How Do We Apply Boundary Conditions in the Euler-Lagrange Equation Derivation?

[homology]

So when deriving the Euler-Lagrange equation for the simple case (the integrand is just a function of x, y(x), and y'(x) where y is defined on [a,b]) we're interested in those functions which connect two points (x_0, y(x_0)) and (x_1, y(x_1)). But these functions don't form a function space in themselves. So are we just looking at all functions on [a,b] with continuous second derivatives and out of those, considering the ones which satisfy the boundary conditions we're interested in?

This must be the case, because in the derivation we then increment y(x) by an h(x) where h(x_0)=h(x_1)=0. This wouldn't make sense unless we were considering a larger space than just those functions satisfying the boundary conditions. Just let me know if I'm on the right track here.

Thanks,

Kevin

 

[fresh_42]

Yes. The variation principle violates the boundary conditions. We first solve the problem in general, obtaining a set of solutions, and the use the boundary conditions to pick the specific solution we are looking for.