\displaystyle (a,\ b)\,\!}$ 內對於所有的${\displaystyle i=1,\ 2,\ \ldots ,\ N\,\!}$ ，歐拉-拉格朗日方程式成立：

${\displaystyle {\frac {d}{dx}}{\frac {\partial }{\partial {\dot {y}}_{i}}}f(\mathbf {y} ,\ {\dot {\mathbf {y} }},\ x)-{\frac {\partial }{\partial y_{i}}}f(\mathbf {y} ,\ {\dot {\mathbf {y} }},\ x)=0\,\!}$

注释

1. ^ 考虑到这一点左右一阶导数符号不改变的情况
2. ^ 考慮到邊界條件