The vector field of a rolling rigid body

TL;DR

This paper derives the equations of motion for a rigid body rolling on a surface using differential geometry and semi-symplectic formalism, extending known results to arbitrary surfaces.

Contribution

It provides a differential-geometric derivation of nonholonomic equations for arbitrary surfaces using semi-symplectic formalism and shape operators.

Findings

Derivation of nonholonomic equations via semi-symplectic formalism

Extension of equations to arbitrary surfaces

Semi-symplectic reduction for horizontal plane case

Abstract

Nonholonomic systems are variational models commonly used for mechanical systems with ideal no-slip constraints. This note provides a differential-geometric derivation of the nonholonomic equations of motion for an arbitrary rigid body rolling on an arbitrary surface, via the semi-symplectic formalism, and in terms of shape operators (a.k.a. Weingarten maps). By a semi-symplectic reduction, the well-known differential equations in the case where the surface is a horizontal plane are shown to be semi-symplectic.