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Pseudo-holomorphic dynamics in the restricted three-body problem

Part of: Symplectic geometry, contact geometry Complex manifolds

Published online by Cambridge University Press:  05 December 2022

AGUSTIN MORENO
AGUSTIN MORENO*
Affiliation:
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton NJ, 08540, U.S.A. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany. e-mail: amoreno@ias.edu
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Abstract

In this paper, we identify the five dimensional analogue of the finite energy foliations introduced by Hofer–Wysocki–Zehnder for the study of three dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.

MSC classification

Primary: 70F07: Three-body problems 70F15: Celestial mechanics
Secondary: 53D30: Symplectic structures of moduli spaces 32Q65: Pseudoholomorphic curves 53D35: Global theory of symplectic and contact manifolds
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 663 - 693
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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