Translations:Dictionary:Calculus of variations/11/es

$L(x+\delta x,{\dot {x}}+\delta {\dot {x}},t)=L(x,{\dot {x}},t)+\left[{\frac {\partial L}{\partial x}}\delta x+{\frac {\partial L}{\partial {\dot {x}}}}\delta {\dot {x}}\right]+{\frac {1}{2!}}\left\{{\frac {\partial ^{2}L}{\partial x^{2}}}\left(\delta x\right)^{2}+2{\frac {\partial ^{2}L}{\partial x\partial {\dot {x}}}}\delta x\delta {\dot {x}}+{\frac {\partial ^{2}L}{\partial {\dot {x}}^{2}}}\left(\delta {\dot {x}}\right)^{2}\right\}+O(\epsilon ^{3}).$