Who's Afraid of the Hill Boundary?
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2014 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2014
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146540 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Who's Afraid of the Hill Boundary?/ R. Montgomery // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. |
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Montgomery, R. 2019-02-09T21:00:43Z 2019-02-09T21:00:43Z 2014 Who's Afraid of the Hill Boundary?/ R. Montgomery // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J50; 58E10; 70H99; 37J45; 53B50 DOI:10.3842/SIGMA.2014.101 https://nasplib.isofts.kiev.ua/handle/123456789/146540 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. I thank Mark Levi and Mikhail Zhitomirskii for helpful e-mail conversations. I acknowledge NSF grant DMS-1305844 for support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Who's Afraid of the Hill Boundary? Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Who's Afraid of the Hill Boundary? |
| spellingShingle |
Who's Afraid of the Hill Boundary? Montgomery, R. |
| title_short |
Who's Afraid of the Hill Boundary? |
| title_full |
Who's Afraid of the Hill Boundary? |
| title_fullStr |
Who's Afraid of the Hill Boundary? |
| title_full_unstemmed |
Who's Afraid of the Hill Boundary? |
| title_sort |
who's afraid of the hill boundary? |
| author |
Montgomery, R. |
| author_facet |
Montgomery, R. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146540 |
| citation_txt |
Who's Afraid of the Hill Boundary?/ R. Montgomery // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. |
| work_keys_str_mv |
AT montgomeryr whosafraidofthehillboundary |
| first_indexed |
2025-12-07T16:27:22Z |
| last_indexed |
2025-12-07T16:27:22Z |
| _version_ |
1850867546600767488 |