Weakly asymptotically quasiperiodic solutions for time-dependent Hamiltonians with a view to celestial mechanics

Abstract

We consider the planar three-body problem perturbed by a celestial body modeled as a time-dependent perturbation that decays in time. We assume that the motion of the celestial body is given and is unbounded with a non-zero asymptotic velocity. We prove the existence of orbits converging in time to some motions that are ``close'' to the quasiperiodic solutions associated with the Hamiltonian of the planar three-body problem. The proof relies on an abstract theorem that contains a substantial portion of the mathematical complexities presented in this work. This theorem is flexible and can be applied to many other physical phenomena. It considers Hamiltonian vector fields that are the sum of two components. The first possesses quasiperiodic solutions, and the second decays polynomially fast as time tends to infinity. We prove the existence of orbits converging in time to some motions that are ``close'' to the quasiperiodic solutions associated with the unperturbed system. It generalizes a previous work where a stronger polynomial decay in time was considered, and solutions converging in time to the quasiperiodic orbits associated with the unperturbed system were proved. In the abstract theorem contained in the present paper, the too-weak decay in time of the perturbation strongly modifies the dynamic at infinity. This serious difficulty requires a deep modification of the proof. This new strategy relies on the application of a Nash-Moser implicit function theorem (the previous result was proved with the fixed point theorem) and the introduction of weak solutions (in this case, the orbits do not converge to the quasiperiodic solutions associated with the unperturbed system).