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Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion
Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes
1. | Unidad asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (1978).
|
[2] |
F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems,, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323.
doi: 10.1017/S0305004101005679. |
[3] |
T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.
doi: 10.2307/2001258. |
[4] |
S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids,, Quart. J. Math. Oxford, 50 (1999), 417.
doi: 10.1093/qjmath/50.200.417. |
[5] |
Y. Fedorov, L. García-Naranjo and J. C. Marrero, Hamiltonian dynamics on skew-symmetric algebroids, unimodularity and preservation of volumes in nonholonomic mechanics,, in preparation., (). Google Scholar |
[6] |
P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids,, J. Algebra, 129 (1990), 194.
doi: 10.1016/0021-8693(90)90246-K. |
[7] |
B. Jovanovic, Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on $SO(4)$,, J. Phys. A: Math. Gen., 31 (1998), 1415.
doi: 10.1088/0305-4470/31/5/011. |
[8] |
V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras,, Funkt. Anal. Prilozh., 22 (1988), 69.
doi: 10.1007/BF01077727. |
[9] |
V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics,, Regular and Chaotic Dynamics, 7 (2002), 161.
doi: 10.1070/RD2002v007n02ABEH000203. |
[10] |
M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005).
doi: 10.1088/0305-4470/38/24/R01. |
[11] |
A. Lewis, Reduction of simple mechanical systems,, Mechanics and symmetry seminars, (1997). Google Scholar |
[12] |
A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées,, J. Differential Geometry, 12 (1977), 253.
|
[13] |
K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series 213, 213 (2005).
|
[14] |
E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.
doi: 10.1023/A:1011965919259. |
[15] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry,", Texts in Applied Mathematics, 17 (1994).
|
[16] |
J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, 222 (2004).
|
[17] |
J. P. Ostrowski, "The Mechanics and Control of Undulatory Robotic Locomotion,", PhD Thesis, (1995). Google Scholar |
[18] |
A. Weinstein, Lagrangian mechanics and groupoids,, Fields Inst. Comm., 7 (1996), 207.
|
[19] |
A. Weinstein, The modular automorphism group of a Poisson manifold,, J. Geom. Phys., 23 (1997), 379.
doi: 10.1016/S0393-0440(97)80011-3. |
[20] |
D. V. Zenkov and A. M. Bloch, Invariant measures of nonholonomic flows with internal degrees of freedom,, Nonlinearity, 16 (2003), 1793.
doi: 10.1088/0951-7715/16/5/313. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (1978).
|
[2] |
F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems,, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323.
doi: 10.1017/S0305004101005679. |
[3] |
T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.
doi: 10.2307/2001258. |
[4] |
S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids,, Quart. J. Math. Oxford, 50 (1999), 417.
doi: 10.1093/qjmath/50.200.417. |
[5] |
Y. Fedorov, L. García-Naranjo and J. C. Marrero, Hamiltonian dynamics on skew-symmetric algebroids, unimodularity and preservation of volumes in nonholonomic mechanics,, in preparation., (). Google Scholar |
[6] |
P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids,, J. Algebra, 129 (1990), 194.
doi: 10.1016/0021-8693(90)90246-K. |
[7] |
B. Jovanovic, Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on $SO(4)$,, J. Phys. A: Math. Gen., 31 (1998), 1415.
doi: 10.1088/0305-4470/31/5/011. |
[8] |
V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras,, Funkt. Anal. Prilozh., 22 (1988), 69.
doi: 10.1007/BF01077727. |
[9] |
V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics,, Regular and Chaotic Dynamics, 7 (2002), 161.
doi: 10.1070/RD2002v007n02ABEH000203. |
[10] |
M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005).
doi: 10.1088/0305-4470/38/24/R01. |
[11] |
A. Lewis, Reduction of simple mechanical systems,, Mechanics and symmetry seminars, (1997). Google Scholar |
[12] |
A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées,, J. Differential Geometry, 12 (1977), 253.
|
[13] |
K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series 213, 213 (2005).
|
[14] |
E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.
doi: 10.1023/A:1011965919259. |
[15] |
J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry,", Texts in Applied Mathematics, 17 (1994).
|
[16] |
J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, 222 (2004).
|
[17] |
J. P. Ostrowski, "The Mechanics and Control of Undulatory Robotic Locomotion,", PhD Thesis, (1995). Google Scholar |
[18] |
A. Weinstein, Lagrangian mechanics and groupoids,, Fields Inst. Comm., 7 (1996), 207.
|
[19] |
A. Weinstein, The modular automorphism group of a Poisson manifold,, J. Geom. Phys., 23 (1997), 379.
doi: 10.1016/S0393-0440(97)80011-3. |
[20] |
D. V. Zenkov and A. M. Bloch, Invariant measures of nonholonomic flows with internal degrees of freedom,, Nonlinearity, 16 (2003), 1793.
doi: 10.1088/0951-7715/16/5/313. |
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