The Equations of Motion of a Charged Particle in the Five-Dimensional Model of the General Relativity Theory with the Four-Dimensional Nonholonomic Velocity Space

摘要

We consider the four-dimensional nonholonomic distribution defined by the4-potential of the electromagnetic field on the manifold. This distribution hasa metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, thecausal structure appears as in the general relativity theory. By means of thePontryagin's maximum principle we proved that the equations of the horizontalgeodesics for this distribution are the same as the equations of motion of acharged particle in the general relativity theory. This is a Kaluza -- Kleinproblem of classical and quantum physics solved by methods of sub-Lorentziangeometry. We study the geodesics sphere which appears in a constant magneticfield and its singular points. Sufficiently long geodesics are not optimalsolutions of the variational problem and define the nonholonomic wavefront.This wavefront is limited by a convex elliptic cone. We also study variationalprinciple approach to the problem. The Euler -- Lagrange equations are the sameas those obtained by the Pontryagin's maximum principle if the restriction ofthe metric tensor on the distribution is the same.