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

The equations of motion of a charged particle in a five-dimensional model of the general theory of relativity with a nonholonomic four-dimensional velocity space are considered. A nonholonomic distribution defined by the differential form omega= A(0)dx(0) + A(1)dx(1) + A(2)dx(2) + A(3)dx(3) + dx(4) on a five- dimensional smooth Lorentzian manifold is studied. By means of the Pontryagin maximum principle, it is proved that the equations of horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general theory of relativity. Thus, a model of the Kaluza-Klein theory is built by means of the sub-Lorentzian geometry. Finally, the geodesic sphere, which appears in a constant magnetic field, is studied, as well as its singular points.