Contact geometry of Hill's approximation in a spatial restricted four-body problem

TL;DR

The paper proves that Hill's approximation in the spatial equilateral circular R4BP possesses the contact type property below the first critical value for all mass ratios $\mu\in[0,\tfrac{1}{2}]$. It achieves this by constructing a Liouville vector field transverse to the energy hypersurface, first in the non-regularized setting and then after Moser regularization, which interchanges position and momentum and uses a stereographic projection to $T^*S^3$. Consequently, the bounded, regularized Hill energy level is fiberwise starshaped and diffeomorphic to $S^*S^3$ with the standard contact structure, while the planar restriction corresponds to $S^*S^2$. This contact-type property opens the door to applying holomorphic curve methods and Floer theory to analyze Reeb dynamics and global surface-of-section questions in the Hill four-body context, mirroring results known for CR3BP and its planar/spatial variants.

Abstract

It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-García and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can "contact" the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.

Figures (3)

• Figure 1: Equilateral circular restricted four-body problem. Left: Case of $m_1 > m_2 > m_3$. Right: Case of $m_2 = m_3$ in a rotating frame of reference; $B_2$ and $B_3$ are located symmetrically with respect to $B_1$.
• Figure 2: The quantities $a$ (red), $b$ (green), $\lambda_1$, $\lambda_2$ (both blue) and $d$ (black).
• Figure 3: Hill's region (gray shaded domains) for planar problem $\{z=0\}$ for $\mu=0.2$. White domains correspond to forbidden regions. Red dots indicate $L_{1/2}$; blue dots indicate $L_{3/4}$. Right: For $c < H(L_{1/2})$. Left: For $H(L_{1/2}) < c < H(L_{3/4})$. In the Hill 3BP ($\mu=0$), when $L_{3/4}$ are sent to infinity, below the critical value the Hill's region consists of one bounded component and two unbounded components.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 2.1
• Definition 3.1
• Definition 3.2
• Example 3.3
• Proposition 3.4
• Lemma 3.5
• proof
• Corollary 3.6
• proof