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March  2015, 7(1): 43-80. doi: 10.3934/jgm.2015.7.43

On the discretization of nonholonomic dynamics in $\mathbb{R}^n$

Fernando Jiménez 1, and  Jürgen Scheurle 1,

1. 

Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen, Germany, Germany

Received  July 2014 Revised  February 2015 Published  March 2015

In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold $D$. Moreover, we show that any $D-$preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system. By means of discretizing the corresponding Lagrange-d'Alembert principle, we construct geometric integrators for the original nonholonomic system. We give precise conditions under which these integrators generate a discrete flow preserving the distribution $D$. Also, we derive corresponding consistency estimates. Finally, we carefully treat the example of the nonholonomic particle, showing how to discretize the equations of motion in a reasonable way, particularly regarding the nonholonomic constraints.
Keywords: discrete variational calculus, Nonholonomic mechanics, geometric integration, differential algebraic equations., ordinary differential equations, discretization as perturbation.
Mathematics Subject Classification: 34C15, 37J15, 37N05, 65P10, 70F2.
Citation: Fernando Jiménez, Jürgen Scheurle. On the discretization of nonholonomic dynamics in $\mathbb{R}^n$. Journal of Geometric Mechanics, 2015, 7 (1) : 43-80. doi: 10.3934/jgm.2015.7.43
References:
[1]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar

[2]

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

[3]

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

[4]

J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lecture Notes in Mathematics, (1793). doi: 10.1007/b84020. Google Scholar

[5]

J. Cortés and E. Martínez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365. Google Scholar

[6]

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

[7]

S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics,, Nonlinearity, 21 (2008), 1911. doi: 10.1088/0951-7715/21/8/009. Google Scholar

[8]

S. Ferraro, D. Iglesias and D. Martín de Diego, Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods,, Discrete and Continuous Dynamical Systems, (2009), 220. Google Scholar

[9]

S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator,, accepted by Nonlinearity, (). Google Scholar

[10]

B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos,, Memoirs of the American Mathematical Society, 119 (1996). doi: 10.1090/memo/0570. Google Scholar

[11]

J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations,, SIAM Journal on Numerical Analysis, 20 (1983), 732. doi: 10.1137/0720049. Google Scholar

[12]

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

[13]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, (2002). doi: 10.1007/978-3-662-05018-7. Google Scholar

[14]

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

[15]

Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics,, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61. Google Scholar

[16]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry,, Arch. Rational Mech. Anal., 118 (1992), 113. doi: 10.1007/BF00375092. Google Scholar

[17]

W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry,, Rep. Math. Phys., 42 (1998), 101. doi: 10.1016/S0034-4877(98)80007-4. Google Scholar

[18]

K. Hüper and F. Silva Leite, On the geometry of rolling and interpolating curves on $S^n$, $SO_n$ and Grassman manifolds,, Journal of Dynamical and Control Systems, 13 (2007), 467. doi: 10.1007/s10883-007-9027-3. Google Scholar

[19]

M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems,, J. Math. Phys., 37 (1996), 3389. doi: 10.1063/1.531571. Google Scholar

[20]

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

[21]

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

[22]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217. doi: 10.1007/BF02352494. Google Scholar

[23]

P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations,, Journal of Differential Equations, 109 (1994), 110. doi: 10.1006/jdeq.1994.1046. Google Scholar

[24]

P. J. Rabier and W. C. Rheinboldt, Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint,, Society for Industrial and Applied Mathematics, (1987). Google Scholar

[25]

S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations,, Circuits, 11 (1992). doi: 10.1007/BF01189230. Google Scholar

[26]

S. Reich, On a geometrical interpretation of differential-algebraic equations,, Circuits, 10 (1991), 343. doi: 10.1007/BF01187550. Google Scholar

[27]

W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds,, Mathematics of Computation, 43 (1984), 473. doi: 10.1090/S0025-5718-1984-0758195-5. Google Scholar

show all references

References:
[1]

A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar

[2]

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

[3]

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

[4]

J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lecture Notes in Mathematics, (1793). doi: 10.1007/b84020. Google Scholar

[5]

J. Cortés and E. Martínez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365. Google Scholar

[6]

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

[7]

S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics,, Nonlinearity, 21 (2008), 1911. doi: 10.1088/0951-7715/21/8/009. Google Scholar

[8]

S. Ferraro, D. Iglesias and D. Martín de Diego, Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods,, Discrete and Continuous Dynamical Systems, (2009), 220. Google Scholar

[9]

S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator,, accepted by Nonlinearity, (). Google Scholar

[10]

B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos,, Memoirs of the American Mathematical Society, 119 (1996). doi: 10.1090/memo/0570. Google Scholar

[11]

J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations,, SIAM Journal on Numerical Analysis, 20 (1983), 732. doi: 10.1137/0720049. Google Scholar

[12]

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

[13]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, (2002). doi: 10.1007/978-3-662-05018-7. Google Scholar

[14]

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

[15]

Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics,, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61. Google Scholar

[16]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry,, Arch. Rational Mech. Anal., 118 (1992), 113. doi: 10.1007/BF00375092. Google Scholar

[17]

W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry,, Rep. Math. Phys., 42 (1998), 101. doi: 10.1016/S0034-4877(98)80007-4. Google Scholar

[18]

K. Hüper and F. Silva Leite, On the geometry of rolling and interpolating curves on $S^n$, $SO_n$ and Grassman manifolds,, Journal of Dynamical and Control Systems, 13 (2007), 467. doi: 10.1007/s10883-007-9027-3. Google Scholar

[19]

M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems,, J. Math. Phys., 37 (1996), 3389. doi: 10.1063/1.531571. Google Scholar

[20]

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

[21]

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

[22]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217. doi: 10.1007/BF02352494. Google Scholar

[23]

P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations,, Journal of Differential Equations, 109 (1994), 110. doi: 10.1006/jdeq.1994.1046. Google Scholar

[24]

P. J. Rabier and W. C. Rheinboldt, Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint,, Society for Industrial and Applied Mathematics, (1987). Google Scholar

[25]

S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations,, Circuits, 11 (1992). doi: 10.1007/BF01189230. Google Scholar

[26]

S. Reich, On a geometrical interpretation of differential-algebraic equations,, Circuits, 10 (1991), 343. doi: 10.1007/BF01187550. Google Scholar

[27]

W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds,, Mathematics of Computation, 43 (1984), 473. doi: 10.1090/S0025-5718-1984-0758195-5. Google Scholar

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