J^+-like invariants and families of periodic orbits in the restricted three-body problem

  • Since Poincaré, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at obvious periodic orbits. On the other hand, given two periodic orbits one might ask if they are connected by an orbit cylinder, i.e., by a one-parameter family of periodic orbits. In this thesis we study this question for the planar circular restricted three-body problem. More precisely, we first consider periodic orbits gamma^{RKP} and alpha^{Euler} in the rotating Kepler problem resp. in the Euler problem: The rotating Kepler problem is obtained by letting the mass ratio in the restricted three-body problem go to zero. One gets the Euler problem from the restricted three-body problem by setting the rotating term equal to zero. We assume that gamma^{RKP} and alpha^{Euler} are connected with periodic orbits gamma^{3BP} and alpha^{3BP} of theSince Poincaré, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at obvious periodic orbits. On the other hand, given two periodic orbits one might ask if they are connected by an orbit cylinder, i.e., by a one-parameter family of periodic orbits. In this thesis we study this question for the planar circular restricted three-body problem. More precisely, we first consider periodic orbits gamma^{RKP} and alpha^{Euler} in the rotating Kepler problem resp. in the Euler problem: The rotating Kepler problem is obtained by letting the mass ratio in the restricted three-body problem go to zero. One gets the Euler problem from the restricted three-body problem by setting the rotating term equal to zero. We assume that gamma^{RKP} and alpha^{Euler} are connected with periodic orbits gamma^{3BP} and alpha^{3BP} of the PCR3BP through Stark-Zeeman homotopies, respectively. We then ask for obstructions to find orbit cylinders in PCR3BP from gamma^{3BP} and alpha^{3BP}. Our strategy is to compare their Cieliebak-Frauenfelder-van Koert invariants which are obstructions to the existence of an orbit cylinder. We will prove that if gamma^{RKP} and alpha^{Euler} are contractible, then the invariants of gamma^{3BP} and alpha^{3BP} do not coincide with each other. Consequently, there exist no orbit cylinders connecting these periodic orbits in the PCR3BP.show moreshow less

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Metadaten
Author:Seongchan KimORCiD
URN:urn:nbn:de:bvb:384-opus4-388165
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/38816
Advisor:Urs Frauenfelder
Type:Doctoral Thesis
Language:English
Year of first Publication:2018
Publishing Institution:Universität Augsburg
Granting Institution:Universität Augsburg, Mathematisch-Naturwissenschaftlich-Technische Fakultät
Date of final exam:2018/07/12
Release Date:2018/11/09
Tag:Eulerproblem
planar circular restricted three-body problem; rotating Kepler problem; Euler problem; dynamical system; periodic orbits
GND-Keyword:Kepler-Bewegung; Eingeschränktes Dreikörperproblem; Periodischer Orbit; Dynamisches System
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Analysis und Geometrie
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht