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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
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
Citation:
Richard Montgomery, “Who's Afraid of the Hill Boundary?”, SIGMA, 10 (2014), 101, 11 pp.
Linking options:
https://www.mathnet.ru/eng/sigma966 https://www.mathnet.ru/eng/sigma/v10/p101
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Abstract page: | 150 | Full-text PDF : | 38 | References: | 58 |
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