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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, 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, arXiv:1312.1587.

[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.

show all references

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, arXiv:1312.1587.

[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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