Euler-Lagrange equations for composition functionals in calculus of variations on time scales

Abstract

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H({\int }_a^{b}f(t,x^{\sigma }(t),x^{\Delta }(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.

Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-2010