American Institute of Mathematical Sciences

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June  2012, 4(2): 137-163. doi: 10.3934/jgm.2012.4.137

Variational Integrators for Hamiltonizable Nonholonomic Systems

Oscar E. Fernandez 1, , Anthony M. Bloch 2, and  P. J. Olver 3,

1. 

Department of Mathematics, Wellesley College, Wellesley, MA 02482, USA Government
2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States
3. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  May 2011 Revised  October 2011 Published  August 2012

We report on new applications of the Poincaré and Sundman time-transformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these non-variational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results.
Keywords: Nonholonomic systems, Hamiltonization, variational integrators..
Mathematics Subject Classification: Primary: 37J60; Secondary: 34K2.
Citation: Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137
References:
[1]

A. M. Bloch, "Nonholonomic Mechanics and Control,'' Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003.  Google Scholar

[2]

A. M. Bloch, O. E. Fernandez and T. Mestdag, Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations, Rep. Math. Phys., 63 (2009), 225-249. doi: 10.1016/S0034-4877(09)90001-5.  Google Scholar

[3]

A. V. Borisov and I. S. Mamaev, Rolling of a rigid body on plane and sphere. Hierarchy of dynamics, Reg. Chaotic Dyn., 7 (2002), 177-200.  Google Scholar

[4]

A. V. Borisov and I. S. Mamaev, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Reg. Chaotic Dyn., 13 (2008), 443-490. doi: 10.1134/S1560354708050079.  Google Scholar

[5]

R. L. Burden and J. D. Faires, "Numerical Analysis,'' 8th edition, Thomson Brooks/Cole, Belmont, CA, 2005. Google Scholar

[6]

S. A. Chaplygin, On a ball's rolling on a horizontal plane, (in Russian), Mat. Sbornik, 24 (1903), 139-168; (in English), Reg. Chaotic Dyn., 7 (2002), 131-148.  Google Scholar

[7]

S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, (in Russian), Mat. Sbornik, 28 (1911), 303-314; (in English), Reg. Chaotic Dyn., 13 (2008), 369-376.  Google Scholar

[8]

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

[9]

J. Cortés Monforte and S. Martĺnez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. doi: 10.1088/0951-7715/14/5/322.  Google Scholar

[10]

Y. N. Fedorov and B. Jovanović, Quasi-Chaplygin systems and nonholonomic rigid body dynamics, Lett. Math. Phys., 76 (2006), 215-230. doi: 10.1007/s11005-006-0069-3.  Google Scholar

[11]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie Groups, Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017.  Google Scholar

[12]

O. E. Fernandez, "The Hamiltonization of Nonholonomic Systems and its Applications,'' Ph.D. Thesis, The University of Michigan, 2009.  Google Scholar

[13]

O. E. Fernandez and A. M. Bloch, The Weitzenböck connection and time reparameterization in nonholonomic mechanics, J. Math. Phys., 52 (2011), 012901, 18 pp. doi: 10.1063/1.3525798.  Google Scholar

[14]

O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data, J. Phys. A, 41 (2008), 344005, 20 pp. doi: 10.1088/1751-8113/41/34/344005.  Google Scholar

[15]

O. E. Fernandez, T. Mestdag and A. M. Bloch, A generalization of Chaplygin's reducibility theorem, Reg. Chaotic Dyn., 14 (2009), 635-655. doi: 10.1134/S1560354709060033.  Google Scholar

[16]

P. Fitzpatrick, "Advanced Calculus,'' 2nd edition, Thomson Brooks/Cole, Belmont, CA, 2006. Google Scholar

[17]

M. R. Flannery, The enigma of nonholonomic constraints, Am. J. of Phys., 73 (2005), 265-272. doi: 10.1119/1.1830501.  Google Scholar

[18]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A, 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6.  Google Scholar

[19]

E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), 219-227. doi: 10.1016/S0168-9274(97)00061-5.  Google Scholar

[20]

D. Iglesias, J. C. Marrero, D. M. de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 221-276. doi: 10.1007/s00332-007-9012-8.  Google Scholar

[21]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries, Discrete and Continuous Dynamical Systems, Series S, 3 (2010), 61-84.  Google Scholar

[22]

D. Korteweg, Über eine ziemlich verbreitete unrichtige Behandlungsweise eines Problemes der rollenden Bewegung und insbesondere Über kleine rollende Schwingungen um eine Gleichgewichtslage, Nieuw Archiefvoor Wiskunde, 4 (1899), 130-155. Google Scholar

[23]

B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics,'' Cambridge Monographs on Applied and Computational Mathematics, 14, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar

[24]

B. Leimkuhler and R. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Computational Phys., 112 (1994), 117-125. doi: 10.1006/jcph.1994.1085.  Google Scholar

[25]

M. Leok and J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numerical Analysis, 31 (2011), 1497-1532.  Google Scholar

[26]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'' 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.  Google Scholar

[27]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

[28]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1.  Google Scholar

[29]

T. Mestdag, A. M. Bloch and O. E. Fernandez, Hamiltonization and geometric integration of nonholonomic mechanical systems, in "Proc. 8th Nat. Congress on Theor. and Applied Mechanics," Brussels, Belgium, (2009), 230-236, arXiv:1105.5223. Google Scholar

[30]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, eds., "NIST Handbook of Mathematical Functions,'' U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge Univ. Press, Cambridge, MA, 2010.  Google Scholar

[31]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geometry and Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[32]

J. Ryckaert, G. Ciccotti and H. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Computational Phs., 23 (1977), 327-341. doi: 10.1016/0021-9991(77)90098-5.  Google Scholar

