Non-avoided Crossings for n-Body Balanced Configurations in $${\mathbb {R}}^3$$ R 3 Near a Central Configuration

Abstract

The balanced configurations are those n-body configurations which admit a relative equilibrium motion in a Euclidean space E of high enough dimension 2p. They are characterized by the commutation of two symmetric endomorphisms of the $$(n-1)$$ -dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism B which encodes the shape and the Wintner–Conley endomorphism A which encodes the forces. In general, p is the dimension d of the configuration, which is also the rank of B. Lowering to $$2(d-1)$$ the dimension of E occurs when the restriction of A to the (invariant) image of B possesses a double eigenvalue. It is shown that, while in the space of all $$d\times d$$ symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition (H) is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if $$d=n-1$$ ), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of four bodies with no three of the masses equal, of exactly three families of balanced configurations which admit relative equilibrium motion in a four dimensional space.