An Inverse Problem of the Calculus of Variations on Arbitrary Time Scales

TL;DR

This paper addresses an inverse extremal problem in the calculus of variations on arbitrary time scales, deriving a general form of functionals that achieve local minima using Euler-Lagrange and Legendre conditions.

Contribution

It introduces a novel approach to characterize variational functionals with prescribed extremal properties on arbitrary time scales.

Findings

Derived a general form for variational functionals with local minima

Extended classical calculus of variations results to arbitrary time scales

Utilized Euler-Lagrange and Legendre conditions in the derivation

Abstract

We consider an inverse extremal problem for variational functionals on arbitrary time scales. Using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variational functional that attains a local minimum at a given point of the vector space.