Any hint to solve Euler–Lagrange equation with Lagrangian $L=\sqrt{\frac{\dot{x}}{1-x^{2}}+(1-x^2)\dot{y}}$?

Question

I encounter it in a textbook with no solution. I calculated it and got a very complicated equation and cannot derive a decoupled first order set of equations from them. I would appreciate any hint about it. Thanks in advance.

Answer 1

First we have

$$\begin{cases}\frac{\partial L}{\partial y} = 0 \\ \frac{\partial L}{\partial \dot{y}} = \frac{(1-x^2)}{2L}\end{cases} \implies \frac{d}{dt}\left(\frac{1-x^2}{2L}\right)=0$$

or in other words, $2L = k(1-x^2)$ for some $k\in\Bbb{R}$. Then similarly we have

$$\frac{\partial L}{\partial \dot{x}} = \frac{1}{1-x^2}\cdot\frac{1}{2L} = \frac{1}{k(1-x^2)^2}\implies \frac{4x\dot{x}}{k(1-x^2)^3} - \frac{1}{2L}\left(\frac{2x\dot{x}}{(1-x^2)^2}-2x\dot{y}\right)=0$$

which simplifies to

$$\dot{x}+(1-x^2)\dot{y}=0 \implies \frac{dy}{dx} = \frac{1}{x^2-1}$$