Calculus of Variation (4) - Functions with Several Dependent Variables

5. Functions with Several Dependent Variables

 

$ f=f(y_{1}(x),y_{1}'(x),y_{2}(x),y_{2}'(x), ... ;x) $

$ \Rightarrow f=f(y_{i}(x),y_{i}'(x);x), i=1,2,...,m $

   $ y_{i}(\alpha ,x)=y_{i}(0,x)+\alpha \eta _{i}(x) $

 

$ \frac{\partial J}{\partial \alpha }=\int_{x_{1}}^{x_{2}}\sum_{i}(\frac{\partial f}{\partial y_{i}}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y_{i}'})\eta _{i}(x) dx=0 $

$ \therefore \frac{\partial f}{\partial y_{i}}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y_{i}'}=0  (i=1,2,...,m) $