Abstract:
Here is presented an analytic study on the stability of hyperbolic and equilibrium of a non-linear dynamical system. This study is applied to the planar, restricted three-body problem, which is an Hamiltonian system with two degrees of freedom and has, on the synodic system five equilibrium points: three colinear points and two triangular points. The triangular points can be hyperbolic or elliptic, depending on the mass ratio between the two more massive bodies. The stability of the elliptic points is studied by applying general theorems from the non-linear Hamiltonian systems with two degrees of freedom, namely the Birkhoff theorem and the Arnold-Moser theorem. The hyperbolic and colinear points instability results from general theorems of non-linear ordinary differential equations. The restricted three body problem is non-integrable, in general. As such, a numerical study of its orbits is presented, through the analysis of the eccentricity, the Poincaré sections and the Lyapunov exponents. These methods are applied to the study of asteroids orbits, in the vicinity of equilibrium points of the Sun-Jupiter system and, to the study of Polydeuces orbit of the Saturn-Dionne system.

Estabilidade do problema de três corpos restrito (pdf) »»