The Least Action and Matter-Field Dynamics in Gravitational Field

We define the gravitation action as an integral over the four-dimensional space of the total curvature density. Integrating by parts, we obtain the Lagrangian density as a function of the Christoffel symbols. From a variation of this Lagrangian density with the metric elements, we obtain the Einstein law of gravitation for vacuum. We define the matter action as an integral over the four-dimensional space of the mass scalar density. With the momentum four-vector, we obtain a Lagrangian density variation with the metric elements and the time-space coordinates. From the total variation of the matter action in a gravitational field, we obtain the Einstein law of gravitation in matter, and the geodesic equations. According to the Maxwell equations, we obtain the electric and magnetic fields as an electromagnetic tensor. We define the electromagnetic action as an integral over the four-dimensional space of the amplitude scalar of this tensor, and obtain a variation of this action with metric elements and the electromagnetic potentials. At the same time, we consider the electric charge action as the scalar product of the charge flux four-vector with the electromagnetic potential four-vector, and obtain its variation with electromagnetic potentials and the time-space coordinates. We obtain a matter-field action variation with three terms: for the variations of the metric elements, of the time-space coordinates, and of the electromagnetic potentials. From the first term, we obtain the matter dynamics in a gravitational-electromagnetic field; from the second term, we obtain the Lorentz force in a gravitational field, and from the third term, the Maxwell equations in the gravitational field.