# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.00714