The role of pseudo-hypersurfaces in non-holonomic motion

TL;DR

This paper extends hypersurface geometry to pseudo-hypersurfaces defined by Pfaff equations and applies this framework to model non-holonomic motion, including a charged particle in an electromagnetic field, revealing the Lorentz equation as a constrained geodesic.

Contribution

It introduces pseudo-hypersurfaces via Pfaff equations and demonstrates their application to non-holonomic motion and electromagnetic dynamics, providing a geometric perspective.

Findings

Lorentz equation as a constrained geodesic

Pseudo-hypersurfaces model non-holonomic constraints

Unified geometric framework for motion in electromagnetic fields

Abstract

The geometry of hypersurfaces is generalized to pseudo-hypersurfaces, which are defined by Pfaff equations. The general methods are then applied to modeling the kinematics of motion constrained by a single linear, non-holonomic constraint. They are then applied to the example of a charge moving in an electromagnetic field, and the Lorentz equation of motion is shown to represent a geodesic that is constrained to lie in a pseudo-hypersurface that is defined by the potential 1-form.