Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$

Question

I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be interesting to know, that we are supposed to bring the solution to this Lagrangian in relation to Fata Morgana mirages, maybe this helps you somehow. Are there any further ways to construct a solution to the given Lagrangian?

Answer 1

Notice that your problem is equivalent to finding the area of the revolution of the graph of the function $y(x)$. You can find a complete worked solution here: http://mathworld.wolfram.com/MinimalSurfaceofRevolution.html

Answer 2

The solutions of the Euler equations for the variational problems associated with Lagrangians of the form $y(x)^\alpha(1+y'(x)^2)^{\frac 12}$ can be subsumed in the family of parametrised curves with parametrisation of the form $(F(t),f(t))$ where $F$ is a primitive of $f$ and the latter has the form $f(t) = p (\cos (d(t-t_0))^{\frac 1 d}$ for suitable parameters $p$ and $t_0$ where $d$ depends on $\alpha$ in a simple fashion. Many of the standard examples of explicit solutions to simple variational problems which can be found in standard texts on the calculus of variations fit into this scheme (see, for example, Weinstock "Calculus of Variations"). The deeper reason behind this fact is expounded in the arXiv paper 1102.1540.