Fujiwara's rendition of some central configurations

Minton's build-yr-own choreography scribbler

isosceles families by Nai-Chia Chen. reminescent of Paul Klee

more choreographies ; beautiful animations by Greis; based on the paper of Montaldi and Steckles

13 orbits -Suvakov & Dmitrasinovi zero angular mom. equal mass 3 bdy; beginnings of a free homotopy coding

Fusco-Gronchi-Negrini 3D Archimedean Solid orbits platonic solid symmetric orbits

A dozen choreographies (via Simó) quick renditions -thanks Charley McDowell!

Pythagorean three-body problem

spherical visualization and animations 345 choreos uncovered by Simó; w 3 bodies programming: Troy Fisher

Carles Simó's 47 choreographies to download. R=requires gnuplot.

burtleburtle's N body orbit site Bob Jenkins site. All kinds of things to explore .

How Simo made his movies. plus ending remarks on which choreographies might be realized.

Ferrario: includes a 60-body orbit w icosahedral symmetry [entered, 2006]

Cris Moore gallery

Fujiwara! three tangents theorem; other geometric 3-body theorems

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The Pythagorean (3-4-5) three body problem, aka Burrau's solution. This is Greg Laughlin's favorite. as a dance performance and: a post on the performance

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3-body problem -Chenciner

N-body choreographies -me

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2011

5 Lectures on the N body Problem, Madrid

(unpublished) ***************************

MORE GOOD SHTUFF !! Vanbderbei

Sverre Aarseth, a champ of N-body integrators

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ARXIVAL MATERIAL: 2003 and early entries:

OPEN PROBLEMS

From the June 2003 AIM/ARCC conference; notes by K-C Chen

modified version of open problem list from 'HAMSYS 2001' Guanajuato, Mexico, March 2001. open problems (postscript) OR pdf (latest correction to list : jan 22, 2003: thank you A. Albouy)

References for Stanford 2003 seminar. This seminar meets Tu 4:15-5:15 , autumn 2003, in Stanford. Write to me or Rafe Mazzeo for room and other details.

my N-body papers can be found here and one unpublished:

``Figure 8s with three bodies'' (an earlier incomplete unpublished version of the eventual paper with Chenciner. See commentary below.)

Cris Moore , (Santa Fe inst.) had earlier found the same, via numerical integration and a gradient search. In a Physical Reviews Letters paper, 1993, vol 70. In 2003 with Fujiwara we found an analytic proof that each of the eight's two lobes are convex.

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.