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Non-avoided crossings for n-body balanced configurations in R^3 near a central configuration | | Alain Chenciner
; | | Date: |
25 Nov 2014 | | 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. 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. | | Source: | arXiv, 1411.6935 | | Services: | Forum | Review | PDF | Favorites | |
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