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Regular and Chaotic Dynamics, 2017, Volume 22, Issue 6, Pages 688–699
DOI: https://doi.org/10.1134/S1560354717060077
(Mi rcd283)
 

This article is cited in 3 scientific papers (total in 3 papers)

The Hyperbolic Plane, Three-Body Problems, and Mnëv’s Universality Theorem

Richard Montgomery

Mathematics Department, University of California, Santa Cruz, Santa Cruz CA 95064
Citations (3)
References:
Abstract: We show how to construct the hyperbolic plane with its geodesic flow as the reduction of a three-problem whose potential is proportional to $I/\Delta^2$ where $I$ is the moment of inertia of this triangle whose vertices are the locations of the three bodies and $\Delta$ is its area. The reduction method follows [11]. Reduction by scaling is only possible because the potential is homogeneous of degree $-2$. In trying to extend the assertion of hyperbolicity to the analogous family of planar N-body problems with three-body interaction potentials we run into Mnëv's astounding universality theorem which implies that the extended assertion is doomed to fail.
Keywords: Jacobi–Maupertuis metric, reduction, Mnev’s Universality Theorem, three-body forces, Hyperbolic metrics.
Funding agency Grant number
National Science Foundation DMS-1305844
I thank NSF grant DMS-1305844 for essential support.
Received: 21.08.2017
Accepted: 27.10.2017
Bibliographic databases:
Document Type: Article
MSC: 70F10, 37N05, 70G45
Language: English
Citation: Richard Montgomery, “The Hyperbolic Plane, Three-Body Problems, and Mnëv’s Universality Theorem”, Regul. Chaotic Dyn., 22:6 (2017), 688–699
Citation in format AMSBIB
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\by Richard Montgomery
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\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 6
\pages 688--699
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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