Optimization: Lagrangian with two functions

**Find so that is minimized.**

My work so far

I set up the Euler-Lagrange equation for each of the functions, giving me a set of second order differential equations:

I tried putting the equations in matrix form and using Wolfram to calculate the eigenvalues and eigenvectors. But solving a 4x4 system with complex eigenvalues seems to be beyond the scope of my calculus of variations course. I suspect there is a simpler way to do it, something that can be done by hand.

According to my textbook, a first integral of a Lagrangian of several functions, not explicitly dependent on , is given by:

Using that property I obtained another equation:

I'm not sure what to do with this though. So basically I'm stuck.

Any help is appreciated!

Re: Optimization: Lagrangian with two functions

Re: Optimization: Lagrangian with two functions