Análise qualitativa da integrabilidade de bilhares

Abstract

A Billiard consists basically of a particle confined in a region of space. We will just dealing Billiards in two dimensions in the absence of extern fields and any type of dissipative forces so that particle collisions with Billiard boundaries are elastic. Furthermore, the boundaries are fixed, ie, they obey an equation of kind g = g(r,θ) = g0, where r and θ are the polar coordinates in the plan and g0 is any constant, without time dependence. The Billiard is an interesting model for several reasons. First, it is a very simple (it have few degrees of freedom) and easy display system. However, it has a non-trivial dynamic with wealth behavior (may have regular, caotic behavior or even mixed, in which case in the phase space of a single Billiard coexist regular and caotic regions). Second, the numerical treatment of these systems does not require numerical integration of differential equations and, therefore, does not consume much runtime. In addition, the Billiards allows we make investigations of fundamental character, for example, we can study how regular systems react to being slightly disturbed.