Then we find the Hamiltonian in its two forms, with the help of 4-velocity and the generalized momentum, and substitute the Hamiltonian into Hamilton equations to verify the motion equations.
mu]]) we shall introduce the 3-vector of generalized momentum with the following components:
According to (19) for continuously distributed matter the rate of change of the generalized momentum of substance and the field is determined by gradients from the following quantities: the energy of the substance unit in gravitational and electromagnetic fields that can be found through the velocity V and the scalar and vector potentials; the integral by volume of the term with scalar spacetime curvature; the integral by volume of energy invariants of the gravitational and electromagnetic fields, which are in the volume of the substance unit, as well as those of their proper fields, which are generated by this substance and interact with it.
z] are the components of the 3-vector of the so-called conjugate generalized momentum P = ([P.
We shall find the components of the generalized momentum from (30), given that the velocity components [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are directly included in the Lagrangian (4) according to (12) and (13) only in three terms, forming part of the Lagrangian L'.
The scalar product of the generalized momentum P and the velocity V, taking into account the relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], gives:
THE EXPRESSION OF THE HAMILTONIAN THROUGH THE GENERALIZED MOMENTUM
We express the component x of the 3-velocity through components of the generalized momentum P=([P.
z], the Hamiltonian will be expressed through the 3-vector of the generalized momentum P, through the scalar potentials [psi], [phi], and vector potentials D, A:
Hamilton's equations according to (30) and (31), with the components of 3-vector coordinate velocity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the components of 3-vector of the generalized momentum P = ([P.
The physical meaning of equation (59) lies in the fact that the gradient of the Hamiltonian as the energy of the system, taken with opposite sign, is equal to the rate of change of the generalized momentum with time.