I asked this question in physics forum but got no reply and I thought maybe the math forum will be able to help. The question is:
For some free particle, use Galilean transformation to show that the solution of the Euler-Lagrange equation is the absolute minimum of the action.
I have several problems understanding the question. For one, I remember that the solution to these optimization problems comes from an extremal value of action, and not neccesarily minimal, could it be that the question is wrong? But if not, I think the direction is to assume there is a different path with smaller action, move to the particles frame of reference using the transformation and reach some contradiction, but so far I couldn't succeed.
Thanks in advance