We study the dynamics of two bodies moving along an immobile straight line with regard for a finite speed of gravity. It is shown that the escape velocity is higher than the corresponding velocity in the classical celestial mechanics. We present estimates for this velocity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Einstein, On Special and General Relativity [Russian translation], Gosizdat, Moscow (1922).
Y. Choquet-Bruhat, General Relativity and Einstein Equations, Oxford Univ. Press, Oxford (2009).
E. B. Fomalont and S. M. Kopeikin, “The measurement of the light deflection from Jupiter: experimental results,” Astrophys. J., 598, 704–711 (2003); Preprint arXiv:astro-ph/0302294 (2003).
B. P. Abbott et al., “Gravitational waves and Gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A,” Astrophys. J., 848, No. 2, L13 (2017); https://doi.org/10.3847/2041-8213/aa920c.
V. Yu. Slyusarchuk, “Non-Keplerian behavior and instability of motion of two bodies caused by a finite velocity of gravitation,” Nelin. Kolyv., 21, No. 3, 397–419 (2018); English translation: J. Math. Sci., 243, No. 3, 467–492 (2019); https://doi.org/10.1007/s10958-019-04550-0
V. Yu. Slyusarchuk, “Mathematical model of the solar system with regard for the velocity of gravitation,” Nelin. Kolyv., 21, No. 2, 238–261 (2018); English translation: J. Math. Sci., 243, No. 2, 287–312 (2019); https://doi.org/10.1007/s10958-019-04540-2.
V. Yu. Slyusarchuk, “Investigation of systems of differential equations with delay and constraints imposed on the delays and derivatives of the solutions,” Ukr. Mat. Zh., 71, No. 5, 677–691 (2019); English translation: Ukr. Math. J., 71, No. 5, 774–791 (2019); https://doi.org/10.1007/s11253-019-01673-0.
Yu. V. Aleksandrov, Celestial Mechanics. A Textbook [in Russian], Kharkov National University, Kharkov (2006).
I. Newton, Philosophiae Naturalis Principia Mathematica (1687).
V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], URSS, Moscow (2002).
V. A. Brumberg, Relativistic Celestial Mechanics [in Russian], Nauka, Moscow (1972).
G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus [in Russian], Vol. 1, Nauka, Moscow (1966).
I. P. Natanson, Theory of Functions of Real Variable [in Russian], Nauka, Moscow (1974).
U. Dini, Fondamenti per la Teorica Delle Funzioni di Variabili Reali, T. Nistri, Pisa (1878).
O. F. Kabardin, Physics: A Handbook [in Russian], Prosveshchenie, Moscow (1991).
V. Yu. Slyusarchuk, Absolute Stability of Dynamical Systems with Aftereffect [in Ukrainian], Rivne National University of Water Management and Utilization of Nature Resources, Rivne (2003).
Yu. A. Belyi, Johannes Kepler (1571–1630) [in Russian], Nauka, Moscow (1971).
F. R. Moulton, An Introduction to Celestial Mechanics, The MacMillian Company, New York (1914).
D. V. Anosov, From Newton to Kepler [in Russian], MTSNMO, Moscow (2006).
O. V. Golubeva, Theoretical Mechanics [in Russian], Vysshaya Shkola, Moscow (1968).
V. Yu. Slyusarchuk, “Dynamics of three bodies located on a straight line for a finite speed of gravity,” Nelin. Kolyv., 23, No. 4, 529–552 (2020); English translation: J. Math. Sci., 263, No. 2, 299–326 (2022).
V. Yu. Slyusarchuk, “Instability of unbounded solutions of evolutionary equations with operator coefficients commutative with operators of rotation,” Bukov. Mat. Zh., 7, No. 1, 99–113 (2019).
V. Yu. Slyusarchuk, “Equations in Hilbert spaces whose sets of solutions are invariant under a group isomorphic to a one-parameter group of unitary operators,” Ukr. Mat. Zh., 72, No. 1, 86–99 (2020); English translation: Ukr. Math. J., 72, No. 1, 98–113 (2020), https://doi.org/10.1007/s11253-020-01765-2
V. Yu. Slyusarchuk, “Equations whose sets of solutions are invariant under a group of mappings isomorphic to a one-parameter group of rotations,” Nelin. Kolyv., 23, No. 1, 112–123 (2020); English translation: J. Math. Sci., 256, No. 5, 689–702 (2021).
J. Chazy, La Théorie de la Relativité et la Mécanique Céleste, Vol. 1, Gauthier-Villars, Paris (1928); Vol. 2 (1930).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 24, No. 2, pp. 249–277, April–June, 2021.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Slyusarchuk, V.Y. Dynamics of Two Bodies with Trajectories on a Fixed Straight Line with Regard for the Finite Speed of Gravity. J Math Sci 270, 353–384 (2023). https://doi.org/10.1007/s10958-023-06351-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06351-y