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Differential geometry of rigid bodies collisions and non-standard billiards
1. | Washington University, Department of Mathematics, One Brookings Dr., Campus Box 1146, St. Louis, MO 63130, United States, United States |
References:
[1] |
V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institue Fourier, tome 16, n1 (1966), 319-361.
doi: 10.5802/aif.233. |
[2] |
A. M. Bloch, Nonholonomic Mechanics and Control, Springer, 2003.
doi: 10.1007/b97376. |
[3] |
D. S. Broomhead and E. Gutkin, The dynamics of billiards with no-slip collisions, Physica D, 67 (1993), 188-197.
doi: 10.1016/0167-2789(93)90205-F. |
[4] |
M. do Carmo, Riemannian Geometry, Birkhäuser, 1993.
doi: 10.1007/978-1-4757-2201-7. |
[5] |
B, Chen, L.-S. Wang, S.-S. Chu and W.-T. Chou, A new classification of nonholonomic constraints, Proc. R. Soc. Lond. A, 453 (1997), 631-642.
doi: 10.1098/rspa.1997.0035. |
[6] |
N. Chernov, R. Markarian, Chaotic billiards, Mathematical Surveys and Monographs, V. 127, American Mathematical Society, 2006.
doi: 10.1090/surv/127. |
[7] |
S. Cook and R. Feres, Random billiards with wall temperature and associated Markov chains, Nonlinearity, 25 (2012), 2503-2541.
doi: 10.1088/0951-7715/25/9/2503. |
[8] |
J. Cortés, M. de León, D. M. de Diego and S. Martínez, Mechanical systems subjected to generalized nonholonomic constraints, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 651-670.
doi: 10.1098/rspa.2000.0686. |
[9] |
J. Cortés and A. M. Vinogradov, Hamiltonian theory of constrained impulse, J. Math. Phy., 47, 042905 (2006), 1-30.
doi: 10.1063/1.2192974. |
[10] |
C. Cox and R. Feres, {No-slip billiards in dimension two, (2016); arXiv:1602.01490 |
[11] |
W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Dover, 2001. |
[12] |
A. Ibort, M. de León, E. A. Lacomba, J. C. Marrero, D. M. de Diego and P. Pitanga, Geometric formulation of Carnot's theorem, J. Phys. A: Math. Gen., 34 (2001), 1691-1712.
doi: 10.1088/0305-4470/34/8/314. |
[13] |
E. A. Lacomba and W. A. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A: Math. Gen., 23 (1990), 2801-2813.
doi: 10.1088/0305-4470/23/13/019. |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer 1999.
doi: 10.1007/978-0-387-21792-5. |
[15] |
S. Tabachnikov, Billiards, Panoramas et Synthèses 1, Société Mathématique de France, 1995. |
[16] |
M. Wojtkowski, The system of two spinning disks in the torus, Physica D, 71 (1994), 430-439.
doi: 10.1016/0167-2789(94)90009-4. |
show all references
References:
[1] |
V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institue Fourier, tome 16, n1 (1966), 319-361.
doi: 10.5802/aif.233. |
[2] |
A. M. Bloch, Nonholonomic Mechanics and Control, Springer, 2003.
doi: 10.1007/b97376. |
[3] |
D. S. Broomhead and E. Gutkin, The dynamics of billiards with no-slip collisions, Physica D, 67 (1993), 188-197.
doi: 10.1016/0167-2789(93)90205-F. |
[4] |
M. do Carmo, Riemannian Geometry, Birkhäuser, 1993.
doi: 10.1007/978-1-4757-2201-7. |
[5] |
B, Chen, L.-S. Wang, S.-S. Chu and W.-T. Chou, A new classification of nonholonomic constraints, Proc. R. Soc. Lond. A, 453 (1997), 631-642.
doi: 10.1098/rspa.1997.0035. |
[6] |
N. Chernov, R. Markarian, Chaotic billiards, Mathematical Surveys and Monographs, V. 127, American Mathematical Society, 2006.
doi: 10.1090/surv/127. |
[7] |
S. Cook and R. Feres, Random billiards with wall temperature and associated Markov chains, Nonlinearity, 25 (2012), 2503-2541.
doi: 10.1088/0951-7715/25/9/2503. |
[8] |
J. Cortés, M. de León, D. M. de Diego and S. Martínez, Mechanical systems subjected to generalized nonholonomic constraints, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 651-670.
doi: 10.1098/rspa.2000.0686. |
[9] |
J. Cortés and A. M. Vinogradov, Hamiltonian theory of constrained impulse, J. Math. Phy., 47, 042905 (2006), 1-30.
doi: 10.1063/1.2192974. |
[10] |
C. Cox and R. Feres, {No-slip billiards in dimension two, (2016); arXiv:1602.01490 |
[11] |
W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Dover, 2001. |
[12] |
A. Ibort, M. de León, E. A. Lacomba, J. C. Marrero, D. M. de Diego and P. Pitanga, Geometric formulation of Carnot's theorem, J. Phys. A: Math. Gen., 34 (2001), 1691-1712.
doi: 10.1088/0305-4470/34/8/314. |
[13] |
E. A. Lacomba and W. A. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A: Math. Gen., 23 (1990), 2801-2813.
doi: 10.1088/0305-4470/23/13/019. |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer 1999.
doi: 10.1007/978-0-387-21792-5. |
[15] |
S. Tabachnikov, Billiards, Panoramas et Synthèses 1, Société Mathématique de France, 1995. |
[16] |
M. Wojtkowski, The system of two spinning disks in the torus, Physica D, 71 (1994), 430-439.
doi: 10.1016/0167-2789(94)90009-4. |
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