Abstract
We consider four bodies in space with same masses forming two binaries, each one symmetric with respect to a fixed axis and moving under Newtonian gravitation in opposite directions about this axis. It is given a direct proof that all singularities of this model are due to collisions, and it is proved that the singularities due to simultaneous double collisions are regularizable.
The set of equilibrium points on the total collision manifold is studied as well as the possible connections among them. We show that the set of initial conditions on a given energy surface going to quadruple collision is a union of twenty submanifolds: twelve of them have dimension 2 and the others have dimension 3. Similarly for ejection orbits from quadruple collision.
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References
Cabral, H. E.: 1973, ‘On the integral manifolds of the N-body problem’, Invent. Math. 20, 59–72.
Devaney, R. L.: 1981, ‘Singularities in classical mechanical systems’, in: A. Katok (ed.), Ergodic Theory and Dynamical Systems I, Proc. Special Year, Maryland, 1979-80, Birkhauser, pp. 211–333.
Easton, R. and McGehee, R.: 1979, ‘Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere’, Indiana Univ. Math. J. 28, 211–240.
Llibre, J. and Simó, C.: 1980, ‘Some homoclinic phenomena in the three-body problem’, J. Diff Equations 33, 444–465.
Llibre, J. and Simó, C.: 1980, ‘Oscillatory solution in the planar restricted three-body problem’, Math. Annalen 248, 153–184.
McGehee, R.: 1974, ‘Triple collision in the collinear three-body problem’, Invent. Math. 27, 191–227.
McGehee, R.: 1973, ‘A stable manifold theorem for degenerate fixed points with applications to celestial mechanics’, J. Diff. Equations 14, 70–88.
Moeckel, R.: 1984 ‘Heteroclinic phenomena in the isosceles three-body problem’, Siam J. Math. Anal. 15, 857–876.
Moser, J.: 1973, Stable and Random Motions in Dynamical Systems, Princeton.
Painlevé, P.: 1897, Leçons sur la thérie analytique des equations différentielles, A. Hermann, Paris.
Saari, D.: 1973, ‘On oscillatory motion in gravitational systems’, J. Diff Equations 14, 275–292.
Saari, D. and Xia, Z.: 1989, 'The existence of oscillatory and super-hyperbolic motions in Newtonian systems, J. Diff Equations 82, 342–355.
Siegel, C. and Moser, J.: 1971, Lectures on Celestial Mechanics, Springer-Verlag, New York, Heildelberg, Berlin.
Simó, C. and Lacomba, E.: 1982, ‘Analysis of some degenerate quadruple collisions’, Celest. Mech. 28, 49–62.
Sitnikov, K.: 1964, ‘The existence of oscillatory motion in the three-body problem’, Soviet Physics Doklady 5, 647–656.
Smale, S.: 1970, ‘Topology and mechanics I’, Invent. Math. 10, 305–331.
Sundman, K.: 1909, ‘Nouvelles recherches sur le problème des trois corps’, Acta Soc. Sci. Fenn. 35(9).
Vidal, C.: 1996, ‘O problema tetraedral simétrico com rotaç ão’, Thesis for Degree of Doctor in Mathematics, Universidade Federal de Pernambuco, Brazil.
Vidal, C. and Delgado, J.: ‘The tetrahedral four-body problem’, Preprint.
Wintner, A.: 1941, The Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton, N. J.
Xia, Z. 1992, ‘The existence of non-collision singularities in Newtonian systems’, Annal. Math. 135, 411–468.
Xia, Z.: ‘A heteroclinic map and the oscillatory, capture, escape motions in the three-body problem’, Hamiltonian Systems and Celestial Mechanics (Guanajuato, 1991), pp. 191–208, Adv. Ser. Nonlinear Dynamics 4, World Sci. Publishing, River Edge, NJ, 1993.
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Vidal, C. The Tetrahedral 4‐Body Problem with Rotation. Celestial Mechanics and Dynamical Astronomy 71, 15–33 (1998). https://doi.org/10.1023/A:1008397202674
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DOI: https://doi.org/10.1023/A:1008397202674