Maple Symbolic Computation in the Calculus of Variations

TL;DR

This paper explores how Maple software can be effectively used to solve complex variational problems, including the Euler-Lagrange equations and classical problems like the brachistochrone, highlighting both its strengths and limitations.

Contribution

It demonstrates the application of Maple in solving nonlinear differential equations in the calculus of variations and reformulates classical problems within this computational framework.

Findings

Maple effectively solves Euler-Lagrange equations.

The brachistochrone problem is successfully addressed using Maple.

Reformulation of variational problems shows potential and limitations.

Abstract

It is the aim of this work to identify and illustrate the potential and weaknesses of the computer algebra system Maple in the area of the Calculus of Variations: a classical area of mathematics that studies the methods for finding maximum and minimum values of functionals of integral type. Variational problems are usually solved with the help of the necessary optimality conditions of Euler-Lagrange, which are, in general, nonlinear and difficult second order differential equations to be solved. We show how the computer algebra system Maple can be useful in the determination and resolution of these equations. We will also present the solution to the celebrated brachistochrone problem from the point of view of the calculus of variations and the Maple system, and a reformulation of the classical problem obtained by restricting the class of admissible functions.