# American Institute of Mathematical Sciences

December  2014, 6(4): 527-547. doi: 10.3934/jgm.2014.6.527

## Nonlinear constraints in nonholonomic mechanics

 1 Department of Applied Mathematics, University of Craiova, Craiova 200585, Str. A.I. Cuza 13, Romania 2 Department of Mathematics and Informatics, University Transilvania of Braşov, Braşov 500091, Str. Iuliu Maniu 50, Romania

Received  December 2013 Revised  June 2014 Published  December 2014

In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3,4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.
Citation: Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
##### References:
 [1] A. Bejancu, Nonholonomic mechanical systems and Kaluza-Klein theory, Journal of Nonlinear Science, 22 (2012), 213-233. doi: 10.1007/s00332-011-9114-1. [2] S. Benenti, Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 203-212. [3] A. M. Bloch, Nonholonomic Mechanics and Control, Vol. 24, Springer, 2003. doi: 10.1007/b97376. [4] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136 (1996), 21-99. doi: 10.1007/BF02199365. [5] I. Bucataru and R. Miron, Finsler-Lagrange geometry: Applications to dynamical systems, Editura Academiei Romane, Bucuresti, 2007. [6] H. Cendra, A. Ibort, M. de Léon and D. M. de Diego, A generalization of Chetaev's principle for a class of higher order nonholonomic constraints, J. Math. Phys., 45 (2004), 2785-2801. doi: 10.1063/1.1763245. [7] J. Cortés, M. de León, J. C. Marrero and E. Martí nez, Non-holonomic Lagrangian systems on Lie algebroids, arXiv preprint math-ph/0512003 (2005). [8] P. Dazord, Mécanique hamiltonienne en présence de contraintes, Illinois Journal of Mathematics, 38 (1994), 148-175. [9] K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204. [10] K. Grabowska, P. Urbański and J. Grabowski, Geometrical mechanics on algebroids, International Journal of Geometric Methods in Modern Physics, 3 (2006), 559-575. doi: 10.1142/S0219887806001259. [11] Y.-X. Guo, J. Li-Yan and Y. Ying, Symmetries of mechanical systems with nonlinear nonholonomic constraints, Chinese Physics, 10 (2001), p181. [12] L. A. Ibort, M. de León, G. Marmo and D. M. de Diego, Non-holonomic constrained systems as implicit differential equations, Rend. Semin. Mat., Torino, 54 (1996), 295-317. [13] M. H. Kobayashi and W. M. Oliva, A note on the conservation of energy and volume in the setting of nonholonomic mechanical systems, Qualitative Theory of Dynamical Systems, 4 (2004), 383-411. doi: 10.1007/BF02970866. [14] O. Krupková, Mechanical systems with nonholonomic constraints, Journal of Mathematical Physics, 38 (1997), 5098-5126. doi: 10.1063/1.532196. [15] O. Krupková, Geometric mechanics on nonholonomic submanifolds, Communications in Mathematics, 18 (2010), 51-77. [16] S. Lang, Differential and Riemannian Manifolds, 3-th ed., Springer Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9. [17] M. de León, A historical review on nonholonomic mechanics, Revista de la Real Academia de Ciencias Exactas, Fisicas Y Naturales (Serie A: Matematicas) 105, 2011. [18] M. de León, J. C. Marrero and D. M. de Diego, Mechanical systems with nonlinear constraints, International Journal of Theoretical Physics, 36 (1997), 979-995. doi: 10.1007/BF02435796. [19] M. de León, D. Martíin de Diego and M. Vaquero, A Hamilton-Jacobi theory on Poisson manifolds, Journal of Geometric Mechanics, 6 (2014), 121-140. doi: 10.3934/jgm.2014.6.121. [20] A. D. Lewis, The geometry of the Gibbs-Appell equations and Gauss' principle of least constraint, Reports on Mathematical Physics, 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0. [21] S.-M. Li and J. Berakdar, A generalization of the Chetaev condition for nonlinear nonholonomic constraints: The velocity-determined virtual displacement approach, Reports on Mathematical Physics, 63 (2009), 179-189. doi: 10.1016/S0034-4877(09)00012-3. [22] C. M. Marle, Kinematic and geometric constraints, servomechanisms and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364. [23] C. M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Reports on Mathematical Physics, 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6. [24] T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, Journal of Physics A: Mathematical and General, 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011. [25] P. Molino, Riemannian Foliations, Birkhäuser, Progr. Math. 73, 1988. doi: 10.1007/978-1-4684-8670-4. [26] P. Popescu and M. Popescu, Lagrangians adapted to submersions and foliations, Differential Geom. Appl., 27 (2009), 171-178. doi: 10.1016/j.difgeo.2008.06.017. [27] W. Sarlet, F. Cantrijn and D. J. Saunders, A geometrical framework for the study of non-holonomic Lagrangian systems, J. Phys. A, 28 (1995), 3253-3268. doi: 10.1088/0305-4470/28/11/022. [28] M. Swaczyna, Several examples of nonholonomic mechanical systems, Communications in Mathematics, 19 (2011), 27-56.

