Halfway between Heaven and Hell
Richard Montgomery
Abstract
We pose several questions for the classical N-body problem inspired by connections between the virial equation and the Jacobi-Maupertuis formulationof mechanics. We answer some.
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Halfway between Heaven and Hell
Authors
Richard Montgomery
Abstract
We pose several questions for the classical N-body problem inspired by connections between the virial equation and the Jacobi-Maupertuis formulationof mechanics. We answer some.
Paper Structure (10 sections, 7 theorems, 11 equations, 2 figures)
This paper contains 10 sections, 7 theorems, 11 equations, 2 figures.
Table of Contents
Key Result
Theorem 1.1
Consider the Newtonian N-body problem at fixed negative energy $E = -h$. Let $q_0 \in \mathbb E$ be a point in the corresponding Hill region $\{U \ge h \}$. There exists an energy $E$ brake orbit passing through $q_0$.
Figures (2)
- Figure 1: A schematic of the Hill region, shaded with a collision brake orbit (green) and non-collision periodic orbit (red) indicated.
- Figure 2: (Courtesy of Rick Moeckel.) The Hill region $\{ U \ge 1\}$ projected onto three-body shape space resembles a plumbing fixture made of three pipes joined at the origin. The origin represents triple collision. The three rays issuing from the origin about which each pipe is centered represent the binary collision locus. For details regarding shape space and this picture see shape.
Theorems & Definitions (19)
- Theorem 1.1: Lost Theorem
- Theorem 1.2: Virial theorem, periodic version
- proof
- Theorem 1.3: Moeckel MoeckelLarge
- Theorem 1.4: Pollard's Virial theorem
- Remark 1.5
- Remark 1.6
- Remark 1.7
- Remark 1.8
- Definition 1.9
- ...and 9 more