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Symmetry, Integrability and Geometry: Methods and Applications, 2014, Volume 10, 101, 11 pp.
DOI: https://doi.org/10.3842/SIGMA.2014.101
(Mi sigma966)
 

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

Who's Afraid of the Hill Boundary?

Richard Montgomery

Math Dept. UC Santa Cruz, Santa Cruz, CA 95064, USA
Full-text PDF (469 kB) Citations (3)
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Abstract: The Jacobi–Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove the conjugate locus of any point near enough to the boundary is a hypersurface tangent to the boundary. Our method of proof is to reduce analysis of geodesics near the boundary to that of solutions to Newton's equations in the simplest model case: a constant force. This model case is equivalent to the beginning physics problem of throwing balls upward from a fixed point at fixed speeds and describing the resulting arcs, see Fig. 2.
Keywords: Jacobi–Maupertuis metric; conjugate points.
Received: August 25, 2014; in final form October 28, 2014; Published online November 2, 2014
Bibliographic databases:
Document Type: Article
Language: English
Citation: Richard Montgomery, “Who's Afraid of the Hill Boundary?”, SIGMA, 10 (2014), 101, 11 pp.
Citation in format AMSBIB
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Symmetry, Integrability and Geometry: Methods and Applications
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    Abstract page:150
    Full-text PDF :38
    References:58
     
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