A Short Tour of famous 3-body solutions ... in suggested, roughly historical, order.

Another Tour of 2 and 3-body solutions made for Feb 18 2015 colloquium at CIMAT, Guanajuato

a tour based on other sites... made for Feb 18 2015 colloquium at CIMAT, Guanajuato

Astounding interactive sites:
Fujiwara's rendition of some central configurations
Minton's build-yr-own choreography scribbler

2013 open problem list: Some Problems on the Classical N-Body Problem , compiled by Albouy, Cabral, and Santos


Animations:
2014 non-collision three-body brake orbit (Nai-Chia Chen) , 4 body collision families (Nai-Chia Chen)

2013:
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
--- Earlier:
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
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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.
--- Earlier:
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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ARTICLES:
Scholarpedia :
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(unpublished) ***************************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:
On the N-body problem
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.)


Commentary on the eight. Alain Chenciner and I rediscovered this surprising new orbit for the Newtonian three body problem with three equal masses. They chase each other around a figure eight in the plane! C. Simo has shown (numerically) that it is (KAM) stable: more precisely: as stable as a general periodic orbit in a 3 degree of freedom Hamiltonian system can be (pure imag. eigenvalues, torsion non-zero).
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.