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.
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Figure 2. Conditions for a transversal period $ 2 $ orbit in dimension $ 3 $. The projection to $ \mathbb{R}^2 $ of the velocity $ u $ of the center of mass and the angular velocity $ \dot{\theta} $ are related by $ |\dot{\theta}| = (m r/\mathcal{I})|u\sin\phi| $, where $ m $ is the projected disc's mass, $ r $ is its radius, and $ \mathcal{I} $ is its moment of inertia for the projected (or marginal) mass distribution
Figure 6. When the transversal rolling impact condition does not hold (see Definition 2.8), the particle acquires an overall acceleration downward. The apparent increase in thickness of the height function graph and of the particle's path is due to a small scale vertical zig-zag motion of increasing amplitude. See also Figure 8
Figure 8. Near grazing paths of no-slip billiard particle. On the left-hand side, the transversal rolling impact condition holds for the initial bounce, but on the right-hand side a small deviation of this condition is introduced. Notice the characteristic zig-zag nature of the curve and that it does not return all the way to the initial height. The starting center of mass position is indicated by the small black dot
Figure 5. Comparison of the height functions for the rolling motion in a circular cylinder under a constant downward force (solid line) and the corresponding no-slip billiard motion satisfying the transverse rolling impact condition (small circles). Initial conditions are chosen so that the two processes rotate around the cylinder at the same rate
Figure 7. These graphs correspond to a fixed initial transversal rolling defect $ -r\omega\cdot e/u\cdot\tau_a = 1.15. $ (When the transversal rolling impact initial condition holds, this value is $ 1 $.) Here $ \omega\cdot e $ is the longitudinal component of the initial angular velocity vector and $ u\cdot \tau_a $ is the tangential (to the boundary of the billiard domain) component of the center of mass velocity. As the intercollision flight becomes shorter and motion grazes the cylinder more and more closely, the height function becomes smooth but the falling rate remains essentially unchanged
Figure 11. Transition from regular to chaotic motion. The moving particle begins from the middle of the lower flat side with linear velocity pointing up and a small angular velocity that causes it to move right after the first collision with the upper flat side. For small values of the angular velocity trajectories never touch the curved sides of the boundary, and the motion along the axis of the cylinder is bounded. If the initial angular velocity is large enough, trajectories move beyond the ends of the flat sides and eventually becomes unstable
Figure 12. Transition from regular to chaotic motion for the transverse dynamics of the stadium cylinder no-slip billiard system, as viewed in the velocity phase portrait. The full velocity space is a disc of radius $ 1 $ as shown in the far right. Initial conditions for the depicted orbits roughly compare to those of Figure 11
Figure 13. Height function for the stadium cylinder for a trajectory in the chaotic regime, corresponding to the short orbit segment on the right of Figure 11 and of Figure 12. To give a sense of the scales involved, the diameter of the stadium is $ 8 $ and velocities are of order $ 1 $. The number of time steps is $ 4\times 10^4 $
[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. |
[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. |
[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. |
[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. |
No-slip billiard system between two parallel plates under gravity have bounded orbits in all dimensions. Far left: a simple periodic orbit with gravity turned off. Orbits shown from left to right are under the influence of increasing force
Conditions for a transversal period
Cross-section of the stadium cylinder and the rolling sphere
Height of center of mass of rolling particle in a cylinder with stadium cross-section
When the transversal rolling impact condition does not hold (see Definition 2.8), the particle acquires an overall acceleration downward. The apparent increase in thickness of the height function graph and of the particle's path is due to a small scale vertical zig-zag motion of increasing amplitude. See also Figure 8
Near grazing paths of no-slip billiard particle. On the left-hand side, the transversal rolling impact condition holds for the initial bounce, but on the right-hand side a small deviation of this condition is introduced. Notice the characteristic zig-zag nature of the curve and that it does not return all the way to the initial height. The starting center of mass position is indicated by the small black dot
Comparison of the height functions for the rolling motion in a circular cylinder under a constant downward force (solid line) and the corresponding no-slip billiard motion satisfying the transverse rolling impact condition (small circles). Initial conditions are chosen so that the two processes rotate around the cylinder at the same rate
These graphs correspond to a fixed initial transversal rolling defect
Notation for the proof of Corollary 1
Cross-sectional projection of initial velocity
Transition from regular to chaotic motion. The moving particle begins from the middle of the lower flat side with linear velocity pointing up and a small angular velocity that causes it to move right after the first collision with the upper flat side. For small values of the angular velocity trajectories never touch the curved sides of the boundary, and the motion along the axis of the cylinder is bounded. If the initial angular velocity is large enough, trajectories move beyond the ends of the flat sides and eventually becomes unstable
Transition from regular to chaotic motion for the transverse dynamics of the stadium cylinder no-slip billiard system, as viewed in the velocity phase portrait. The full velocity space is a disc of radius
Height function for the stadium cylinder for a trajectory in the chaotic regime, corresponding to the short orbit segment on the right of Figure 11 and of Figure 12. To give a sense of the scales involved, the diameter of the stadium is