The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

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].Comment: This work was partially presented at the Workshop in Control, Nonsmooth Analysis and Optimization, celebrating Francis Clarke's and Richard Vinter's 60th birthday, Porto, May 4-8, 2009. Submitted 26-May-2009; Revised 12-Jan-2010; Accepted 29-March-2010 in revised form; for publication in the European Journal of Contro

Behavior of solutions of a third-order dynamic equation on time scales

In this paper, we will establish some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equatio