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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146540 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862696476031844352 |
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| author | Montgomery, R. |
| author_facet | Montgomery, R. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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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| first_indexed | 2025-12-07T16:27:22Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146540 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:27:22Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | 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 |
| spellingShingle | Who's Afraid of the Hill Boundary? Montgomery, R. |
| title | 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_short | Who's Afraid of the Hill Boundary? |
| title_sort | who's afraid of the hill boundary? |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146540 |
| work_keys_str_mv | AT montgomeryr whosafraidofthehillboundary |