American Institute of Mathematical Sciences

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March  2011, 3(1): 23-40. doi: 10.3934/jgm.2011.3.23

Reduction of invariant constrained systems using anholonomic frames

Mike Crampin 1, and  Tom Mestdag 1,

1. 

Department of Mathematics, Ghent University, Krijgslaan 281, S9, B-9000 Gent, Belgium, Belgium

Received  October 2010 Revised  April 2011 Published  April 2011

We analyze two reduction methods for nonholonomic systems that are invariant under the action of a Lie group on the configuration space. Our approach for obtaining the reduced equations is entirely based on the observation that the dynamics can be represented by a second-order differential equations vector field and that in both cases the reduced dynamics can be described by expressing that vector field in terms of an appropriately chosen anholonomic frame.
Keywords: reduction, anholonomic frames, nonholonomic constraints, symmetry group., Lagrangian system.
Mathematics Subject Classification: 34A26, 37J60, 70G45, 70H0.
Citation: Mike Crampin, Tom Mestdag. Reduction of invariant constrained systems using anholonomic frames. Journal of Geometric Mechanics, 2011, 3 (1) : 23-40. doi: 10.3934/jgm.2011.3.23
References:
[1]

A. M. Bloch with the collaboration of J. Baillieul, P. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control,'' Springer, 2003.  Google Scholar

[2]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[3]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasi-velocities and symmetries in nonholonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.  Google Scholar

[4]

F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Camb. Phil. Soc., 132 (2002), 323-351. doi: 10.1017/S0305004101005679.  Google Scholar

[5]

H. Cendra, J. E. Marsden and T. S. Ratiu, "Lagrangian Reduction by Stages,'' Memoirs of the American Mathematical Society 152, AMS 2001.  Google Scholar

[6]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, in "Mathematics Unlimited -- 2001 and Beyond'' (eds. B. Engquist and W. Schmid), Springer (2001), 221-273.  Google Scholar

[7]

J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lecture Notes in Mathematics 1793, Springer, 2002.  Google Scholar

[8]

M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901. doi: 10.1063/1.2885077.  Google Scholar

[9]

M. Crampin and T. Mestdag, Reduction and reconstruction aspects of second-order dynamical systems with symmetry, Acta Appl. Math., 105 (2009), 241-266. doi: 10.1007/s10440-008-9274-7.  Google Scholar

[10]

M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dynamical Systems, 25 (2010), 159-187. doi: 10.1080/14689360903360888.  Google Scholar

[11]

M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' LMS Lecture Notes 59, Cambridge University Press, 1988.  Google Scholar

[12]

R. H. Cushman, H. Duistermaat and J. Śniatycki, "Geometry of Nonholonomically Constrained Systems,'' Advanced Series in Nonlinear Dynamics 26, World Scientific, 2010.  Google Scholar

[13]

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.  Google Scholar

[14]

K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, "The Breadth of Symplectic Geometry'' (eds. J. E. Marsden and T. S. Ratiu), Birkhäuser, 2005, 75-116.  Google Scholar

[15]

B. Jovanovic, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567. doi: 10.1088/0951-7715/14/6/308.  Google Scholar

[16]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Arch. Rat. Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.  Google Scholar

[17]

O. Krupková, Mechanical systems with non-holonomic constraints, J. Math. Phys., 38 (1997), 5098-5126. doi: 10.1063/1.532196.  Google Scholar

[18]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Texts in Applied Mathematics 17, Springer, 1999.  Google Scholar

[19]

J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41 (2000), 3379-3429. doi: 10.1063/1.533317.  Google Scholar

[20]

T. Mestdag and M. Crampin, Invariant Lagrangians, mechanical connections and the Lagrange-Poincaré equations, J. Phys. A: Math. Theor., 41 (2008), 344015. doi: 10.1088/1751-8113/41/34/344015.  Google Scholar

[21]

T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, J. Phys. A: Math. Gen., 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011.  Google Scholar

[22]

