A Discretization of the Nonholonomic Chaplygin Sphere Problem
The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete...
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| Datum: | 2007 |
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| Sprache: | English |
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Інститут математики НАН України
2007
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| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/147819 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Discretization of the Nonholonomic Chaplygin Sphere Problem / Y.N. Fedorov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1478192019-02-17T01:23:41Z A Discretization of the Nonholonomic Chaplygin Sphere Problem Fedorov, Y.N. The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the n-dimensional generalization of the Chaplygin sphere problem, which preserves the same first integrals as the continuous model, except the energy. We then study the discretization of the classical 3-dimensional problem for a class of special initial conditions, when an analog of the energy integral does exist and the corresponding map is given by an addition law on elliptic curves. The existence of the invariant measure in this case is also discussed. 2007 Article A Discretization of the Nonholonomic Chaplygin Sphere Problem / Y.N. Fedorov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 20 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J60; 37J35; 70H45 http://dspace.nbuv.gov.ua/handle/123456789/147819 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the n-dimensional generalization of the Chaplygin sphere problem, which preserves the same first integrals as the continuous model, except the energy. We then study the discretization of the classical 3-dimensional problem for a class of special initial conditions, when an analog of the energy integral does exist and the corresponding map is given by an addition law on elliptic curves. The existence of the invariant measure in this case is also discussed. |
| format |
Article |
| author |
Fedorov, Y.N. |
| spellingShingle |
Fedorov, Y.N. A Discretization of the Nonholonomic Chaplygin Sphere Problem Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Fedorov, Y.N. |
| author_sort |
Fedorov, Y.N. |
| title |
A Discretization of the Nonholonomic Chaplygin Sphere Problem |
| title_short |
A Discretization of the Nonholonomic Chaplygin Sphere Problem |
| title_full |
A Discretization of the Nonholonomic Chaplygin Sphere Problem |
| title_fullStr |
A Discretization of the Nonholonomic Chaplygin Sphere Problem |
| title_full_unstemmed |
A Discretization of the Nonholonomic Chaplygin Sphere Problem |
| title_sort |
discretization of the nonholonomic chaplygin sphere problem |
| publisher |
Інститут математики НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147819 |
| citation_txt |
A Discretization of the Nonholonomic Chaplygin Sphere Problem / Y.N. Fedorov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 20 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-11T02:54:12Z |
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2025-07-11T02:54:12Z |
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1837317439247876096 |