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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2007 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2007
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| Online Zugang: | https://nasplib.isofts.kiev.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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862665911610114048 |
|---|---|
| author | Fedorov, Y.N. |
| author_facet | Fedorov, Y.N. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T15:18:41Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147819 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:18:41Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fedorov, Y.N. 2019-02-16T08:43:32Z 2019-02-16T08:43:32Z 2007 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 https://nasplib.isofts.kiev.ua/handle/123456789/147819 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. This paper is a contribution to the Vadim Kuznetsov Memorial Issue ‘Integrable Systems and Related Topics’. I am grateful to the anonymous referees whose remarks helped to improve the text. The research was partially supported by Spanish Ministry of Science and Technology grant BFM 2003-09504-C02-02. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Discretization of the Nonholonomic Chaplygin Sphere Problem Article published earlier |
| spellingShingle | A Discretization of the Nonholonomic Chaplygin Sphere Problem Fedorov, Y.N. |
| title | 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_short | A Discretization of the Nonholonomic Chaplygin Sphere Problem |
| title_sort | discretization of the nonholonomic chaplygin sphere problem |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147819 |
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