Calculus of Variation (6) - Delta Notation

7. Delta Notation

 

$ \frac{\partial J}{\partial \alpha }d\alpha =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\frac{\partial y}{\partial \alpha }d\alpha dx $

$ \Rightarrow \delta J=\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\delta y dx $ ($ \frac{\partial J}{\partial \alpha }d\alpha\equiv \delta J, \frac{\partial y}{\partial \alpha }d\alpha \equiv \delta y $)

 

extreme condition: $ \delta J=\delta \int_{x_{1}}^{x_{2}}f(y,y';x)dx=0 $

$ \delta J=\int_{x_{1}}^{x_{2}}\delta f dx $

   $ =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}\delta y+\frac{\partial f}{\partial y'}\delta y')dx $

   $ =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}\delta y+\frac{\partial f}{\partial y'}\frac{\mathrm{d} }{\mathrm{d} x}(\delta y))dx $ ($ \because \delta y'=\delta (\frac{\mathrm{d} y}{\mathrm{d} x})=\frac{\mathrm{d} }{\mathrm{d} x}(\delta y) $)

   $ =\int_{x_{1}}^{x_{2}}\frac{\partial f}{\partial y}\delta y dx+(\frac{\partial f}{\partial y'}\delta y-\int_{x_{1}}^{x_{2}}\delta y\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'} dx) $

   $ =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\delta y dx=0 $

$ \delta y $: arbitrary

$ \therefore \frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'}=0 $: Euler Equation