A practical proposal to obtain solutions of certain variational problems avoiding Euler formalism

U Filobello-Nino,H Vazquez-Leal,J Huerta-Chua,R.A Callejas-Molina,M.A Sandoval-Hernandez

doi:10.1016/j.heliyon.2020.e03703

Abstract

The aim of this article is to show the way to get both, exact and analytical approximate solutions for certain variational problems with moving boundaries but without resorting to Euler formalism at all, for which we propose two methods: the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) and the Moving Boundary Condition Employing Transversality Conditions (METC). It is worthwhile to mention that the first of them avoids the concept of transversality condition, which is basic for this kind of problems, from the point of view of the known Euler formalism. While it is true that the second method will utilize the above mentioned conditions, it will do through a systematic elementary procedure, easy to apply and recall; in addition, it will be seen that the Generalized Bernoulli Method (GBM) will turn out to be a fundamental tool in order to achieve these objectives.

Highlights

In this brief introduction the required aspects of the variational calculus for this work are presented, we will begin by exposing the case of problems with fixed boundaries and later on we will expose what concerns to variational problems with moving boundaries [1, 2, 3, 4]
The aim of this article is to show the way to get both, exact and analytical approximate solutions for certain variational problems with moving boundaries but without resorting to Euler formalism at all, for which we propose two methods: the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) and the Moving Boundary Condition Employing Transversality Conditions (METC)
Despite to the fact that some variational problems were solved through especial methods (like for instance, Bernoulli’s solution to the Brachistochrone problem, which is of particular interest for this work [6, 7] (see (4)), it was Euler who presented the variational calculus as a coherent branch of the analysis by discovering the basic differential equation for an extremization curve

Summary

Introduction

In this brief introduction the required aspects of the variational calculus for this work are presented, we will begin by exposing the case of problems with fixed boundaries and later on we will expose what concerns to variational problems with moving boundaries [1, 2, 3, 4]. Calculus of variations is relevant from theoretical point of view, but many laws of physics are expressed in terms of a variational principle. In this case, a certain functional has to reach its maximum or minimum value in the physical considered process. Despite to the fact that some variational problems were solved through especial methods (like for instance, Bernoulli’s solution to the Brachistochrone problem, which is of particular interest for this work [6, 7] (see (4)), it was Euler who presented the variational calculus as a coherent branch of the analysis by discovering the basic differential equation for an extremization curve. It is clear that the amplitude and relevance of the variational calculus justify the research on the subject; especially the one that contributes to facilitate both the variational problems formulation as well as the solution methods of these

Objectives

Discussion

Conclusion