Abstract
A local integral that allows us to immediately determine the velocity field for certain specific initial conditions was introduced by Antonov and Shamshiev in 1992. We consider cases where the integrability of the trajectory can be advanced further. We then obtain an analytical relationship between \(x\), \(y\), \(z\), \(t\) on a given trajectory. Possible cases are classified and one of them is analyzed.
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REFERENCES
V. A. Antonov, Vestnik LGU 19, 97 (1981).
V. A. Antonov and F. T. Shamshiev, Astron. Zh. 69 (5), 971 (1992).
V. A. Antonov and F. T. Shamshiev, Celestial Mechanics and Dynamical Astronomy 56 (3), 451 (1993). https://doi.org/10.1007/BF00691813
V. A. Antonov and F. T. Shamshiev, Celestial Mechanics and Dynamical Astronomy 59 (3), 209 (1994). https://doi.org/10.1007/BF00692872
V. V. Golubev, Lectures on integrating the equations of motion of a heavy rigid body around a fixed point (Gosudarsnvennoe izdatel’stvo naucho-tekhnicheskoj literatury, Moscow, 1953) [in Russian].
L. D. Landau and E. M. Lifshitz, Field theory, 6th ed. (Nauka, Moscow, 1973).
D. Lynden-Bell, Monthly Notices Royal Astron. Soc. 124, 95 (1962). https://doi.org/10.1093/mnras/124.2.95
D. Lynden-Bell, Monthly Notices Royal Astron. Soc. 458 (1), 726 (2016). https://doi.org/10.1093/mnras/stw229
F. T. Shamshiev, Astronomical and Astrophysical Transactions 7 (4), 269 (1995). https://doi.org/10.1080/10556799508203273
F. T. Shamshiev, in Proc. All-Russian Conf. on Astronomy at the Epoch of Multimessenger Studies, Moscow, Russia, 2021, Ed. by A. M. Cherepashchuk (Janus-K, Moscow, 2022), pp. 468–470. https://doi.org/10.51194/VAK2021.2022.1.1.195
I. S. Sokolnikov, Tensor analysis: Theory and applications in geometry and continuum mechanics (Nauka, Moscow, 1971) [in Russian].
ACKNOWLEDGMENTS
This article is dedicated to the blessed memory of Vadim Anatolyevich Antonov. The author expresses gratitude to A.S. Rastorguev and O.Y. Malkov, organizers of the conference ‘‘Modern Stellar Astronomy’’, held at the Caucasian Mountain Observatory of Moscow State University named after M.V. Lomonosov on November 8–10, 2022 and at Volgograd State University on November 8–10, May 15–19, 2023, as well as their participants for their interest in this work and discussion of the results proved useful.
Funding
The work was partially supported by grant FZ-20200929344, allocated by the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan.
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Shamshiev, F.T. Cases of Further Integrability of the Equation of Motion in the Presence of the Local Integral in the Space Model. I. Astrophys. Bull. 79, 151–158 (2024). https://doi.org/10.1134/S1990341323600308
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DOI: https://doi.org/10.1134/S1990341323600308