Relative Critical Points
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures – symplectic, Poisson, or variational – generating dynamical systems from such...
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| Видавець: | Інститут математики НАН України |
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| Дата: | 2013 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2013
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149195 |
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| Цитувати: | Relative Critical Points / Lewis D. // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 53 назв. — англ. |
Репозиторії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149195 |
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dspace |
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irk-123456789-1491952019-02-20T01:25:18Z Relative Critical Points Lewis, D. Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures – symplectic, Poisson, or variational – generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems – the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids – and generalizations of these systems. 2013 Article Relative Critical Points / Lewis D. // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 53 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J15; 53D20; 58E09; 70H33 DOI: http://dx.doi.org/10.3842/SIGMA.2013.038 http://dspace.nbuv.gov.ua/handle/123456789/149195 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 |
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures – symplectic, Poisson, or variational – generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems – the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids – and generalizations of these systems. |
| format |
Article |
| author |
Lewis, D. |
| spellingShingle |
Lewis, D. Relative Critical Points Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Lewis, D. |
| author_sort |
Lewis, D. |
| title |
Relative Critical Points |
| title_short |
Relative Critical Points |
| title_full |
Relative Critical Points |
| title_fullStr |
Relative Critical Points |
| title_full_unstemmed |
Relative Critical Points |
| title_sort |
relative critical points |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149195 |
| citation_txt |
Relative Critical Points / Lewis D. // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 53 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT lewisd relativecriticalpoints |
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2023-05-20T17:32:12Z |
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2023-05-20T17:32:12Z |
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