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Dynamics of a point in the axisymmetric gravitational potential of a massive fixed ring and center. (English. Russian original) Zbl 1476.70052

Theor. Math. Phys. 207, No. 2, 678-688 (2021); translation from Teor. Mat. Fiz. 207, No. 2, 319-330 (2021).
Summary: We consider the problem of three-dimensional motion of a passively gravitating point in the potential created by a homogeneous thin fixed ring and a massive point located in the center of the ring. The motion of a passively gravitating point admits two first integrals. We first consider the integrable case of an invariant motion in the equatorial plane and then consider the general case of three-dimensional motion, where we classify the possible trajectories of a point depending on the values of the first integrals. Finally, some previous results for similar problems are compared.

MSC:

70F15 Celestial mechanics
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[1] Binney, J.; Tremaine, S., Galactic Dynamics (1994), Princeton, NJ: Princeton Univ. Press, Princeton, NJ · Zbl 1136.85001
[2] Kutuzov, S. A., Orbits in ‘Disk + Halo’ galaxy model, Order and Chaos in Stellar and Planetary Systems, 0, 37-42 (2004)
[3] Tresaco, E.; Elipe, A.; Riaguas, A., Dynamics of a particle under the gravitational potential of a massive annulus: Properties and equilibrium description, Celestial Mech. Dynam. Astronom., 111, 431-447 (2011) · Zbl 1266.70039 · doi:10.1007/s10569-011-9371-1
[4] Tresaco, E.; Elipe, A.; Riaguas, A., Computation of families of periodic orbits and bifurcations around a massive annulus, Astrophys. Space Sci., 338, 23-33 (2012) · Zbl 1238.85012 · doi:10.1007/s10509-011-0925-1
[5] Tresaco, E.; Riaguas, A.; Elipe, A., Numerical analysis of periodic solutions and bifurcations in the planetary annulus problem, Appl. Math. Comput., 225, 645-655 (2013) · Zbl 1358.70020
[6] Alberti, A.; Vidal, C., Dynamics of a particle in a gravitational field of a homogeneous annulus disk, Celestial Mech. Dynam. Astronom., 98, 75-93 (2007) · Zbl 1330.70071 · doi:10.1007/s10569-007-9071-z
[7] Fukushima, T., Precise computation of acceleration due to uniform ring or disk, Celestial Mech. Dynam. Astronom., 108, 339-356 (2010) · Zbl 1223.70040 · doi:10.1007/s10569-010-9304-4
[8] Alberti, A.; Vidal, C., Singularities and dynamics aspects of a particle in a gravitational field of a central punctual body surrounded by a solid circular ring, SIAM J. Appl. Dyn. Syst., 18, 1-32 (2019) · Zbl 1446.70032 · doi:10.1137/18M1179274
[9] Alberti, A.; Vidal, C., Singularities in the gravitational attraction problem due to massive bodies, Discrete Contin. Dyn. Syst., 26, 805-822 (2009) · Zbl 1379.70044 · doi:10.3934/dcds.2010.26.805
[10] Sakharov, A. V., Some trajectories of a point in the potential of a fixed ring and center, Russ. J. Nonlinear Dyn., 15, 587-592 (2019) · Zbl 1439.70023
[11] Najid, N.-E.; Zegoumou, M.; Haj, E. El, Dynamical behavior in the vicinity of a circular anisotropic ring, Open Astr. J., 5, 54-60 (2012) · doi:10.2174/1874381101205010054
[12] Duboshin, G. N., Theory of Attraction [in Russian] (1961), Fizmatgiz: Moscow, Fizmatgiz · Zbl 0102.39201
[13] Lass, H.; Blitzer, L., The gravitational potential due to uniform disks and rings, Celest. Mech., 30, 225-228 (1983) · Zbl 0561.70005 · doi:10.1007/BF01232189
[14] Broucke, R. A.; Elipe, A., The dynamics of orbits in a potential field of a solid circular ring, Regul. Chaotic Dyn., 10, 129-143 (2005) · Zbl 1128.70308 · doi:10.1070/RD2005v010n02ABEH000307
[15] Sidorenko, V. V., Dynamics of ‘jumping’ Trojans: A perturbative treatment, Celestial Mech. Dynam. Astronom., 130, 67 (2018) · Zbl 1448.70078 · doi:10.1007/s10569-018-9860-6
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