A collection of animations, sites, pictures, a few papers ... concerning the classical N-body problem

The figure eight solution:


Three planets of equal masses chase each other around a figure-eight shaped curve under the influence of their mutual gravitational attraction.
initial conditions:
positions: (x1,y1) = (-0.97000436, 0.24308753), (x2,y2) = (-x1, -y1), (x3,y3) = (0,0)
velocites: (vx1,vy1) = (vx2, vy2) = -(vx3, vy3)/2; where (vx3,vy3) = (0.93240737, 0.86473146)
masses: 1. gravitational constant: 1. i.c.s found by C Sim

N-body tours built for lectures

Minton's build-your-own Interactive choreographies

Celtic knot choreographies illustrating the equivariant homotopic encodings of: Montaldi and Steckles

Open Problems , compiled by Albouy, Cabral, and Santos

More than 300 planar choreographies found by Carles Simó ; implemented by Paul Masson

3D Platonic and Archimedean Solid symmetric orbits found by Gronchi and Fusco. .. burtleburtle's N body orbit site Bob Jenkins site. All kinds of things ....
13 orbits -Suvakov & Dmitrasinovi zero angular mom. equal mass 3 bdy; beginnings of a free homotopy coding ;
100s of orbits -Suvakov & Dmitrasinovi zero angular mom. equal mass 3 bdy; all starting from Euler.


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.
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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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(unpublished) ***************************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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An unpublished arxival version
``Figure 8s with three bodies'' (an earlier incomplete unpublished version of the eventual paper with Chenciner.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 orbit for the Newtonian three body problem: with three equal point masses chase each other 'round 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 Fujiwara and I gave an analytic proof that each of the eight's two lobes are convex.

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.


**** overflow... non-collision 3-body brake (Nai-Chia Chen) 4 body collision families (Nai-Chia Chen) (both coded in 2014 by Nai-Chia Chen) isosceles families (Nai-Chia Chen; 2013). reminescent of Paul Klee