# 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, (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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##### 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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