TITLE
Dynamical properties of the figure eight solution of the three-body problem
AUTHOR
Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi
Universitat de Barcelona
Gran Via 585, 08007 Barcelona
e-mail: carles@maia.ub.es
ABSTRACT
At the Conference on Celestial Mechanics held in Evanston, A.~Chenciner and
R.~Montgomery gave a proof of the existence of a remarkable periodic solution of
the 3--body problem, the "figure eight orbit", with all the bodies having
the same mass and following, with time shift equal to 1/3 of the period, the
same path on the plane. This solution lives on the zero level of the angular
momentum. See the references for a description of methods and results and
for some historical remarks.
The figure eight solution has "extremely remarkable properties". This paper
is devoted to describe some of them. The key property is stability. A part of
these results was presented at the above mentioned Conference, as a complement
to Chenciner and Montgomery lectures.
The second question we can address is "how exceptional" is the figure eight
solution. Are there other choreographies of the 3-body problem letting aside
the Lagrange and figure eight solutions? The question, posed like this, was
answered in a previous paper, together with Chenciner, Gerver and Montgomery.
Indeed, there are satellite choreographies of the eight and relative
choreographies (choreographies in a rotating frame) which can give rise to true
choreographies in fixed axes.
These choreographies are a direct consequence of the existence of the eight.
We ask for other choreographies in fixed axes, not related to the eight. The
answer is positive and, in fact, hundreds of new choreographies of the 3--body
problem have been found. They give, definitively, the numerical evidence of the
existence of several countable families. Still the figure eight preserves his
unique character due to its simplicity and because it is the only one found to
be stable.