Abstract
We consider \((n+1)\) bodies moving under their mutual gravitational attraction in spaces with constant Gaussian curvature \(\kappa \). In this system, \(n\) primary bodies with equal masses form a relative equilibrium solution with a regular polygon configuration; the remaining body of negligible mass does not affect the motion of the others. We show that the singularity due to binary collision between the negligible mass and the primaries can be regularized locally and globally through suitable changes of coordinates (Levi-Civita and Birkhoff type transformations).
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References
Andrade, J., Pérez-Chavela, E., Vidal, C.: Regularization of the circular restricted three body problem on surfaces of constant curvature. J. Dyn. Differ. Equ. 30, 1607–1626 (2017)
Andrade, J., Perez-Chavela, E., Vidal, C.: The restricted three body problem on surfaces of constant curvature. J. Differ. Equ. 265, 4486–4529 (2018)
Birkhoff, G.: The restricted problem of three bodies. Rend. Circ. Mat. Palermo 39, 1 (1915)
Bolyai, W., Bolyai, J.: In: Stäckel, P. (ed.) Geometrische Untersuchungen. Teubner, Leipzig (1913)
Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Two body problem on a sphere. Reduction, stochasticity, periodic orbits. Regul. Chaotic Dyn. 9, 3 (2004)
Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: The spatial problem of two bodies on the sphere. Reduction and stochasticity. Regul. Chaotic Dyn. 21, 5 (2016)
Diacu, F.: Relative Equilibria of the Curved N-Body Problem, vol. 1. Atlantis Press, Amsterdam (2012)
Diacu, F., Pérez-Chavela, E., Santoprete, M.: The \(n\)-body problem in spaces of constant curvature. Part I: relative equilibria. J. Nonlinear Sci. 22, 247–266 (2012a)
Diacu, F., Pérez-Chavela, E., Santoprete, M.: The \(n\)-body problem in spaces of constant curvature. Part II: singularities. J. Nonlinear Sci. 22, 267–275 (2012b)
Diacu, F., Sánchez-Cerritos, J.M., Zhu, S.: Stability of fixed points and associated relative equilibria of the 3-body problem on \(S^{1}\) and \(S^{2}\). J. Dyn. Differ. Equ. 30, 209–225 (2018)
Lovachevski, N.: The new foundations of geometry with full theory of parallels. In: Collected Works, vol. 2, p. 159. GITTL, Moscow (1949). In Russian 1835–1838
MacGregor, M.A., et al.: A complete map of the Fomalhaut debris disk (2017). arXiv:1705.05867v1 [astro-ph.EP]
Marsden, J.E.: Basic Complex Analysis. W.H. Freeman and Company, San Francisco (1973)
Martínez, R., Simó, C.: Relative equilibria of the restricted three body problem in curved spaces. Celest. Mech. Dyn. Astron. 128, 221–259 (2017)
Maxwell, J.C.: On the stability of the motion of Saturn’s rings. In: Niven, W.D. (ed.) The Scientific Papers of James Clerk Maxwell, pp. 288–376. Cambridge University Press, Cambridge (1890)
Maxwell, J.C.: In: Brush, S.G., Everitt, C.W.F., Garber, E. (eds.) Maxwell on Saturn’s Rings. MIT Press, Cambridge (1983)
Pérez-Chavela, E., Reyes-Victoria, J.G.: An intrinsic approach in the curved \(n\)-body problem. The positive curvature case. Trans. Am. Math. Soc. 364(7), 3805–3827 (2012)
Pérez-Chavela, E., Sánchez-Cerritos, J.M.: Euler-type relative equilibria in spaces of constant curvature and their stability. Can. J. Math. 70(2), 426–450 (2018a)
Pérez-Chavela, E., Sánchez-Cerritos, J.M.: Hyperbolic relative equilibria for the negative curved \(n\)-body problem. Commun. Nonlinear Sci. Numer. Simul. (2018b). https://doi.org/10.1016/j.cnsns.2018.07.022
Richter, K., Tanner, G., Wintgen, D.: Classical mechanics of two-electron atoms. Phys. Rev. A 48(6), 4182–4196 (1993)
Roman, R., Szücs-Csillik, I.: New regularization of the restricted problem. Rom. Astron. J. 22(2) (2012)
Roman, R., Szücs-Csillik, I.: Generalization of Levi-Civita regularization in the restricted three-body problem. Astrophys. Space Sci. 349, 117–123 (2014)
Stiefel, L., Scheifele, G.: Linear and Regular Celestial Mechanics. Springer, Berlin (1971)
Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York (1967)
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We thank the anonymous referee for comments and criticism, which helped us to improve this work. The first author has been partially supported by Asociación Mexicana de Cultura A.C.
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Pérez-Chavela, E., Sánchez-Cerritos, J.M. Regularization of the restricted \((n+1)\)-body problem on curved spaces. Astrophys Space Sci 364, 170 (2019). https://doi.org/10.1007/s10509-019-3655-4
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DOI: https://doi.org/10.1007/s10509-019-3655-4