Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes

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

Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes

Juan Carlos Marrero ^1,

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

Abstract
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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.

Keywords: Hamiltonian mechanical systems, linear Poisson structure, unimodularity, Lie algebroids, modular class, volume forms..

Mathematics Subject Classification: Primary: 70G45, 70G65, 70H05; Secondary: 17B66, 53D1.

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.

show all references

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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