The calculus of variations on time scales is considered. We propose a new
approach to the subject that consists in applying a differentiation tool called
the contingent epiderivative. It is shown that the contingent epiderivative
applied to the calculus of variations on time scales is very useful: it allows
to unify the delta and nabla approaches previously considered in the
literature. Generalized versions of the Euler-Lagrange necessary optimality
conditions are obtained, both for the basic problem of the calculus of
variations and isoperimetric problems.