TITLE
Periodic orbits of the planar $N$--body problem with equal masses and all
bodies on the same path
AUTHOR
Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi
Universitat de Barcelona
Gran Via de les Corts Catalanes, 585 08007 Barcelona
e-mail:carles@maia.ub.es
To appear in
ABSTRACT:
Very few things are known about the solutions of the $N$--body problem, either
under the action of the Newtonian potential or other homogeneous potentials
of the form $1/r^a,\, a>0.$ Only some partial results about central
configurations are available.
In this lecture we review some recent results concerning the existence of
periodic solutions of the planar $N$--body problem, with all masses equal,
such that the $N$ bodies travel along the some path on the plane. These
orbits are denoted as {\em choreographies}. A huge amount of families
have been found numerically. Existence proofs are based on variational
methods and require, up to now, the exponent $a\ge 2$ in the potential.