The law of universal gravitation is given, taking into account the finite speed of gravity, the special case of which is the law of universal gravitation of Newtonian mechanics, which coincides with the law in the limiting case. Using this law and Newton’s second law, a mathematical model of the motion of a system of arbitrary numbers of material points is constructed, in particular, a mathematical model of the solar system with a finite speed of gravity, which does not coincide with the corresponding mathematical model of classical celestial mechanics. The mathematical model of the solar system of Newtonian celestial mechanics is a special case of the mathematical model of the solar system constructed and coincides with it in the extreme case. The basis for the construction of these models is based on nonlinear differential equations with a delayed argument and nonlinear functional equations. 

Also studies of the motion of two bodies of the same mass with finite speed of gravity are given. It is shown that the movement of these bodies is not carried out by Kepler’s laws. In the study of the motion of bodies, it is essential to use nonlinear differential equations with a delayed argument for the law on increasing the sectoral velocity of relative motion of bodies caused by the finite speed of gravity.

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