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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

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, (1978).   Google Scholar

[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.  Google Scholar

[3]

T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.  doi: 10.2307/2001258.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras,, Funkt. Anal. Prilozh., 22 (1988), 69.  doi: 10.1007/BF01077727.  Google Scholar

[9]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics,, Regular and Chaotic Dynamics, 7 (2002), 161.  doi: 10.1070/RD2002v007n02ABEH000203.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series 213, 213 (2005).   Google Scholar

[14]

E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.  doi: 10.1023/A:1011965919259.  Google Scholar

[15]

J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry,", Texts in Applied Mathematics, 17 (1994).   Google Scholar

[16]

J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, 222 (2004).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (1978).   Google Scholar

[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.  Google Scholar

[3]

T. J. Courant, Dirac manifolds,, Trans. Amer. Math. Soc., 319 (1990), 631.  doi: 10.2307/2001258.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras,, Funkt. Anal. Prilozh., 22 (1988), 69.  doi: 10.1007/BF01077727.  Google Scholar

[9]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics,, Regular and Chaotic Dynamics, 7 (2002), 161.  doi: 10.1070/RD2002v007n02ABEH000203.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

K. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series 213, 213 (2005).   Google Scholar

[14]

E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.  doi: 10.1023/A:1011965919259.  Google Scholar

[15]

J. E. Marsden and T. Ratiu, "Introduction to Mechanics with Symmetry,", Texts in Applied Mathematics, 17 (1994).   Google Scholar

[16]

J. P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, 222 (2004).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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