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Rolling and no-slip bouncing in cylinders
1. | Department of Mathematics and Statistics, Mount Holyoke College, 50 College St, South Hadley, MA 01075, USA |
2. | Department of Mathematics, Tarleton State University, Box T-0470, Stephenville, TX 76401, USA |
3. | Department of Mathematics and Statistics, Washington University, Campus Box 1146, St. Louis, MO 63130, USA |
We compare a classical non-holonomic system—a sphere rolling against the inner surface of a vertical cylinder under gravity—with certain discrete dynamical systems called no-slip billiards in similar configurations. A feature of the former is that its height function is bounded and oscillates harmonically up and down. We investigate whether similar bounded behavior is observed in the no-slip billiard counterpart. For circular cylinders in dimension $ 3 $, no-slip billiards indeed have bounded orbits, and very closely approximate rolling motion, for a class of initial conditions we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to overall downward acceleration. Concerning different cross-sections, we show that no-slip billiards between two parallel hyperplanes in arbitrary dimensions are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept relying on a factorization of the motion into transversal and longitudinal components) has period two, the motion, under no forces, is generically not bounded. This commonly occurs in planar no-slip billiards.
References:
[1] |
A. M. Bloch, Nonholonomic Mechanics and Control, vol. 24 of Interdisciplinary Applied Mathematics, 2nd edition, Springer, New York, 2015.
doi: 10.1007/978-1-4939-3017-3. |
[2] |
A. V. Borisov, I. S. Mamaev and A. A. Kilin,
Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Regul. Chaotic Dyn., 7 (2002), 201-219.
doi: 10.1070/RD2002v007n02ABEH000205. |
[3] |
D. S. Broomhead and E. Gutkin,
The dynamics of billiards with no-slip collisions, Phys. D, 67 (1993), 188-197.
doi: 10.1016/0167-2789(93)90205-F. |
[4] |
C. Cox and R. Feres, No-slip billiards in dimension two, in Dynamical Systems, Ergodic theory, and Probability: In Memory of Kolya Chernov, vol. 698 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, 91–110.
doi: 10.1090/conm/698/14032. |
[5] |
C. Cox, R. Feres and H.-K. Zhang,
Stability of periodic orbits in no-slip billiards, Nonlinearity, 31 (2018), 4443-4471.
doi: 10.1088/1361-6544/aacc43. |
[6] |
C. Cox and R. Feres,
Differential geometry of rigid bodies collisions and non-standard billiards, Discrete Contin. Dyn. Syst., 36 (2016), 6065-6099.
doi: 10.3934/dcds.2016065. |
[7] |
R. L. Garwin, Kinematics of an ultraelastic rough ball, American Journal of Physics, 37 (1969), 88-92. Google Scholar |
[8] |
M. Gualtieri, T. Tokieda, L. Advis-Gaete, B. Carry, E. Reffet and C. Guthmann, Golfer's dilemma, American Journal of Physics, 74 (2006), 497-501. Google Scholar |
[9] |
H. Larralde, F. Leyvraz and C. Mejía-Monasterio,
Transport properties of a modified Lorentz gas, J. Statist. Phys., 113 (2003), 197-231.
doi: 10.1023/A:1025726905782. |
[10] |
J. M. Lee, Introduction to Smooth Manifolds, vol. 218 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[11] |
C. Mejía-Monasterio, H. Larralde and F. Leyvraz, Coupled normal heat and matter transport in a simple model system, Phys. Rev. Lett., 86 (2001), 5417-5420. Google Scholar |
[12] |
J. I. Neǐmark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1972. |
[13] |
M. P. Wojtkowski,
The system of two spinning disks in the torus, Phys. D, 71 (1994), 430-439.
doi: 10.1016/0167-2789(94)90009-4. |
show all references
References:
[1] |
A. M. Bloch, Nonholonomic Mechanics and Control, vol. 24 of Interdisciplinary Applied Mathematics, 2nd edition, Springer, New York, 2015.
doi: 10.1007/978-1-4939-3017-3. |
[2] |
A. V. Borisov, I. S. Mamaev and A. A. Kilin,
Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Regul. Chaotic Dyn., 7 (2002), 201-219.
doi: 10.1070/RD2002v007n02ABEH000205. |
[3] |
D. S. Broomhead and E. Gutkin,
The dynamics of billiards with no-slip collisions, Phys. D, 67 (1993), 188-197.
doi: 10.1016/0167-2789(93)90205-F. |
[4] |
C. Cox and R. Feres, No-slip billiards in dimension two, in Dynamical Systems, Ergodic theory, and Probability: In Memory of Kolya Chernov, vol. 698 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, 91–110.
doi: 10.1090/conm/698/14032. |
[5] |
C. Cox, R. Feres and H.-K. Zhang,
Stability of periodic orbits in no-slip billiards, Nonlinearity, 31 (2018), 4443-4471.
doi: 10.1088/1361-6544/aacc43. |
[6] |
C. Cox and R. Feres,
Differential geometry of rigid bodies collisions and non-standard billiards, Discrete Contin. Dyn. Syst., 36 (2016), 6065-6099.
doi: 10.3934/dcds.2016065. |
[7] |
R. L. Garwin, Kinematics of an ultraelastic rough ball, American Journal of Physics, 37 (1969), 88-92. Google Scholar |
[8] |
M. Gualtieri, T. Tokieda, L. Advis-Gaete, B. Carry, E. Reffet and C. Guthmann, Golfer's dilemma, American Journal of Physics, 74 (2006), 497-501. Google Scholar |
[9] |
H. Larralde, F. Leyvraz and C. Mejía-Monasterio,
Transport properties of a modified Lorentz gas, J. Statist. Phys., 113 (2003), 197-231.
doi: 10.1023/A:1025726905782. |
[10] |
J. M. Lee, Introduction to Smooth Manifolds, vol. 218 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[11] |
C. Mejía-Monasterio, H. Larralde and F. Leyvraz, Coupled normal heat and matter transport in a simple model system, Phys. Rev. Lett., 86 (2001), 5417-5420. Google Scholar |
[12] |
J. I. Neǐmark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1972. |
[13] |
M. P. Wojtkowski,
The system of two spinning disks in the torus, Phys. D, 71 (1994), 430-439.
doi: 10.1016/0167-2789(94)90009-4. |













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