Estabilidade normal de sistemas hamiltonianos e aplicações ao problema restrito circular dos três corpos - Santos, Alessandro Hipólito dos (2022)
The main objective of this dissertation is to provide necessary and sufficient conditions for normal stability of a Hamiltonian system with n degrees of freedom and to apply them in the study of the stability of pound points L4 and L5 of the restricted circular spatial problem of the three bodies. To achieve our goal, we have included a chapter of foreplay, where we provide the basic theory of autonomous Hamiltonian systems with n degrees of freedom, understandings of simpleton transformations, providing definitions and relevant results, the concepts of equilibrium points of a system of ordinary differential equations, presenting the notion of stability in the sense of Lyapunov, as well as linear systems with constant coefficients, Lyapunov’s direct method for the study of stability and Lie normal form theory. In chapter 2 we discuss a new concept of stability called normal stability of linear Hamiltonian systems, where we have proved several necessary and sufficient conditions, among them a new condition on the quadratic part of the Hamiltonian function called the Moser-Weinstein condition. We present in the last chapter the restricted spatial circular problem of the three bodies, where we studied the linear stability of the pounding points L1, L2, L3, L4 and L5, and in the case of L4 and L5 we provide conditions for the normal stability of the linearized system, which implies a formal stability of the nonlinear system.