Noether's theorem for higher-order variational problems of Herglotz type

TL;DR

This paper extends classical variational calculus by deriving generalized Euler-Lagrange equations, transversality conditions, and Noether's theorem for higher-order Herglotz-type problems using optimal control theory.

Contribution

It introduces a novel optimal control framework to analyze higher-order Herglotz variational problems, including new necessary conditions and symmetry results.

Findings

Derived generalized Euler-Lagrange equations for Herglotz problems

Established transversality and DuBois-Reymond conditions

Proved Noether's theorem for higher-order Herglotz variational problems

Abstract

We approach higher-order variational problems of Herglotz type from an optimal control point of view. Using optimal control theory, we derive a generalized Euler-Lagrange equation, transversality conditions, a DuBois-Reymond necessary optimality condition and Noether's theorem for Herglotz's type higher-order variational problems, valid for piecewise smooth functions.