Minimum calculus of variation 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
 A: You are searching for sufficient conditions for a minimum. The Euler-Lagrange equations are just necessary conditions. They can give you each, maximum or minimum. You can read about sufficient conditions in Gelfand and Fomin, Calculus of variations in chapters 5 and 6. Also in Giaquinta and Hildebrand, Calculus of Variations, Vol 1 in chapter 4. In the case you need to prove that you have a weak minimum or maximum you need to invoke the Jacobi contion, also called Legendre-Clebsh (if I'm not wrong, but you can verify it). That is a necessary condition for a weak minimum/maximum, and can be stated as follows:
Consider the variational problem 
$
\min \int_0^T F(t,x,\dot x)\; dt
$
where $x \in \mathbb{R}^n$, and $F \in \mathcal{C}^2$. Now, the solution of the Euler-Lagrange equation gives a weak minimum if the matrix  which components $\frac{\partial^2 F}{\partial \dot{x}^i \partial \dot{x}^j}$ is positive definite.
In other words you have to compute the "Hessian" of the lagrangian w.r.t. the time derivatives of your variables. Then verify if it is positive definite.
