# American Institute of Mathematical Sciences

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

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

 1 Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen, 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.
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, 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [2] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365. [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), 323-351. doi: 10.1017/S0305004101005679. [4] J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, 1793, Springer-Verlag, Berlin, 2002. doi: 10.1007/b84020. [5] J. Cortés and E. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. [6] 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. [7] S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928. doi: 10.1088/0951-7715/21/8/009. [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), 220-229. [9] S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator,, accepted by Nonlinearity, (). [10] B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos, Memoirs of the American Mathematical Society, 119 (1996), viii+79 pp. doi: 10.1090/memo/0570. [11] J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations, SIAM Journal on Numerical Analysis, 20 (1983), 732-746. doi: 10.1137/0720049. [12] Z. Ge and J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A., 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6. [13] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7. [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), 221-276. doi: 10.1007/s00332-007-9012-8. [15] Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61-81. [16] J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092. [17] W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134. doi: 10.1016/S0034-4877(98)80007-4. [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), 467-502. doi: 10.1007/s10883-007-9027-3. [19] M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414. doi: 10.1063/1.531571. [20] R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Science, 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1. [21] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [22] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. [23] P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, Journal of Differential Equations, 109 (1994), 110-146. doi: 10.1006/jdeq.1994.1046. [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. [25] S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations, Circuits, Systems and Signal Processing, 11 (1992), p281. doi: 10.1007/BF01189230. [26] S. Reich, On a geometrical interpretation of differential-algebraic equations, Circuits, Systems and Signal Processing, 10 (1991), 343-359. doi: 10.1007/BF01187550. [27] W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds, Mathematics of Computation, 43 (1984), 473-482. doi: 10.1090/S0025-5718-1984-0758195-5.

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##### References:
 [1] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [2] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365. [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), 323-351. doi: 10.1017/S0305004101005679. [4] J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, 1793, Springer-Verlag, Berlin, 2002. doi: 10.1007/b84020. [5] J. Cortés and E. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. [6] 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. [7] S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928. doi: 10.1088/0951-7715/21/8/009. [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), 220-229. [9] S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator,, accepted by Nonlinearity, (). [10] B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos, Memoirs of the American Mathematical Society, 119 (1996), viii+79 pp. doi: 10.1090/memo/0570. [11] J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations, SIAM Journal on Numerical Analysis, 20 (1983), 732-746. doi: 10.1137/0720049. [12] Z. Ge and J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A., 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6. [13] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7. [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), 221-276. doi: 10.1007/s00332-007-9012-8. [15] Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61-81. [16] J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092. [17] W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134. doi: 10.1016/S0034-4877(98)80007-4. [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), 467-502. doi: 10.1007/s10883-007-9027-3. [19] M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414. doi: 10.1063/1.531571. [20] R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Science, 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1. [21] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. [22] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. [23] P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, Journal of Differential Equations, 109 (1994), 110-146. doi: 10.1006/jdeq.1994.1046. [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. [25] S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations, Circuits, Systems and Signal Processing, 11 (1992), p281. doi: 10.1007/BF01189230. [26] S. Reich, On a geometrical interpretation of differential-algebraic equations, Circuits, Systems and Signal Processing, 10 (1991), 343-359. doi: 10.1007/BF01187550. [27] W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds, Mathematics of Computation, 43 (1984), 473-482. doi: 10.1090/S0025-5718-1984-0758195-5.
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