  We deal with direct and inverse problems of the calculus of variations on
arbitrary time scales. Firstly, using the Euler-Lagrange equation and the
strengthened Legendre condition, we give a general form for a variational
functional to attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.
