## 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

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Krym, V. R.

Petrov, N. N.

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

Comment: 14 pages, 4 figures

Comment: 14 pages, 4 figures

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Mathematics - Differential Geometry, General Relativity and Quantum Cosmology, Mathematics - Metric Geometry, Mathematics - Optimization and Control, Physics - Classical Physics, 37J60, 37J55, 53B30, 53B50, 53D50, 58A30, 70F25, 81S05