Third Year Major Oral: Rolling Equations with Force

Abstract: Rolling motion is traditionally modeled as a non-holonomic system, often studied in the context of classical geometry and mechanics. In this talk, I will present a framework for deriving and analyzing rolling equations with force on nice sub-manifold of Euclidean space—extending the familiar no-slip constraints of rolling to include external or internal forces such as gravity, electromagnetic fields, or control inputs. Starting from fundamental principles of Euclidean group actions and the associated configuration manifold, I will show how the acceleration, constraint, and force “pairs” can be expressed in a unified geometric setting. This leads naturally to the Bowei equation for rolling, now generalized to incorporate additional forces. I will also highlight how our rolling flow parallels to geodesic flow, magnetic flow, and frame flow, illustrating how rolling dynamics bridges multiple areas of geometric and ergodic theory. Finally, I will discuss potential applications and future directions, including rolling-bouncing analogies in higher dimensions and diffusion processes in nanoscale systems, emphasizing how introducing forces significantly broadens the scope and real-world relevance of these rolling models.

Host: Renato Feres