Unstable motions in the three body problem
(English) The 3 Body Problem (3BP) models the motion of three bodies interacting via Newtonian gravitation. It is called restricted when one body has zero mass and the other two, the primaries, have strictly positive masses. In the region of the phase space where one body is far from the other two (...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/690046 |
| Acceso en línea: | http://hdl.handle.net/10803/690046 https://dx.doi.org/10.5821/dissertation-2117-402146 |
| Access Level: | acceso abierto |
| Palabra clave: | Àrees temàtiques de la UPC::Matemàtiques i estadística 517 |
| Sumario: | (English) The 3 Body Problem (3BP) models the motion of three bodies interacting via Newtonian gravitation. It is called restricted when one body has zero mass and the other two, the primaries, have strictly positive masses. In the region of the phase space where one body is far from the other two (the primaries for the restricted case) both models can be studied as a nearly integrable Hamiltonian system. This is the so-called hierarchical regime. The present thesis deals with the existence of unstable motions, in the 3BP and/or its restricted versions. More concretely, we analyze the existence of topological instability, non trivial hyperbolic sets and oscillatory motions (complete orbits which are unbounded but return infinitely often to some bounded region). On one hand, the existence of (a strong form of) topological instability in the N Body Problem was coined by Herman to be "the oldest question in dynamical systems". On the other hand, oscillatory motions are the unique type of complete motions for the 3BP which are not present in the integrable approximation. Their connection with the existence of non trivial hyperbolic sets have lead to the formulation of fundamental, yet unsolved, conjectures about their abundance.We establish the existence of Arnold diffusion, a robust mechanism leading to topological instability, in the Restricted 3BP for any value of the masses of the primaries. The transition chain leading to Arnold diffusion is built in the hierarchical region. We extend a previous result by Kaloshin, Delshams, De la Rosa and Seara, which applied to arbitrarily small mass ratio. Their setting, which exploits the trick, used by Arnold in his original paper, of making use of two perturbative parameters, lead to an a priori unstable model. In our setting, we face some of the challenges present in a priori stable systems.We present several results concerning the existence of oscillatory motions and non trivial hyperbolic sets in the restricted and non restricted 3 Body Problem. First, we develop new tools which blend geometric ideas with variational techniques to prove that there exist oscillatory motions in the restricted 3BP in a non nearly integrable regime. Second we show the existence of non trivial hyperbolic sets and oscillatory motions in the 3BP for all values of the masses. The non trivial hyperbolic set, contained in a subset of the hierarchical region where the inner bodies perform approximately circular motions, is associated to a transverse intersection between the stable and unstable manifolds of a Normally Hyperbolic Invariant Manifold. The existence of center directions complicates heavily both the analysis of existence of transverse intersections between these invariant manifolds and the construction of the horseshoe. The contribution of the author focuses on completing the first of these two steps.Finally, we study the existence of Arnold diffusion in the 3BP for all values of the masses. The robustness of the mechanism which we use to prove the existence of Arnold Diffusion in the Restricted 3BP implies that the obtained transition chain admits a continuation in the 3BP if one mass is sufficiently small. The substantial difference when the masses are fixed is that one can construct a transition chain along which there is a significant exchange of momentum between the inner and outer bodies, resulting in a large change of the eccentricity of the inner bodies. This requires considerably more work than in our construction of the transition chain in the Restricted 3BP and our construction of hyperbolic sets for the 3BP. The first step towards establishing this result, which constitutes the subject of the last chapter of this thesis, is the analysis of the so called Melnikov approximation associated to the aforementioned transition chain. |
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