Direct and Inverse Variational Problems on Time Scales: A Survey

TL;DR

This survey explores direct and inverse calculus of variations problems on arbitrary time scales, providing general conditions, new results for discrete and quantum cases, and addressing Helmholtz's problem.

Contribution

It offers a comprehensive overview of variational problems on time scales, including new results for discrete and quantum cases and general formulations for complex variational functionals.

Findings

Derived a general form for variational functionals with local minima.

Provided necessary conditions for Euler-Lagrange equations on time scales.

Obtained new results for discrete and quantum variational problems.

Abstract

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation (Helmholtz's problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.