Calculus of Variation (1) - Calculus of Variation

Calculus of Variation 카테고리는 <Classical Dynamics of Particles and Systems>5th (Marion)의 chapter 6 순서를 따라갑니다

 

1. Calculus of Variation

 

find an extreme solution (max or min)

$ J=\int_{x_{1}}^{x_{2}}f(y(x),y'(x);x)dx $ ($ y'\equiv \frac{\mathrm{d} y}{\mathrm{d} x} $ )

(x: independent variable,

 y: dependent variable)

 

$ y=y(\alpha ,x) $

$ \alpha =0 \rightarrow y=y(0,x)=y(x) $: function that yields an extreme for J

$ \Rightarrow y=y(\alpha ,x)=y(0,x)+\alpha \eta (x) $ ($ \eta (x_{1})=\eta (x_{2})=0 $: at the endpoint)

 

$ J(\alpha )=\int_{x_{1}}^{x_{2}}f(y(\alpha ,x),y'(\alpha ,x);x)dx $

$ \frac{\partial J}{\partial \alpha }|_{\alpha =0}=0 $: necessary condition (not sufficient)