Determinação de constantes de movimento polinomiais (polinômios de Darboux) em campos vetoriais no plano

Abstract

In this work we present a procedure for calculating Darboux polynomials of vector fields in the plane, i.e., we build a methodology to solve the problem: Find the polynomials p(x; y) that satisfy the equation D[p(x; y)] = p(x; y) q(x; y); where D is the differential operator defined by D := N(x; y) @x + M(x; y) @y (M and N polynomials) and q(x; y) is a polynomial2. The Darboux polynomials define algebraic solutions (algebraic invariant curves) for the rational first order ordinary differential equation (rational 1ODE) given by dy = M(x; y) dx N(x; y) These algebraic solutions are fundamental in obtaining the general solution of rational 1ODEs in closed form or, equivalently, in the construction of first integrals for the dynamical system given by dx = N(x; y) , dy = M(x; y): dt dt