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

Detalhes bibliográficos
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/
Resumo: 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.
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spelling Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon systemVariedades invariantes em sistemas hamiltonianos com aplicações ao sistema Terra-LuaCaosChaosHamiltonian systemsInvariant manifoldsProblema de três corposSistemas hamiltonianosThree-body problemVariedades invariantesInvariant 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.Variedades invariantes são o esqueleto da dinâmica caótica em sistemas hamiltonianos. Em Mecânica Celeste, essas estruturas geométricas são aplicadas em uma multitude de problemas físicos e práticos, como na descrição do transporte natural de asteróides e na construção de trajetórias de satélites artificiais. Neste trabalho, nós focamos nossa investigação no movimento de um corpo com massa desprezível, o qual se move devido à atração gravitacional de ambas a Terra e a Lua. Como modelo, nós adotamos o problema planar, circular e restrito de três corpos, um sistema hamiltoniano quase-integrável com dois graus de liberdade, e consideramos uma situação onde todas as órbitas nos domínios da Terra e da Lua têm a liberdade de se moverem entre essas regiões porém estão presas dentro do sistema. Nós derivamos as equações de movimento do problema e explicamos, em detalhes, os métodos numéricos utilizados, desde a determinação de órbitas periódicas até o cálculo de variedades invariantes bidimensionais. Ao variar a constante de Jacobi, nós observamos que o sistema sofre uma transição partindo de um espaço de fases misto com um efeito de stickiness de longo alcance, para um cenário de caos global, e de volta para um espaço de fases misto, porém desta vez com um stickiness localizado. Durante este processo, as variedades das órbitas de Lyapunov se espalham pelo espaço de fases, exibindo uma relação próxima com o formato e a localização das regiões regulares, e também com o transporte de órbitas entre os domínios, enquanto que as variedades associadas a certas órbitas periódicas instáveis, formadas pela destruição da última curva KAM das regiões regulares, estão relacionadas ao comportamento do stickiness e, consequentemente, ao aprisionamento dinâmico das órbitas de trânsito. Nossos resultados fornecem uma descrição visual da influência das variedades invariantes nas propriedades dinâmicas do sistema Terra-Lua e podem contribuir para o entendimento da conexão entre dinâmica e geometria em sistemas hamiltonianos.Biblioteca Digitais de Teses e Dissertações da USPCaldas, Ibere LuizOliveira, Vitor Martins de2021-04-28info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/43/43134/tde-12052021-152814/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2021-05-20T20:27:04Zoai:teses.usp.br:tde-12052021-152814Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212021-05-20T20:27:04Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false
dc.title.none.fl_str_mv Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
Variedades invariantes em sistemas hamiltonianos com aplicações ao sistema Terra-Lua
title Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
spellingShingle Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
Oliveira, Vitor Martins de
Caos
Chaos
Hamiltonian systems
Invariant manifolds
Problema de três corpos
Sistemas hamiltonianos
Three-body problem
Variedades invariantes
title_short Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
title_full Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
title_fullStr Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
title_full_unstemmed Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
title_sort Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system
author Oliveira, Vitor Martins de
author_facet Oliveira, Vitor Martins de
author_role author
dc.contributor.none.fl_str_mv Caldas, Ibere Luiz
dc.contributor.author.fl_str_mv Oliveira, Vitor Martins de
dc.subject.por.fl_str_mv Caos
Chaos
Hamiltonian systems
Invariant manifolds
Problema de três corpos
Sistemas hamiltonianos
Three-body problem
Variedades invariantes
topic Caos
Chaos
Hamiltonian systems
Invariant manifolds
Problema de três corpos
Sistemas hamiltonianos
Three-body problem
Variedades invariantes
description 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.
publishDate 2021
dc.date.none.fl_str_mv 2021-04-28
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/doctoralThesis
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dc.rights.driver.fl_str_mv Liberar o conteúdo para acesso público.
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Liberar o conteúdo para acesso público.
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reponame_str Biblioteca Digital de Teses e Dissertações da USP
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