The role of pseudo-hypersurfaces in non-holonomic motion (Q2129850)

The aim of this paper is to generalize the geometry of hypersurfaces 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 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. This paper is organized as follows : Section 1 is an introduction to the subject. Section 2 deals with the geometry of hypersurfaces. The treatment here is a generalization of the classical geometry of surfaces, but a specialization of the more modern treatment. Section 3 is devoted to the geometry of pseudo-hypersurfaces. Many of the geometric constructions that were just made for hypersurfaces are based upon tangent and cotangent objects, which can exist independently of whether the hypersurface is represented as an embedded submanifold. Hence, one must consider its definition as an envelope, not a locus, and not assume that the envelope is completely integrable. In Section 4 the author studies kinematics in pseudo-hypersurfaces. Section 5 deals with an example concerning a charge moving in an electromagnetic field. In Section 6, the author discusses some aspects from what was developed in this study, the scope of the applications of pseudohypersurfaces to physics is essentially identical to the scope of application of the theory of Pfaff equation to physics, which the author has previously discussed [\textit{S. Schroll} and \textit{H. Treffinger}, ``A $\tau$-tilting approach to the first Brauer-Thrall conjecture'', Preprint, \url{arXiv:1210.4976}]. Hence, there are still many other physical applications of pseudo-hypersurfaces to explore. Section 7 is devoted to conclusions about the results obtained.