Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system

• Autor(a) principal: Oliveira, Vitor Martins de
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/43/43134/tde-12052021-152814/
• topic: Caos; Chaos; Hamiltonian systems; Invariant manifolds; Problema de três corpos; Sistemas hamiltonianos; Three-body problem; Variedades invariantes

Invariant manifolds are the skeleton of the chaotic dynamics in Hamiltonian systems. In Celestial Mechanics, these geometrical structures are applied to a multitude of physical and practical problems, such as to the description of the natural transport of asteroids, and to the construction of trajectories for artificial satellites. In this work, we focus our investigation on the motion of a body with negligible mass that moves due to the gravitational attraction of both the Earth and the Moon. As a model, we adopt the planar circular restricted three-body problem, a near-integrable Hamiltonian system with two degrees of freedom, and consider a situation where all orbits inside the Earth\'s or the Moon\'s realm are free to move between these regions but are bounded within the system. We derive the equations of motion for the problem and explain in detail all the numerical procedures that are carried out, from the determination of periodic orbits to the calculation of two-dimensional invariant manifolds. By varying the Jacobi constant of motion, we observe that the system undergoes a transition from a mixed phase space with a far-reaching stickiness effect, to a global chaos scenario, and back to a mixed phase space, although now with localized stickiness. During this process, the Lyapunov orbit manifolds spread throughout the phase space, displaying a close relationship with the shape and location of regular regions, and also with the transport of orbits between the realms, while the invariant manifolds associated with certain unstable periodic orbits, formed by the destruction of the last KAM torus of the regular regions, are related to the behavior of stickiness and, consequently, to dynamically trapping transit orbits. Our results provide a visual description of the influence of invariant manifolds in the dynamical properties of the Earth-Moon system and could contribute to the understanding of the connection between dynamics and geometry in Hamiltonian systems.