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Second-kind symmetric periodic orbits for planar perturbed Kepler problems and applications

Part of: Qualitative theory

Published online by Cambridge University Press:  19 May 2023

Angelo Alberti Open the ORCID record for Angelo Alberti [Opens in a new window]  and
Claudio Vidal Open the ORCID record for Claudio Vidal [Opens in a new window]
Angelo Alberti
Affiliation:
Departamento de Matemática, Universidade Federal de Sergipe, Cidade Universitária Prof. José Aloísio de Campos, Jardim Rosa Elze, São Cristovão-SE, Brasil ([email protected])
Claudio Vidal
Affiliation:
Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, VIII Región, Chile ([email protected])
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Abstract

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

Keywords

periodic orbits continuationsecond-kind periodic orbitsDelaunay variablessymmetric periodic solutionsstabilitygeneralized Størmer problemstark and quadratic Zeeman effectanisotropic Seeligers two-body problem

MSC classification

Primary: 34C14: Symmetries, invariants
Secondary: 34C25: Periodic solutions 70F05: Two-body problems 70F15: Celestial mechanics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 3 , June 2024 , pp. 961 - 992
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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