Differential geometry of rigid bodies collisions and non-standard billiards

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November 2016, 36(11): 6065-6099. doi: 10.3934/dcds.2016065

Differential geometry of rigid bodies collisions and non-standard billiards

Christopher Cox ^1, and Renato Feres ^1,

1.	Washington University, Department of Mathematics, One Brookings Dr., Campus Box 1146, St. Louis, MO 63130, United States, United States

Received October 2015 Revised June 2016 Published August 2016

Abstract
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The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$ is a manifold with boundary (typically with singularities.) A full description of the system requires boundary conditions specifying how orbits should be continued after collisions, that is, the assignment of a collision map at each tangent space on the boundary of $M$ giving the post-collision state of the system for each pre-collision state. We give a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility, and the natural requirement that impulse forces only act at the point of contact of the colliding bodies. These assumptions are stated geometrically in terms of a family of vector subbundles of the tangent bundle to $\partial M$: the diagonal, non-slipping, and impulse subbundles. Collision maps are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace. We then make a few observations of a dynamical nature about non-standard billiard systems, among which is a sufficient condition for the billiard map on the space of boundary states to preserve the canonical measure on constant energy hypersurfaces.

Keywords: Rigid body collisions, no-slip billiards, impulse., billiard dynamics, generalized non-holonomic constraints.

Mathematics Subject Classification: Primary: 37F35; Secondary: 37D5.

Citation: Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065

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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