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The rhomboidal four body problem. Global flow on the total collision manifold. (English) Zbl 0733.70007

The geometry of Hamiltonian systems, Proc. Workshop, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 22, 97-110 (1991).
Summary: [For the entire collection see Zbl 0733.00016.]
In this work, we have considered a particular case of the planar four- body problem, obtained when the masses form a rhomboidal configuration. If we take the ratio of the masses \(\alpha\) as a parameter, this problem is a one parameter family of non-integrable Hamiltonian systems with two degrees of freedom. We use the blow up method introduced by R. McGehee [Inventiones Math. 27, 191-227 (1974; Zbl 0297.70011)] to study total collision. This singuarity is replaced by an invariant two- dimensional manifold, called the total collision manifold. Using numerical methods we prove first that there are two equilibrium points for the flow on this manifold, and second, that there are only two values of \(\alpha\) for which there is a connection between the invariant submanifolds of the equilibrium points. For these values of \(\alpha\) the problem is not regularizable.

MSC:

70F10 \(n\)-body problems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70F35 Collision of rigid or pseudo-rigid bodies