The figure eight orbit
initial conditions: (x1,y1) = (-0.97000436, 0.24308753), (x2,y2) = (-x1, -y1), (x3,y3) = (0,0)
initial velocites (vx1,vy1) = (vx2, vy2) = -(vx3, vy3)/2; where (vx3,vy3) = (0.93240737, 0.86473146)
masses: 1. gravitational constant: 1.
thanks to
Carles Simo
numerical wizard.
(The initial positions are chosen to be collinear, with center of mass zero.
The initial velocities are chosen so that the total linear and angular momenta
are zero, and so that the total moment of inertia $\Sigma m_a (x_a ^2 + y_a ^2)$
is extremized. This leaves two free parameters represented by the velocity of mass 3.)
A bit of history.
The unpublished paper mentioned
above predated the Annals paper with Chenciner.
The `eight' of its title was in shape space not in inertial space.
and was a different orbit from the eventual eight,
an orbit whose existence is still in question.
Chenciner and Venturelli discovered an error in the proof of `theorem 1'
which claimed the existence of this orbit.
Theorem 2 led to my paper
with Chenciner. Venturelli, in
his thesis, made use of some ideas here
concerning using
local perturbation
analysis to delete collisions and thereby decreasing
action, and concerning the constancy of energy
along collision minimizers.
A popular accounting of the eight and more new orbits
can be found in the
Notices article plus Casselman's commentary and pictures
Choreographies, are
orbits in which N planets chase each other around the same
planar curve. The eight and Lagrange's orbit are choreographies.
Soon after we rediscovered the
eight solution, a horde of new solutions were
discovered using the same ideas: variational methods
plus symmetry. For the Newtonian potential
case we lack existence proofs for all new solutions
except the eight. In 2003
Ferrario and Terrracini
proved the existence of some infinite
families of choreographies.
The breakthrough was Marchall's averaging of
perturbations idea, as exposed by
Chenciner's ICM notes
All of these new variational minimizers
EXCEPT the eight are unstable dynamically.