Is there any standard approach to solve the following kind of variational problem?
Maximize $F=\int_0 ^1 L(x,y,y')dx $ subject to the constraint $|\int_0 ^1 M(x,y,y')dx| \lt k$ where $y$ is the function of $x$ to be solved for.
I can think immediately of maximizing $\int_0 ^1 Ldx + \lambda(\int_0^1 Mdx-k)^2 $ but don't know whether that will yield the right solution. I can't seem to find this kind of problem solved in the usual references, e.g. Bruce Van Brunt's Calculus of Variations.