Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum.
it is about the following functional$$ \int_0^b \sqrt\frac{x'(t)^2+y'(t)^2}{y(t)} dt $$ Please do not tell me that one you could write this as a functional of only one function. I solved both Euler Lagrange equations for x and y and want to check now that my solution is an actual minimum.
Maybe this can be done with a second derivative but I do not know how this one would look like in this case