J. I. Neĭmark and N. A. Fufaev, "Dynamics of Nonholonomic Systems,'' Transl. of Math. Monographs 33, AMS, 1972. Google Scholar

[23]

W. Sarlet, F. Cantrijn and D. J. Saunders, A geometrical framework for the study of non-holonomic Lagrangian systems, J. Phys. A: Math. Gen., 28 (1995), 3253-3268. doi: 10.1088/0305-4470/28/11/022.  Google Scholar

[24]

J. Vilms, Connections on tangent bundles, J. Diff. Geom., 1 (1967), 235-243.  Google Scholar

show all references

References:
[1]

A. M. Bloch with the collaboration of J. Baillieul, P. Crouch and J. E. Marsden, "Nonholonomic Mechanics and Control,'' Springer, 2003.  Google Scholar

[2]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[3]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasi-velocities and symmetries in nonholonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.  Google Scholar

[4]

F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Camb. Phil. Soc., 132 (2002), 323-351. doi: 10.1017/S0305004101005679.  Google Scholar

[5]

H. Cendra, J. E. Marsden and T. S. Ratiu, "Lagrangian Reduction by Stages,'' Memoirs of the American Mathematical Society 152, AMS 2001.  Google Scholar

[6]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, in "Mathematics Unlimited -- 2001 and Beyond'' (eds. B. Engquist and W. Schmid), Springer (2001), 221-273.  Google Scholar

[7]

J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lecture Notes in Mathematics 1793, Springer, 2002.  Google Scholar

[8]

M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901. doi: 10.1063/1.2885077.  Google Scholar

[9]

M. Crampin and T. Mestdag, Reduction and reconstruction aspects of second-order dynamical systems with symmetry, Acta Appl. Math., 105 (2009), 241-266. doi: 10.1007/s10440-008-9274-7.  Google Scholar

[10]

M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dynamical Systems, 25 (2010), 159-187. doi: 10.1080/14689360903360888.  Google Scholar

[11]

M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' LMS Lecture Notes 59, Cambridge University Press, 1988.  Google Scholar

[12]

R. H. Cushman, H. Duistermaat and J. Śniatycki, "Geometry of Nonholonomically Constrained Systems,'' Advanced Series in Nonlinear Dynamics 26, World Scientific, 2010.  Google Scholar

[13]

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.  Google Scholar

[14]

K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, "The Breadth of Symplectic Geometry'' (eds. J. E. Marsden and T. S. Ratiu), Birkhäuser, 2005, 75-116.  Google Scholar

[15]

B. Jovanovic, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567. doi: 10.1088/0951-7715/14/6/308.  Google Scholar

[16]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Arch. Rat. Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.  Google Scholar

[17]

O. Krupková, Mechanical systems with non-holonomic constraints, J. Math. Phys., 38 (1997), 5098-5126. doi: 10.1063/1.532196.  Google Scholar

[18]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Texts in Applied Mathematics 17, Springer, 1999.  Google Scholar

[19]

J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41 (2000), 3379-3429. doi: 10.1063/1.533317.  Google Scholar

[20]

T. Mestdag and M. Crampin, Invariant Lagrangians, mechanical connections and the Lagrange-Poincaré equations, J. Phys. A: Math. Theor., 41 (2008), 344015. doi: 10.1088/1751-8113/41/34/344015.  Google Scholar

[21]

T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, J. Phys. A: Math. Gen., 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011.  Google Scholar

[22]

J. I. Neĭmark and N. A. Fufaev, "Dynamics of Nonholonomic Systems,'' Transl. of Math. Monographs 33, AMS, 1972. Google Scholar

[23]

W. Sarlet, F. Cantrijn and D. J. Saunders, A geometrical framework for the study of non-holonomic Lagrangian systems, J. Phys. A: Math. Gen., 28 (1995), 3253-3268. doi: 10.1088/0305-4470/28/11/022.  Google Scholar

[24]

J. Vilms, Connections on tangent bundles, J. Diff. Geom., 1 (1967), 235-243.  Google Scholar

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