Title: The dances of the N bodies

Author: Carles Sim\'o
        Departament de Matem\`atica Aplicada i An\`alisi
        Universitat de Barcelona, Barcelona, Catalunya
        {\tt carles@maia.ub.es}

Talk delivered at opening lecture of the Curs Lagrange 2013-2014,
Facultat de Matem\`atiques i Estad\'{\i}stica,
Universitat Polit\`ecnica de Catalunya
Barcelona, October 2, 2013.

Abstract:

There are periodic solutions of the planar N-body problem, with bodies of
equal masses, in which all the bodies move along the same path, with a
phase shift from one body to next one equal to T/N, where T is the period.
These solutions are named choreographies. If one has two such solutions and
one can pass from one to the other by change of scale, rotation or symmetry,
they are considered as equivalent.
We display several examples, the formulation and numerical solution of the
problem and give evidence that, even in the case N=3, there are infinitely
many non-equivalent choreographies.