In this paper, we carried out a numerical study of the planar restricted four-body problem with repulsive Manev potential and perturbations in the Coriolis and centrifugal forces such that the peripherals possess Eulerian configuration. We have presented the equations of motion in the rotating frame and investigated the existence and location of the equilibrium points. We have found that there exist six equilibrium points all of which lie along the coordinate axes and shift in positions as the perturbation parameter is varied. We have also examined the linear stability of these equilibrium points and they are found unstable. The dynamical behavior of this system is also investigated using the Lyapunov Characteristic Exponents and the system is found to be chaotic.
Cite this paper
Singh, J. and Omale, S. O. (2019). Perturbed Planar Restricted Four-Body Problem with Repulsive Manev Potential. Open Access Library Journal, 6, e4980. doi: http://dx.doi.org/10.4236/oalib.1104980.
Maneff, G. (1924) La gravitation et le principe de l’egalie de l’action et de la reaction. Comptes Rendus de l’Académie des Sciences de Paris, 178, 2159-2161.
Maneff, G. (1925) Die Gravitation und das Prinzip von Wirkung und Gegenwirkung. Zeitschrift für Physik, 31, 786-802. https://doi.org/10.1007/BF02980633
Blaga, C. (2015) Prescessing Orbits, Central Forces and Manev Po-tential. In: Gerdjikov, V. and Tsetkov, M., Eds., Prof. G. Manev’s Legacy in Contempo-rary Aspects of Astronomy, Theoretical and Gravitational Physics, Heron Press Ltd., Chicago, IL, 134-139.
Ivanov, R. and Prodanov, E. (2005) Manev Potential and General Relativity. In: Gerdjikov, V. and Tsetkov, M., Eds., Prof. G. Manev’s Legacy in Contemporary Aspects of Astronomy, Theoretical and Gravitational Physics, Heron Press Ltd., Chicago, IL, 148-154.
Kirk, S., Haranas, I. and Gkigkitzis, I. (2013) Satellite Motion in a Manev Potential with Drag. Astrophysics and Space Science, 344, 313-320. https://doi.org/10.1007/s10509-012-1330-0
Barrabes, E., Cors, J. and Vidal, C. (2017) Spatial Collinear Restricted Four Body Problem with Repulsive Manev Potential. Celestial Mechanics and Dynamical Astronomy, 129, 153-176. https://doi.org/10.1007/s10569-017-9771-y
Bhatnager, K.P. and Hallan, P.P. (1978) Effect of Perturbations in Coriolis and Centrifugal Forces on the Stability of Liberation Points in the Restricted Problem. Celestial mechanics, 18, 105-112. https://doi.org/10.1007/BF01228710
Singh, J. and Vincent, A.E. (2015) Effect of Perturbations in the Coriolis and Centrifugal Forces on the Stability of Equilibrium Points in the Restricted Four-Body Problem. Few-Body Systems, 56, 713-723. https://doi.org/10.1007/s00601-015-1019-3
Abdul Raheem, A. and Singh, J. (2006) Combined Effects of Perturbations, Radiation, and Oblateness on Stability of Equilibrium Points in the Restricted Three-Body Problem. The Astronomical Journal, 131, 1880-1885. https://doi.org/10.1086/499300
Abouelmagad, I.A., Asiri, H.M. and Sharaf, M.A. (2013) The Effect of Oblateness in the Perturbed Restricted Three-Body Problem. Meccanica, 48, 2479-2490. https://doi.org/10.1007/s11012-013-9762-3
Arribas, M., Elipe, A. and Kalvouridis, T. (2007) Periodic Solutions in the Planar (n 1) Ring Problem with Ob-lateness. Journal of Guidance, and Dynamics, 30, 1640-1648. https://doi.org/10.2514/1.29524
Fakis, D.G. and Kalvouridis, T.J. (2013) Dy-namics of a Small Body under the Action of a Maxwell Ring-Type N-Body System with a Spheroidal Central Body. Celestial Mechanics and Dynamical Astronomy, 116, 229-240. https://doi.org/10.1007/s10569-013-9484-9
Kumari, R. and Kushvah, B.S. (2013) Equilibrium Points and Zero Velocity Surfaces in the Restricted Four-Body Problem with Solar Wind Drag. Astrophysics and Space Science, 344, 347-359. https://doi.org/10.1007/s10509-012-1340-y
Dubeibe, F.L. and Bermudez-Almanza, L.D. (2013) Optimal Conditions for the Numerical Calculation of the Largest Lyapunov Exponent for Systems of Ordinary Differential Equations. International Journal of Modern Physics C, 25, Article ID: 1450024. https://doi.org/10.1142/S0129183114500247