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Invent math (2011) 185:283332
DOI 10.1007/s00222-010-0306-3
G. Fusco G.F. Gronchi P. Negrini
Received: 3 July 2010 / Accepted: 28 November 2010 / Published online: 16 December 2010 Springer-Verlag 2010
Abstract We prove the existence of a number of smooth periodic motions u of the classical Newtonian N-body problem which, up to a relabeling of
the N particles, are invariant under the rotation group R of one of the ve
Platonic polyhedra. The number N coincides with the order |R| of R and
the particles have all the same mass. Our approach is variational and u is
a minimizer of the Lagrangian action A on a suitable subset K of the H1
T -periodic maps u :
R
G. Fusco
Dipartimento di Matematica Pura ed Applicata, Universit di LAquila, 67100 LAquila, Italye-mail: mailto:[email protected]
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G.F. Gronchi ( )
Dipartimento di Matematica, Universit di Pisa, 56127 Pisa, Italy e-mail: mailto:[email protected]
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P. Negrini
Dipartimento di Matematica, Universit di Roma La Sapienza, 00185 Rome, Italy e-mail: mailto:[email protected]
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Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem
R3N. The set K is a cone and is determined by
imposing on u both topological and symmetry constraints which are dened in terms of the rotation group R. There exist innitely many such cones K,
all with the property that A|K is coercive. For a certain number of them, using
level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometrickinematic structure.
284 G. Fusco et al.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
2 An example of a cone K . . . . . . . . . . . . . . . . . . . . . . . . 290
3 Cones K and Platonic polyhedra I . . . . . . . . . . . . . . . . . . . 291
4 Cones K and Platonic polyhedra II . . . . . . . ....