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...
Збережено в:
| Дата: | 2014 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2014
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146540 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Who's Afraid of the Hill Boundary?/ R. Montgomery // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-146540 |
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irk-123456789-1465402019-02-10T01:25:02Z Who's Afraid of the Hill Boundary? Montgomery, R. 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. 2014 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/146540 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Montgomery, R. |
| spellingShingle |
Montgomery, R. Who's Afraid of the Hill Boundary? Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Montgomery, R. |
| author_sort |
Montgomery, R. |
| title |
Who's Afraid of the Hill Boundary? |
| 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? |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:25:02Z |
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2023-05-20T17:25:02Z |
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