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26 April 2024 |
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The Lagrange reduction of the N-body problem, a survey | | Alain Chenciner
; | | Date: |
5 Nov 2011 | | Abstract: | In his fondamental "Essay on the 3-body problem", Lagrange, well before
Jacobi’s "reduction of the node", carries out the first complete reduction of
symetries. Discovering the so-called homographic motions, he shows that they
necessarily take place in a fixed plane. The true nature of this reduction is
revealed if one considers the n-body problem in an euclidean space of arbitrary
dimension. The actual dimension of the ambiant space then appears as a
constraint, namely the angular momentum bivector’s degeneracy. The main part of
this survey is a detailed description of the results obtained in a joint paper
with Alain Albouy published in french (Inventiones 1998): for a non homothetic
homographic motion to exist, it is necessary that the space of motion be even
dimensional. Two cases are possible: either the configuration is "central"
(that is a critical point of the potential among configurations with a given
moment of inertia) and the space where the motion takes place is endowed with
an hermitian structure, or it is "balanced" (that is a critical point of the
potential among configurations with a given inertia spectrum) and the motion is
a new type, quasi-periodic, of relative equilibrium. Hip-Hops, which are
substitutes to the non-existing homographic solutions with odd dimensional
space of motion, are also discussed. | | Source: | arXiv, 1111.1334 | | Services: | Forum | Review | PDF | Favorites | |
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