A Scale Variational Principle of Herglotz

TL;DR

This paper extends the calculus of variations by developing a scale variational principle for the Herglotz problem, incorporating non-differentiable functions and scale derivatives to derive necessary optimality conditions.

Contribution

It introduces a novel scale variational framework for the Herglotz problem, including non-differentiable functions and multi-dimensional cases.

Findings

Derived necessary conditions for optimal solutions using scale derivatives

Extended the framework to transversality and higher-order derivatives

Addressed multi-dimensional and multi-variable problems

Abstract

The Herglotz problem is a generalization of the fundamental problem of the calculus of variations. In this paper, we consider a class of non-differentiable functions, where the dynamics is described by a scale derivative. Necessary conditions are derived to determine the optimal solution for the problem. Some other problems are considered, like transversality conditions, the multi-dimensional case, higher-order derivatives and for several independent variables.