We announce several theorems which suggest a minimal classification of relative equilibria in the planar «-body problem. These theorems also answer several questions on the nature of degenerate relative equilibria classes which were asked recently by S. Smale [3]. A summary of previous results can be found in an earlier paper [1]. It is a pleasure to thank S. Smale for encouragement in this work.1. Morse theory and relative equilibria. We study the critical set of a real analytic function V m < 0 on a real analytic manifold X m where n > 3 and m = (m 1 , . . . , m n ) GR"_ are fixed. Critical points of V m correspond in a 1-1 fashion to classes of relative equilibria. V m always has a compact critical set which we may investigate by Morse theory even when degenerate critical points exist [2].The integral singular homology of X m (a manifold which is homeomorphic to a Stein manifold P n _ 2 (Q ~ A w _ 2 ) is given by a recurrence relation [1]. This suggests that there is a uniform lower bound on the number of critical points of each index of V m which is given by recurrence. As a first step toward classifying relative equilibria Theorem 1 gives such a relation.In Theorem 2 we assert that V m is a Morse function for any n > 3 and for almost all m G R^_ (in the sense of Lebesgue measure).Theorem 3 answers the question: Is V m always a Morse function? Finally, we examine the case of four masses to show how a degeneracy of V m arises. An interpretation of Theorem 1 in the degenerate case sheds light on the creation and annihilation of relative equilibria.2. Main theorems. In this paragraph for any /, 0 < / < In -4, let M/00 denote a uniform lower bound to the number of critical points of V m AMS (MOS) subject classifications (1970). Primary 70F10; Secondary 57D70.

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An old problem of the evolution of finitely many interacting point vortices in the plane is shown to be amenable to investigation by critical point theory in a way that is identical to the study of the planar n-body problem of celestial mechanics. For any choice of positive circulations of the vortices it is shown by critical point theory applied to Kirchhoff's function that there are many relative equilibria configurations. Each of these configurations gives rise to a stationary configuration of the vortices in a suitably chosen rotating coordinate system. A sharp lower bound on the number of stationary vortex configurations for the problem of point vortices interacting in the plane is given. The problem of point vortices in a circular disk is defined and it is shown that these estimates hold for stationary configurations of small size.An old problem of fluid dynamics that was introduced as a Hamiltonian system by Kirchhoff in 1876 is to determine the evolution of finitely many interacting point vortices in the plane (see ref. 1). A recent experiment on superfluid 4He has demonstrated the existence of stationary configurations of vortices in a rotating reference frame (see ref.2).Here a stationary vortex configuration is identified as a relative equilibrium configuration. A sharp lower bound is given on the numbers of relative' equilibria classes that exist for an arbitrary choice of positive circulations (K, Kn) E R + for which the Hamiltonian function is nondegenerate. These estimates hold in the plane as well as for the problem of the motion ofrectilinear vortices in a circular cylinder. These lower bounds are derived from the topology ofthe domain ofKirchhoff's function. Compare the statements in celestial mechanics (see refs. 3 and 4).The idea is to use topological methods to investigate the sta-*tionary vortex configurations in the same way that the relative equilibria ofpoint masses in the n-body problem is investigated in celestial mechanics. The reason that many ofthe results from the celestial mechanics setting are valid immediately in this fluid dynamics setting is that the topology of the domain of the respective functions is identical. In Section 1 we view the stationary vortex configurations as relative equilibria configurations and the way in which critical points of the Hamiltonian arise is shown. Main theorems are stated in Section 2. The problem of point vortices in a circular disk is defined and investigated in Section 3. Finally, the observations are summarized.

The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.