Prof. Dr. Alain Chenciner, Universität Paris VII, Vortrag: Angular momentum and Horn's problem

Wednesday, 04.05.2016 16:30 im Raum M5

Mathematik und Informatik

Abstract: The central configurations of n point masses are those configurations x = (r 1 , · · · , r n ) which admit periodic rigid motions when submitted to newtonian attraction. For example, Lagrange has proved that the only non colinear central configuration of 3 positive masses is the equilateral triangle. Such rigid motions necessarily take place in an euclidean space E of even dimension 2p and, the initial configuration x 0 being given, they are of the form r i (t) = e ωtJ r i (0)), where J is a complex structure on E compatible with the euclidean structure, that is an isometry such that J 2 = −Id ([C1]). The study of the angular momentum of such motions leads to the following purely algebraic question : Let S 0 be a symmetric non negative 2p × 2p matrix (the inertia matrix of the configuration x 0 ). What can be said of the mapping F which, to each J, associates the ordered spectrum {ν 1 ≥ ν 2 ≥ · · · ≥ ν p } of the J-hermitian matrix S 0 + J −1 S 0 J, considered as a complex p × p matrix ? On the other hand, Horn’s problem asks for the possible spectra of ma- trices C = A + B, where A and B are hermitian (or real symmetric) with given spectra. Introducing two Horn’s problems, one in dimension p and one in dimen- sion 2p, and using a deep combinatorial lemma, proved in [FFLP], one proves [C1, C-JP] that the image of F is a convex polytope which can be described.

Angelegt am Wednesday, 24.02.2016 09:19 von shupp_01
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