Computação Algébrica no Cálculo das Variações: Determinação de Simetrias e Leis de Conservação

Abstract

English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking, nonlinear, and very hard to solve. One way to address the problem is to obtain conservation laws of lower order than those of the corresponding Euler-Lagrange equations. While in Physics and Economics the question of existence of conservation laws is treated in a rather natural way, because the application itself suggest the conservation laws (e.g., conservation of energy, income/health law), from a strictly mathematical point of view, given a problem of the calculus of variations, it is not obvious how one might derive a conservation law or, for that matter, if it even has a conservation law. The question we address is thus to develop computational facilities, based on a systematic method, which permits to identify functionals that have conservation laws. The central result we use is the celebrated Noether's theorem. This theorem links conservation laws with the invariance properties of the problem (with symmetries), and provides an algorithm for finding conservation laws. Thus the problem is reduced to the one of finding the variational symmetries. We show how a Computer Algebra System can help to find the symmetries and the conservation laws in the calculus of variations. Several illustrative examples are given.