# 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

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
 [1] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 [2] Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 [3] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [4] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [5] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [6] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [7] Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355 [8] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [9] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [10] Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39. [11] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 [12] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [13] Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 [14] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 [15] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [16] Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 [17] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [18] Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73 [19] Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. [20] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

2019 Impact Factor: 0.649