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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 |
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. |
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. |
[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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