Reduction of invariant constrained systems using anholonomic frames

Previous Article
Clebsch optimal control formulation in mechanics
JGM Home
This Issue
Next Article
Superposition rules and second-order Riccati equations

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

Abstract
Related Papers

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.
[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.
[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.
[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.
[5]	H. Cendra, J. E. Marsden and T. S. Ratiu, "Lagrangian Reduction by Stages,'' Memoirs of the American Mathematical Society 152, AMS 2001.
[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.
[7]	J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lecture Notes in Mathematics 1793, Springer, 2002.
[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.
[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.
[10]	M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dynamical Systems, 25 (2010), 159-187. doi: 10.1080/14689360903360888.
[11]	M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' LMS Lecture Notes 59, Cambridge University Press, 1988.
[12]	R. H. Cushman, H. Duistermaat and J. Śniatycki, "Geometry of Nonholonomically Constrained Systems,'' Advanced Series in Nonlinear Dynamics 26, World Scientific, 2010.
[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.
[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.
[15]	B. Jovanovic, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567. doi: 10.1088/0951-7715/14/6/308.
[16]	J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Arch. Rat. Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.
[17]	O. Krupková, Mechanical systems with non-holonomic constraints, J. Math. Phys., 38 (1997), 5098-5126. doi: 10.1063/1.532196.
[18]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Texts in Applied Mathematics 17, Springer, 1999.
[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.
[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.
[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.
[22]	J. I. Neĭmark and N. A. Fufaev, "Dynamics of Nonholonomic Systems,'' Transl. of Math. Monographs 33, AMS, 1972.
[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.
[24]	J. Vilms, Connections on tangent bundles, J. Diff. Geom., 1 (1967), 235-243.

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.
[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.
[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.
[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.
[5]	H. Cendra, J. E. Marsden and T. S. Ratiu, "Lagrangian Reduction by Stages,'' Memoirs of the American Mathematical Society 152, AMS 2001.
[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.
[7]	J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'' Lecture Notes in Mathematics 1793, Springer, 2002.
[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.
[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.
[10]	M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dynamical Systems, 25 (2010), 159-187. doi: 10.1080/14689360903360888.
[11]	M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' LMS Lecture Notes 59, Cambridge University Press, 1988.
[12]	R. H. Cushman, H. Duistermaat and J. Śniatycki, "Geometry of Nonholonomically Constrained Systems,'' Advanced Series in Nonlinear Dynamics 26, World Scientific, 2010.
[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.
[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.
[15]	B. Jovanovic, Geometry and integrability of Euler-Poincaré-Suslov equations, Nonlinearity, 14 (2001), 1555-1567. doi: 10.1088/0951-7715/14/6/308.
[16]	J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Arch. Rat. Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.
[17]	O. Krupková, Mechanical systems with non-holonomic constraints, J. Math. Phys., 38 (1997), 5098-5126. doi: 10.1063/1.532196.
[18]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,'' Texts in Applied Mathematics 17, Springer, 1999.
[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.
[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.
[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.
[22]	J. I. Neĭmark and N. A. Fufaev, "Dynamics of Nonholonomic Systems,'' Transl. of Math. Monographs 33, AMS, 1972.
[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.
[24]	J. Vilms, Connections on tangent bundles, J. Diff. Geom., 1 (1967), 235-243.

[1]	Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029
[2]	Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69
[3]	Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001
[4]	Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699
[5]	E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
[6]	Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49
[7]	Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
[8]	Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213
[9]	José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204
[10]	Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967
[11]	Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of discrete mechanical systems by stages. Journal of Geometric Mechanics, 2016, 8 (1) : 35-70. doi: 10.3934/jgm.2016.8.35
[12]	L. Bakker. A reducible representation of the generalized symmetry group of a quasiperiodic flow. Conference Publications, 2003, 2003 (Special) : 68-77. doi: 10.3934/proc.2003.2003.68
[13]	Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185
[14]	Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014
[15]	L. Búa, T. Mestdag, M. Salgado. Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory. Journal of Geometric Mechanics, 2015, 7 (4) : 395-429. doi: 10.3934/jgm.2015.7.395
[16]	Bertuel Tangue Ndawa. Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures. Journal of Geometric Mechanics, 2022 doi: 10.3934/jgm.2022006
[17]	Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[18]	Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
[19]	Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the existence of extended perfect binary codes with trivial symmetry group. Advances in Mathematics of Communications, 2009, 3 (3) : 295-309. doi: 10.3934/amc.2009.3.295
[20]	Carolin Kreisbeck. A note on $3$d-$1$d dimension reduction with differential constraints. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 55-73. doi: 10.3934/dcdss.2017003

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Mike Crampin Tom Mestdag
on Google Scholar
Mike Crampin Tom Mestdag

[Back to Top]