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

TL;DR

This paper establishes the second Euler-Lagrange necessary condition for variational problems on time scales, providing a new proof of the Noether theorem in this context, thus advancing the theoretical framework of calculus of variations.

Contribution

It introduces the second Euler-Lagrange equation on time scales and offers a simplified proof of the Noether theorem in this setting.

Findings

Proved the second Euler-Lagrange necessary condition on time scales.

Provided an alternative proof of the Noether theorem on time scales.

Enhanced the theoretical foundation of variational calculus on time scales.

Abstract

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].