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

Nonlinear constraints in nonholonomic mechanics

Paul Popescu 1, and  Cristian Ida 2,

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.
Keywords: Nonlinear, nonholonomic, Chetaev principle., constraints.
Mathematics Subject Classification: Primary: 70F25, 37J60, 70H45; Secondary: 37C9.
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.  Google Scholar

[2]

S. Benenti, Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 203-212.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Vol. 24, Springer, 2003. doi: 10.1007/b97376.  Google Scholar

[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.  Google Scholar

[5]

I. Bucataru and R. Miron, Finsler-Lagrange geometry: Applications to dynamical systems, Editura Academiei Romane, Bucuresti, 2007.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[8]

P. Dazord, Mécanique hamiltonienne en présence de contraintes, Illinois Journal of Mathematics, 38 (1994), 148-175.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y.-X. Guo, J. Li-Yan and Y. Ying, Symmetries of mechanical systems with nonlinear nonholonomic constraints, Chinese Physics, 10 (2001), p181. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

O. Krupková, Mechanical systems with nonholonomic constraints, Journal of Mathematical Physics, 38 (1997), 5098-5126. doi: 10.1063/1.532196.  Google Scholar

[15]

O. Krupková, Geometric mechanics on nonholonomic submanifolds, Communications in Mathematics, 18 (2010), 51-77.  Google Scholar

[16]

S. Lang, Differential and Riemannian Manifolds, 3-th ed., Springer Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. M. Marle, Kinematic and geometric constraints, servomechanisms and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. Molino, Riemannian Foliations, Birkhäuser, Progr. Math. 73, 1988. doi: 10.1007/978-1-4684-8670-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Swaczyna, Several examples of nonholonomic mechanical systems, Communications in Mathematics, 19 (2011), 27-56.  Google Scholar

show all references

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.  Google Scholar

[2]

S. Benenti, Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 203-212.  Google Scholar

[3]

A. M. Bloch, Nonholonomic Mechanics and Control, Vol. 24, Springer, 2003. doi: 10.1007/b97376.  Google Scholar

[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.  Google Scholar

[5]

I. Bucataru and R. Miron, Finsler-Lagrange geometry: Applications to dynamical systems, Editura Academiei Romane, Bucuresti, 2007.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[8]

P. Dazord, Mécanique hamiltonienne en présence de contraintes, Illinois Journal of Mathematics, 38 (1994), 148-175.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y.-X. Guo, J. Li-Yan and Y. Ying, Symmetries of mechanical systems with nonlinear nonholonomic constraints, Chinese Physics, 10 (2001), p181. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

O. Krupková, Mechanical systems with nonholonomic constraints, Journal of Mathematical Physics, 38 (1997), 5098-5126. doi: 10.1063/1.532196.  Google Scholar

[15]

O. Krupková, Geometric mechanics on nonholonomic submanifolds, Communications in Mathematics, 18 (2010), 51-77.  Google Scholar

[16]

S. Lang, Differential and Riemannian Manifolds, 3-th ed., Springer Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. M. Marle, Kinematic and geometric constraints, servomechanisms and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. Molino, Riemannian Foliations, Birkhäuser, Progr. Math. 73, 1988. doi: 10.1007/978-1-4684-8670-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Swaczyna, Several examples of nonholonomic mechanical systems, Communications in Mathematics, 19 (2011), 27-56.  Google Scholar

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