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!
Great, thanks! That was just what I needed.
I first solved for , then differentiated twice (and divided by two) to get . Using the boundary conditions, I obtained the four constants from a set of four equations. The final solution was:
,
where
I still used Wolfram to solve the first system, but the solution was much shorter and easier to work with