[33]

B. van Brunt, "The Calculus of Variations,'' Universitext, Springer-Verlag, New York, 2004.  Google Scholar

[34]

L. Verlet, Computer experiments on classical fluids, Phys. Rev., 159 (1967), 98-103. doi: 10.1103/PhysRev.159.98.  Google Scholar

show all references

References:
[1]

A. M. Bloch, "Nonholonomic Mechanics and Control,'' Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003.  Google Scholar

[2]

A. M. Bloch, O. E. Fernandez and T. Mestdag, Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations, Rep. Math. Phys., 63 (2009), 225-249. doi: 10.1016/S0034-4877(09)90001-5.  Google Scholar

[3]

A. V. Borisov and I. S. Mamaev, Rolling of a rigid body on plane and sphere. Hierarchy of dynamics, Reg. Chaotic Dyn., 7 (2002), 177-200.  Google Scholar

[4]

A. V. Borisov and I. S. Mamaev, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Reg. Chaotic Dyn., 13 (2008), 443-490. doi: 10.1134/S1560354708050079.  Google Scholar

[5]

R. L. Burden and J. D. Faires, "Numerical Analysis,'' 8th edition, Thomson Brooks/Cole, Belmont, CA, 2005. Google Scholar

[6]

S. A. Chaplygin, On a ball's rolling on a horizontal plane, (in Russian), Mat. Sbornik, 24 (1903), 139-168; (in English), Reg. Chaotic Dyn., 7 (2002), 131-148.  Google Scholar

[7]

S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, (in Russian), Mat. Sbornik, 28 (1911), 303-314; (in English), Reg. Chaotic Dyn., 13 (2008), 369-376.  Google Scholar

[8]

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

[9]

J. Cortés Monforte and S. Martĺnez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. doi: 10.1088/0951-7715/14/5/322.  Google Scholar

[10]

Y. N. Fedorov and B. Jovanović, Quasi-Chaplygin systems and nonholonomic rigid body dynamics, Lett. Math. Phys., 76 (2006), 215-230. doi: 10.1007/s11005-006-0069-3.  Google Scholar

[11]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie Groups, Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017.  Google Scholar

[12]

O. E. Fernandez, "The Hamiltonization of Nonholonomic Systems and its Applications,'' Ph.D. Thesis, The University of Michigan, 2009.  Google Scholar

[13]

O. E. Fernandez and A. M. Bloch, The Weitzenböck connection and time reparameterization in nonholonomic mechanics, J. Math. Phys., 52 (2011), 012901, 18 pp. doi: 10.1063/1.3525798.  Google Scholar

[14]

O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data, J. Phys. A, 41 (2008), 344005, 20 pp. doi: 10.1088/1751-8113/41/34/344005.  Google Scholar

[15]

O. E. Fernandez, T. Mestdag and A. M. Bloch, A generalization of Chaplygin's reducibility theorem, Reg. Chaotic Dyn., 14 (2009), 635-655. doi: 10.1134/S1560354709060033.  Google Scholar

[16]

P. Fitzpatrick, "Advanced Calculus,'' 2nd edition, Thomson Brooks/Cole, Belmont, CA, 2006. Google Scholar

[17]

M. R. Flannery, The enigma of nonholonomic constraints, Am. J. of Phys., 73 (2005), 265-272. doi: 10.1119/1.1830501.  Google Scholar

[18]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A, 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6.  Google Scholar

[19]

E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), 219-227. doi: 10.1016/S0168-9274(97)00061-5.  Google Scholar

[20]

D. Iglesias, J. C. Marrero, D. M. de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 221-276. doi: 10.1007/s00332-007-9012-8.  Google Scholar

[21]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries, Discrete and Continuous Dynamical Systems, Series S, 3 (2010), 61-84.  Google Scholar

[22]

D. Korteweg, Über eine ziemlich verbreitete unrichtige Behandlungsweise eines Problemes der rollenden Bewegung und insbesondere Über kleine rollende Schwingungen um eine Gleichgewichtslage, Nieuw Archiefvoor Wiskunde, 4 (1899), 130-155. Google Scholar

[23]

B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics,'' Cambridge Monographs on Applied and Computational Mathematics, 14, Cambridge Univ. Press, Cambridge, 2004.  Google Scholar

[24]

B. Leimkuhler and R. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Computational Phys., 112 (1994), 117-125. doi: 10.1006/jcph.1994.1085.  Google Scholar

[25]

M. Leok and J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numerical Analysis, 31 (2011), 1497-1532.  Google Scholar

[26]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'' 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.  Google Scholar

[27]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

[28]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1.  Google Scholar

[29]

T. Mestdag, A. M. Bloch and O. E. Fernandez, Hamiltonization and geometric integration of nonholonomic mechanical systems, in "Proc. 8th Nat. Congress on Theor. and Applied Mechanics," Brussels, Belgium, (2009), 230-236, arXiv:1105.5223. Google Scholar

[30]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, eds., "NIST Handbook of Mathematical Functions,'' U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge Univ. Press, Cambridge, MA, 2010.  Google Scholar

[31]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geometry and Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[32]

J. Ryckaert, G. Ciccotti and H. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Computational Phs., 23 (1977), 327-341. doi: 10.1016/0021-9991(77)90098-5.  Google Scholar

[33]

B. van Brunt, "The Calculus of Variations,'' Universitext, Springer-Verlag, New York, 2004.  Google Scholar

[34]

L. Verlet, Computer experiments on classical fluids, Phys. Rev., 159 (1967), 98-103. doi: 10.1103/PhysRev.159.98.  Google Scholar

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