The transition was done from the 4-vector of the generalized velocity to a new 4-vector of the Hamiltonian, specifying the energy and the momentum of substance in fundamental fields.

We shall introduce 4-vector of the generalized velocity with the covariant index:

delta]] is the rotor of 4-vector of generalized velocities [S.

mu]] according to [1], and the 4-vector of generalized velocity (66):

or -]] particle is described by a 4-vector function [W.

is not related to a Hamiltonian because (13) is a Lorentz scalar whereas the Hamiltonian is a 0-component of a 4-vector.

The first term on the right side of (2) is the relativistic damping-force

4-vector that leads to (1) in the nonrelativistic limit.

Now, density is a 0-component of a 4-vector and the same is true for energy.

3]] and be a 0-component of a 4-vector satisfying the continuity equation

Notice that with the

4-vector formulation Maxwell equation are written in terms of two sets of equations.

mu]] is the 4-vector potential of the electromagnetic field and [j.

k] a magnetic interaction field that is separate from the 4-vector [A.