Non-avoided crossings for n-body balanced configurations in R^3 near a central configuration

TL;DR

This paper investigates the conditions under which n-body balanced configurations in R^3 exhibit non-avoided crossings, revealing new geometric and spectral properties that lead to the existence of specific families of relative equilibria.

Contribution

It demonstrates that the codimension of double eigenvalues in the force-related endomorphism is reduced under certain conditions, enabling new families of balanced configurations near a regular tetrahedron.

Findings

Double eigenvalues have codimension 1 under condition (H).

Existence of 3 families of balanced configurations near a regular tetrahedron.

Configurations of maximal dimension always satisfy condition (H).

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 dxd-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 4 bodies with no three of the masses equal, of exactly 3 families of balanced configurations which admit relative equilibrium motion in a four dimensional space.