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

TL;DR

This paper develops Euler-Lagrange equations and optimality conditions for a class of calculus of variations problems involving composition functionals on arbitrary time scales, unifying discrete and continuous cases.

Contribution

It introduces a general framework for variational problems with composition functionals on time scales, deriving new Euler-Lagrange equations and boundary conditions.

Findings

Derived Euler-Lagrange equations for composition functionals on time scales.

Established natural boundary conditions and isoperimetric optimality conditions.

Provided multiple examples illustrating the application of the results.

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.