Constants of motion for Isoperimetric Variational Problems with Time Delay

TL;DR

This paper derives constants of motion for isoperimetric variational problems involving time delays, extending Noether's theorem to nonsmooth cases and providing new optimality conditions.

Contribution

It introduces a nonsmooth extension of Noether's theorem for delayed isoperimetric problems, establishing new conservation laws and optimality conditions.

Findings

Derived isoperimetric Euler-Lagrange and DuBois-Reymond conditions for delayed problems

Extended Noether's theorem to nonsmooth delayed variational problems

Identified new constants of motion for problems with time delay

Abstract

In the present work, we obtain the constants of motion for isoperimetric variational problems with time delay. We consider a constrained optimization problem where the Lagrangian function defining the functional depends on time delayed arguments. We prove the isoperimetric Euler--Lagrange and DuBois--Reymond type optimality conditions and, in order to investigate the constants of motion for this problem, we obtain a nonsmooth extension of Noether's symmetry theorem for isoperimetric variational problems with delayed arguments.