2014/11/25 by Alain Chenciner, Chenciner, Alain
Engineering · Mathematics · Physics and Astronomy · #Robotic Mechanisms and Dynamics #Scientific Research and Discoveries #Spacecraft Dynamics and Control #math-ph #math.DS #math.MP #msc:15B57 #msc:37J15 #msc:70F10 #msc:70H12
paper · pdf · doi:10.48550/arxiv.1411.6935
35 pages, 1 diagram, 6 figures Section 1.5.2 is new: it introduces the condition (H) which had been overlooked in the first version
arxiv created 2015/08/08 · arxiv updated 2015/08/11
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.