Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, w...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2016 |
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Інститут математики НАН України
2016
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| Zitieren: | Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature / K. Rajaratnam, R.G. McLenaghan, C. Valero // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 41 назв. — англ. |
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Rajaratnam, K. McLenaghan, R.G. Valero, C. 2019-02-18T14:47:25Z 2019-02-18T14:47:25Z 2016 Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature / K. Rajaratnam, R.G. McLenaghan, C. Valero // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C15; 70H20; 53A60 DOI:10.3842/SIGMA.2016.117 https://nasplib.isofts.kiev.ua/handle/123456789/148531 We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems. This paper is a contribution to the Special Issue on Analytical Mechanics and Dif ferential Geometry in honour of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html. We would like to thank the referees for their helpful comments and suggestions. This work was supported in part by a QEII-Graduate Scholarship in Science and Technology (KR), Natural Sciences and Engineering Research Council of Canada Discovery Grant (RGM) and Undergraduate Student Research Award (CV). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature |
| spellingShingle |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature Rajaratnam, K. McLenaghan, R.G. Valero, C. |
| title_short |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature |
| title_full |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature |
| title_fullStr |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature |
| title_full_unstemmed |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature |
| title_sort |
orthogonal separation of the hamilton-jacobi equation on spaces of constant curvature |
| author |
Rajaratnam, K. McLenaghan, R.G. Valero, C. |
| author_facet |
Rajaratnam, K. McLenaghan, R.G. Valero, C. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148531 |
| citation_txt |
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature / K. Rajaratnam, R.G. McLenaghan, C. Valero // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 41 назв. — англ. |
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2025-12-07T13:32:20Z |
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2025-12-07T13:32:20Z |
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