The contact geometry of the spatial circular restricted 3-body problem
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The contact geometry of the spatial circular restricted 3‑body problem WanKi Cho1 · Hyojin Jung1 · GeonWoo Kim1 Received: 24 September 2019 © The Author(s), under exclusive licence to Mathematisches Seminar der Universität Hamburg 2020
Abstract We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range. Keywords Spatial circular restricted 3-body problem · Contact geometry · Celestial mechanics Mathematics Subject Classification 53D35 (70F07, 70G45)
1 Introduction In [2], it was proved that the regularized, planar, circular restricted three-body problem is of contact type for energies below and also slightly above the first critical value. Such a result is significant, because it enables the use of holomorphic curve techniques in this classical problem. These techniques have been of great importance in the understanding of the dynamics of Hamiltonian dynamical systems; we mention the work of Floer on the Arnold conjecture, of Hofer and Taubes on the Weinstein conjecture and of Hofer–Wysocki–Zehnder on global surfaces of section. For some of the original works and their improvements using holomorphic curves see [6, 8–10]. The purpose of this paper is to prove that the spatial, circular restricted three-body problem also has the contact property, hence enabling Floer theoretic techniques to be used in this problem. As a direct dynamical application, we prove that the spatial, circular restricted three-body problem does not undergo so-called blue sky catastrophes, Communicated by Janko Latschev. * Hyojin Jung [email protected] WanKi Cho [email protected] GeonWoo Kim [email protected] 1
Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea
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i.e., bifurcations where the period and length of a smooth family of periodic orbits blow up, in the energy range where the contact property holds. In the restricted three-body problem (RTBP from now on) we consider two massive primaries, which we will refer to as the Earth and Moon, respectively, and a massless satellite that interact with each other via Newtonian gravity. The Earth and Moon obey Keplerian two-body dynamics, which are well-understood: solutions are described by conic sections. The satellite is influenced by both the Earth and Moon, and this results in more complicated dynamics that are known to be chaotic for almost all mass ratios of the Earth and Moon. In the circular RTBP, we make the further assumption that the Earth and Moon rotate around each other in circular orbits. This results in an extra integral, the so-called Jacobi integral, which we will describe now. Consider the circular RTBP in a frame that is uniformly rotating with constant speed such that the Earth and Moon are fixed on
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