Abstract
We discuss numerical solution of a system of nonlinear ordinary differential equations with a time-varying delay. This system describes the mutual influence between mass bodies in the case of a finite speed of the gravitational interaction. We show that this fact has to be taken into account in interplanetary flights, because minor changes in the velocity are continuously cumulated during a long time, which affects the position of a satellite. A method to determine an approximate value of the speed of gravitational interaction is introduced. We observed that in two-body problems with delays, there are no periodic solutions as in the classical two-body problem. We also demonstrate that the finite speed of gravitational interaction contributes to the expansion of the universe, slightly protects stars against collisions and thus makes, e.g., globular clusters more stable.
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