摇摆和滚动：一个看似摇摆的罐子实际上可能会滚动。
Rocking and rolling: a can that appears to rock might actually roll
摘要：
桌上的啤酒瓶或苏打水罐，当轻轻倾斜并释放时，会垂直落下，然后摇晃起来，倾斜方向有点相反。从表面上看，这种摇摆运动涉及到容器的扁平圆形底座在摇摆到相反的倾斜之前与桌子碰撞。敏锐的眼睛会注意到，拍后上升的倾斜通常不是正好与初始倾斜完全相反，而是转向一侧或另一侧。Cushman和Duistermaat（2006）最近注意到当具有滚动边界条件的平盘几乎平坦地下落时会发生这种转向。在这里，我们推广这些滚动盘结果任意轴对称机构和无摩擦滑动。更具体地说，我们研究的议案，几乎但不完全涉及一个正面朝下的碰撞的圆形容器的底部与桌面。这些运动包括接触点围绕圆形基座的突然快速运动。令人惊讶的是，像滚动盘一样，该接触点的净运动角几乎与初始条件无关。这个转动角仅仅取决于几何形状和质量分布，而不取决于绕对称轴的转动惯量。我们推导出简单的渐近公式的接触点的这个“转角”，并检查结果与数值计算和简单的实验。对于高的容器（高度比半径大得多），转动角度刚好超过$π$，并且突然的滚动运动表面上表现为几乎对称的碰撞，导致倾斜在底部边缘上的几乎完全相反的点上。
英文摘要：
A beer bottle or soda can on a table, when slightly tipped and released, falls to an upright position and then rocks up to a somewhat opposite tilt. Superficially this rocking motion involves a collision when the flat circular base of the container slaps the table before rocking up to the opposite tilt. A keen eye notices that the after-slap rising tilt is not generally just diametrically opposite the initial tilt but is veered to one side or the other. Cushman and Duistermaat (2006) recently noticed such veering when a flat disk with rolling boundary conditions is dropped nearly flat. Here, we generalize these rolling disk results to arbitrary axi-symmetric bodies and to frictionless sliding. More specifically, we study motions that almost but do not quite involve a face-down collision of the round container's bottom with the table-top. These motions involve a sudden rapid motion of the contact point around the circular base. Surprisingly, like for the rolling disk, the net angle of motion of this contact point is nearly independent of initial conditions. This angle of turn depends simply on the geometry and mass distribution but not on the moment of inertia about the symmetry axis. We derive simple asymptotic formulas for this "angle of turn" of the contact point and check the result with numerics and with simple experiments. For tall containers (height much bigger than radius) the angle of turn is just over $π$ and the sudden rolling motion superficially appears as a nearly symmetric collision leading to leaning on an almost diametrically opposite point on the bottom rim.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
