# American Institute of Mathematical Sciences

March  2020, 12(1): 53-84. doi: 10.3934/jgm.2020004

## 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

* Corresponding author: Renato Feres

Received  February 2019 Revised  June 2019 Published  January 2020

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.

Citation: Timothy Chumley, Scott Cook, Christopher Cox, Renato Feres. Rolling and no-slip bouncing in cylinders. Journal of Geometric Mechanics, 2020, 12 (1) : 53-84. doi: 10.3934/jgm.2020004
##### References:

show all references

##### References:
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 $2$ 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
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 $-r\omega\cdot e/u\cdot\tau_a = 1.15.$ (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
Notation for the proof of Corollary 1
Cross-sectional projection of initial velocity $\bar{u}_0$ and definition of $\theta$ and $\delta$
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 $1$ as shown in the far right. Initial conditions for the depicted orbits roughly compare to those of Figure 11
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] Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065 [2] E. Minguzzi. A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena. Journal of Geometric Mechanics, 2015, 7 (4) : 473-482. doi: 10.3934/jgm.2015.7.473 [3] Panayotis G. Kevrekidis, Vakhtang Putkaradze, Zoi Rapti. Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models. Conference Publications, 2015, 2015 (special) : 696-704. doi: 10.3934/proc.2015.0696 [4] Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109 [5] W. Patrick Hooper, Richard Evan Schwartz. Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2009, 3 (2) : 159-231. doi: 10.3934/jmd.2009.3.159 [6] Serge Tabachnikov. Birkhoff billiards are insecure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035 [7] Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893 [8] Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255 [9] Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048 [10] Giancarlo Benettin, Massimiliano Guzzo, Anatoly Neishtadt. A new problem of adiabatic invariance related to the rigid body dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 959-975. doi: 10.3934/dcds.2008.21.959 [11] Hong-Kun Zhang. Free path of billiards with flat points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445 [12] W. Patrick Hooper, Richard Evan Schwartz. Erratum: Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2014, 8 (1) : 133-137. doi: 10.3934/jmd.2014.8.133 [13] Richard Evan Schwartz. Unbounded orbits for outer billiards I. Journal of Modern Dynamics, 2007, 1 (3) : 371-424. doi: 10.3934/jmd.2007.1.371 [14] Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271 [15] Reimund Rautmann. Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1101-1109. doi: 10.3934/dcdss.2014.7.1101 [16] Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks & Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597 [17] Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132 [18] Daniel Genin. Research announcement: Boundedness of orbits for trapezoidal outer billiards. Electronic Research Announcements, 2008, 15: 71-78. doi: 10.3934/era.2008.15.71 [19] Richard Evan Schwartz. Research announcement: unbounded orbits for outer billiards. Electronic Research Announcements, 2007, 14: 1-6. doi: 10.3934/era.2007.14.1 [20] Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112

2018 Impact Factor: 0.525

## Tools

Article outline

Figures and Tables