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Rolling and no-slip bouncing in cylinders. (English) Zbl 1441.37070

Summary: 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.

MSC:

37J60 Nonholonomic dynamical systems
70F25 Nonholonomic systems related to the dynamics of a system of particles
70F35 Collision of rigid or pseudo-rigid bodies
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