# American Institute of Mathematical Sciences

September  2010, 2(3): 243-263. doi: 10.3934/jgm.2010.2.243

## 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

Received  March 2010 Revised  August 2010 Published  November 2010

In this paper we discuss the relation between the unimodularity of a Lie algebroid $\tau_{A}: A \to Q$ and the existence of invariant volume forms for the dynamics of hamiltonian mechanical systems on the dual bundle $A$*. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.
Citation: Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243
##### References:
 [1] R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings, Reading, Massachusetts, 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), 323-351. doi: 10.1017/S0305004101005679. [3] T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. 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), 417-436. 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. [6] P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230. 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), 1415-22. 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 69-70 (Russian); English trans.: Funct. Anal. Appl., 22 (1988), 58-59. doi: 10.1007/BF01077727. [9] V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regular and Chaotic Dynamics, 7 (2002), 161-176. 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), R241-R308. doi: 10.1088/0305-4470/38/24/R01. [11] A. Lewis, Reduction of simple mechanical systems, Mechanics and symmetry seminars, University of Warwick, 1997, http://www.mast.queensu.ca/~andrew/notes/abstracts/1997a.html. [12] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geometry, 12 (1977), 253-300. [13] K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series 213, Cambridge University Press, 2005. [14] E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259. [15] J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, 1994. [16] J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. [17] J. P. Ostrowski, "The Mechanics and Control of Undulatory Robotic Locomotion," PhD Thesis, California Institute of Technology, 1995. [18] A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231. [19] A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394. 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), 1793-1807. doi: 10.1088/0951-7715/16/5/313.

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##### References:
 [1] R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings, Reading, Massachusetts, 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), 323-351. doi: 10.1017/S0305004101005679. [3] T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. 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), 417-436. 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. [6] P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230. 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), 1415-22. 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 69-70 (Russian); English trans.: Funct. Anal. Appl., 22 (1988), 58-59. doi: 10.1007/BF01077727. [9] V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regular and Chaotic Dynamics, 7 (2002), 161-176. 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), R241-R308. doi: 10.1088/0305-4470/38/24/R01. [11] A. Lewis, Reduction of simple mechanical systems, Mechanics and symmetry seminars, University of Warwick, 1997, http://www.mast.queensu.ca/~andrew/notes/abstracts/1997a.html. [12] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geometry, 12 (1977), 253-300. [13] K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series 213, Cambridge University Press, 2005. [14] E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259. [15] J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, 1994. [16] J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. [17] J. P. Ostrowski, "The Mechanics and Control of Undulatory Robotic Locomotion," PhD Thesis, California Institute of Technology, 1995. [18] A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231. [19] A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394. 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), 1793-1807. doi: 10.1088/0951-7715/16/5/313.
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