|Date: Friday, February 12, 2021
Location: (Zoom) East Hall (3:00 PM to 4:00 PM)
Title: Geometry of Impact and Nonholonomic Systems
Abstract: Constraints are ubiquitous when studying mechanical systems: the simple pendulum requires that the bob maintains a constant distance from the pivot, an ice skate requires that the skate cannot slide perpendicular to its heading, and a billiard ball is required to remain within the confines of the billiard table. The constraint for the pendulum is called holonomic as it can be expressed as a function on the positions, while the ice skate is nonholonomic as it can only be expressed with velocities. Unlike the previous cases where the constraints are continual, the constraint for the billiard ball is an impact-type that only appears when the ball strikes the edge of the table.
In this talk we will derive the equations of motion for both nonholonomic and impact systems, as well as study a few of their properties. All of these systems will remain energy-preserving but the question of volume-preservation becomes much more difficult: It turns out that the existence of an invariant volume for impact systems severely inhibits Zeno behavior (systems experiencing infinite impacts in a finite amount of time). We will demonstrate necessary and sufficient conditions for when such an invariant volume exists and see that this invariant volume does not depend on the location of the impacts, e.g. the billiard ball is volume-preserving for any compact table-top.
Speaker: Will Clark
Institution: Cornell University