Fractional Variational Principle of Herglotz

TL;DR

This paper integrates Herglotz's generalized variational principle with fractional calculus, deriving Euler-Lagrange equations, transversality conditions, and a Noether-type theorem for fractional non-conservative systems.

Contribution

It introduces a fractional variational principle of Herglotz, extending classical variational methods to fractional derivatives for non-conservative systems.

Findings

Derived Euler-Lagrange equations for fractional Herglotz problems

Established transversality conditions in the fractional context

Proved a fractional Noether-type theorem for conservation laws

Abstract

The aim of this paper is to bring together two approaches to non-conservative systems -- the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought, by differential equations that involve Caputo fractional derivatives. The Euler--Lagrange equations are obtained for the fractional variational problems of Herglotz-type and the transversality conditions are derived. The fractional Noether-type theorem for conservative and non-conservative physical systems is proved.