Fractional variational principle of Herglotz for a new class of problems with dependence on the boundaries and a real parameter

Ricardo Almeida,Natália Martins

doi:10.1063/5.0021373

Abstract

The fractional variational problem of Herglotz type for the case where the Lagrangian depends on generalized fractional derivatives, the free endpoints conditions, and a real parameter is studied. This type of problem generalizes several problems recently studied in the literature. Moreover, it allows us to unify conservative and non-conservative dynamical processes in the same model. The dependence of the Lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard Herglotz’s variational problem into a problem of a different nature.

Highlights

The variational problem of Herglotz was introduced by Herglotz in 193017 and constitutes an important generalization of the classical variational problem
The fractional variational problem of Herglotz type for the case where the Lagrangian depends on generalized fractional derivatives, the free endpoints conditions, and a real parameter is studied
We focus our attention to variational problems of Herglotz type in the context of fractional calculus

Summary

INTRODUCTION

The variational problem of Herglotz was introduced by Herglotz in 193017 and constitutes an important generalization of the classical variational problem. We will consider the more general case in which the Lagrangian depends on the free endpoint conditions and a real parameter, creating what we call non-standard fractional Herglotz’s variational problems. This new type of generalized variational problems allows us to obtain new results that generalize several important results recently published in the literature. For the importance of the Lagrangian dependence of a real parameter and the free endpoint conditions, we refer to physical problems and to economics problems, respectively This dependency is effective and transforms the standard fractional Herglotz’s variational problem into a problem of a different nature.

PRELIMINARIES

MAIN RESULTS

Herglotz fractional variational principle with time delay

Herglotz fractional variational principle with several independent variables

ILLUSTRATIVE EXAMPLES