  The fundamental problem of the calculus of variations on time scales concerns
the minimization of a delta-integral over all trajectories satisfying given
boundary conditions. In this paper we prove the second Euler-Lagrange necessary
optimality condition for optimal trajectories of variational problems on time
scales. As an example of application of the main result, we give an alternative
and simpler proof to the Noether theorem on time scales recently obtained in
[J. Math. Anal. Appl. 342 (2008), no. 2, 1220-1226].