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##### References:
 [1] A. Bejancu, Nonholonomic mechanical systems and Kaluza-Klein theory, Journal of Nonlinear Science, 22 (2012), 213-233. doi: 10.1007/s00332-011-9114-1. [2] S. Benenti, Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 203-212. [3] A. M. Bloch, Nonholonomic Mechanics and Control, Vol. 24, Springer, 2003. doi: 10.1007/b97376. [4] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136 (1996), 21-99. doi: 10.1007/BF02199365. [5] I. Bucataru and R. Miron, Finsler-Lagrange geometry: Applications to dynamical systems, Editura Academiei Romane, Bucuresti, 2007. [6] H. Cendra, A. Ibort, M. de Léon and D. M. de Diego, A generalization of Chetaev's principle for a class of higher order nonholonomic constraints, J. Math. Phys., 45 (2004), 2785-2801. doi: 10.1063/1.1763245. [7] J. Cortés, M. de León, J. C. Marrero and E. Martí nez, Non-holonomic Lagrangian systems on Lie algebroids, arXiv preprint math-ph/0512003 (2005). [8] P. Dazord, Mécanique hamiltonienne en présence de contraintes, Illinois Journal of Mathematics, 38 (1994), 148-175. [9] K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204. [10] K. Grabowska, P. Urbański and J. Grabowski, Geometrical mechanics on algebroids, International Journal of Geometric Methods in Modern Physics, 3 (2006), 559-575. doi: 10.1142/S0219887806001259. [11] Y.-X. Guo, J. Li-Yan and Y. Ying, Symmetries of mechanical systems with nonlinear nonholonomic constraints, Chinese Physics, 10 (2001), p181. [12] L. A. Ibort, M. de León, G. Marmo and D. M. de Diego, Non-holonomic constrained systems as implicit differential equations, Rend. Semin. Mat., Torino, 54 (1996), 295-317. [13] M. H. Kobayashi and W. M. Oliva, A note on the conservation of energy and volume in the setting of nonholonomic mechanical systems, Qualitative Theory of Dynamical Systems, 4 (2004), 383-411. doi: 10.1007/BF02970866. [14] O. Krupková, Mechanical systems with nonholonomic constraints, Journal of Mathematical Physics, 38 (1997), 5098-5126. doi: 10.1063/1.532196. [15] O. Krupková, Geometric mechanics on nonholonomic submanifolds, Communications in Mathematics, 18 (2010), 51-77. [16] S. Lang, Differential and Riemannian Manifolds, 3-th ed., Springer Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9. [17] M. de León, A historical review on nonholonomic mechanics, Revista de la Real Academia de Ciencias Exactas, Fisicas Y Naturales (Serie A: Matematicas) 105, 2011. [18] M. de León, J. C. Marrero and D. M. de Diego, Mechanical systems with nonlinear constraints, International Journal of Theoretical Physics, 36 (1997), 979-995. doi: 10.1007/BF02435796. [19] M. de León, D. Martíin de Diego and M. Vaquero, A Hamilton-Jacobi theory on Poisson manifolds, Journal of Geometric Mechanics, 6 (2014), 121-140. doi: 10.3934/jgm.2014.6.121. [20] A. D. Lewis, The geometry of the Gibbs-Appell equations and Gauss' principle of least constraint, Reports on Mathematical Physics, 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0. [21] S.-M. Li and J. Berakdar, A generalization of the Chetaev condition for nonlinear nonholonomic constraints: The velocity-determined virtual displacement approach, Reports on Mathematical Physics, 63 (2009), 179-189. doi: 10.1016/S0034-4877(09)00012-3. [22] C. M. Marle, Kinematic and geometric constraints, servomechanisms and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364. [23] C. M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Reports on Mathematical Physics, 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6. [24] T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, Journal of Physics A: Mathematical and General, 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011. [25] P. Molino, Riemannian Foliations, Birkhäuser, Progr. Math. 73, 1988. doi: 10.1007/978-1-4684-8670-4. [26] P. Popescu and M. Popescu, Lagrangians adapted to submersions and foliations, Differential Geom. Appl., 27 (2009), 171-178. doi: 10.1016/j.difgeo.2008.06.017. [27] W. Sarlet, F. Cantrijn and D. J. Saunders, A geometrical framework for the study of non-holonomic Lagrangian systems, J. Phys. A, 28 (1995), 3253-3268. doi: 10.1088/0305-4470/28/11/022. [28] M. Swaczyna, Several examples of nonholonomic mechanical systems, Communications in Mathematics, 19 (2011), 27-56.
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