气缸内的滚动和无滑移跳动
Rolling and no-slip bouncing in cylinders
摘要：
本文的目的是比较一个经典的非完整系统-一个球体在重力作用下在垂直圆柱的内表面上滚动-和一类离散的动力系统，称为无滑移台球在类似的配置。非完整系统的一个众所周知的显著特征是滚动球不会下落;它的高度函数是有界的，并且上下谐振。目前的工作的中心问题是是否可以观察到类似的有界行为在无滑台球对应。我们的主要结果如下：对于$3$维圆柱，无滑移台球具有有界轨道性质，并且非常接近于滚动运动，对于一类初始条件，我们称之为横向滚动碰撞.当这个条件不成立时，轨道会经历垂直振荡，叠加到整体向下的加速度上。考虑不同截面形状的圆柱体，证明了任意维数的欧几里得空间中两平行超平面间的无滑移台球，即使在平行于板的常外力作用下，也总是有界的;对于普通气缸，当横系的轨道（一个概念，取决于分解成横向和纵向分量的运动）有周期二-在平面无滑动台球中非常常见的现象-在没有力的情况下，纵向方向上的运动一般没有边界。这是使用一个纵向线性漂移，我们证明在任意尺寸的公式。虽然我们可以证明有界轨道的存在的系统有相对简单的横向动力学，我们也简要地探讨了数值无滑台球系统，即体育场圆柱台球，可以表现出混沌横向动力学。
英文摘要：
The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar configurations. A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down. The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart. Our main results are as follows: for circular cylinders in dimension $3$, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration. Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two---a very common occurrence in planar no-slip billiards---the motion in the longitudinal direction, under no forces, is generically not bounded. This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions. While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a no-slip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics.
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