Kepler's laws and the two-body problem with a finite gravitational velocity

Slyusarchuk  Vasyl Yukhimovych

doi:https://doi.org/10.31861/bmj2018.03.134

Slyusarchuk Vasyl Yukhimovych ¹

¹ Department of Higher Mathematics, National University of Water and Environmental Engineering, Rivne, 33028, Ukraine

DOI: https://doi.org/10.31861/bmj2018.03.134

Keywords: Kepler’s laws, the problem of two bodies, the problem of two bodies with a finitespeed of gravity, mathematical model of the solar system

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Abstract

In classical celestial mechanics the kinematic picture of the motion of bodies is determined by Kepler’s three laws.

From the moment of the opening of the law of universal gravitation by I. Newton, ordinary differential equations were used to study the motion of bodies, since it was assumed that the velocity of gravity is infinite and the gravitational field extends instantaneously from the source, however far from it.

In the real world, the velocity of gravitation can not be infinite, which is consistent with the theory of relativity A. Einstein, in which it is postulated that the rate of gravity coincides with the speed of light, and with the research S. Kopeikin and E. Fomalont on the fundamental limit of the velocity of gravity. Therefore, in order to study the dynamics of the motion of bodies in real space, methods of the theory of ordinary differential equations are insufficient.

Due to the delay of the gravitational field for studying the motion of bodies, the mathematical apparatus based on which is laid differential equations with delay argument.

In the article a mathematical model of motion of two bodies with finite velocity of gravity with the use of these equations is constructed.

It is shown that the motion of these bodies is not carried out in accordance with Kepler’s laws. In the study of body motion, the use of the system of nonlinear differential equations with a lagging argument and the law of increasing the sector velocity of the relative motion of bodies due to the finite velocity of gravity is essential.

References

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Cite

ACS Style: Slyusarchuk , V.Y. Kepler's laws and the two-body problem with a finite gravitational velocity. Bukovinian Mathematical Journal. 2019, 6 https://doi.org/https://doi.org/10.31861/bmj2018.03.134
AMA Style: Slyusarchuk VY. Kepler's laws and the two-body problem with a finite gravitational velocity. Bukovinian Mathematical Journal. 2019; 6(3-4). https://doi.org/https://doi.org/10.31861/bmj2018.03.134
Chicago/Turabian Style: Vasyl Yukhimovych Slyusarchuk . 2019. "Kepler's laws and the two-body problem with a finite gravitational velocity". Bukovinian Mathematical Journal. 6 no. 3-4. https://doi.org/https://doi.org/10.31861/bmj2018.03.134

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