Find the extremals for the problem with the functional $$ I[x] = \int^1_0 [\dot{x}^2 - x^2 ] dt $$ and auxillary and boundary conditions $$ \int^1_0 \sqrt{1+\dot{x}^2} dt = \sqrt{2}\hspace{5pt} ,\hspace{5pt} x(0)=0 \hspace{5pt} ,\hspace{5pt}x(1) = 1$$ What I have done: Using $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} - \lambda \left[\frac{d}{dt}\left(\frac{\partial G}{\partial \dot{x}}\right) -\frac{\partial G}{\partial x} \right]=0$$ where $L$ is the functional and $G$ the auxillary condition, I get $$ 2\ddot{x}+2x - \lambda \left[ \frac{d}{dt}\left(\frac{\dot{x}}{\sqrt{1+\dot{x}^2}} \right) \right] =0 $$ Taking the derivative $$ 2\ddot{x}+2x - \lambda \left[ \ddot{x}(1+\dot{x})^{-\frac{1}{2}} - \dot{x}^2\ddot{x}(1+\dot{x})^{-\frac{3}{2}} \right] =0 $$ I don't know how to simplify this to be able to integrate. The best I could do was multiply through by $(1+\dot{x})^{\frac{3}{2}} $ giving $$ (2\ddot{x} + 2x)(1+\dot{x}^2)^{\frac{3}{2}} - \lambda\ddot{x}=0 $$ but all my attempts to simplify it have lead to it being way more complicated.


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