A collection of animations, sites, pictures, a few papers ... concerning the classical N-body problem
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.)
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
Carles Simó's 47 choreographies
to download.
R=requires gnuplot.
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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Scholarpedia :
3-body problem
-Chenciner
N-body choreographies
-me
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2011
5 Lectures on the N body Problem, Madrid
(unpublished)
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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.
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.
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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