
Planar Choreographies: odd orbital mechanics - ColinWright
http://www.maths.manchester.ac.uk/~jm/Choreographies/
======
ColinWright
The paper is here: <http://arxiv.org/abs/1305.0470>

    
    
        Classification of symmetry groups for planar n-body choreographies
        James Montaldi, Katrina Steckles
        (Submitted on 2 May 2013)
    
        Since the foundational work of Chenciner and Montgomery in 2000 there
        has been a great deal of interest in choreographic solutions of the
        n-body problem: periodic motions where the n bodies all follow one
        another at regular intervals along a closed path. The principal approach
        combines variational methods with symmetry properties. In this paper, we
        give a systematic treatment of the symmetry aspect. In the first part we
        classify all possible symmetry groups of planar n-body, collision-free
        choreographies. These symmetry groups fall in to 2 infinite families and,
        if n is odd, three exceptional groups. In the second part we develop the
        equivariant fundamental group and use it to determine the topology of the
        space of loops with a given symmetry, which we show is related to certain
        cosets of the pure braid group in the full braid group, and to centralizers
        of elements of the corresponding coset.

~~~
moondowner
"periodic motions where the n bodies all follow one another at regular
intervals along a closed path"

Out of curiosity, does someone has any idea where is this applicable (&
already is applied)? I'm already starting to imagine some different scenarios
(maybe in hydraulics, computer networks, etc)...

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VLM
That's cool, and attractive, and visually appealing.

I do have a request. Look at stuff like "Belbruno Orbits" "Low energy transfer
orbits" "ITN Interplanetary Transport Network". Then look at the nice clean
easy to use visualizer we're linked to. Then do the obvious merger of the
two...

Yes I already know there's a way to visualize Belbruno orbits with ORSA or
other full fledged simulation packages, but its not quite as easy and
convenient as this webpage.

You know what would make an interesting web standard or maybe startup idea? A
universal internet standard free dynamics system. Not just for orbits but even
physics 101 basic kinematics. Think of like the animated drawing blueprints on
"Mythbusters" but simply include a javascript package of some type (or
whatever) and then the end user merely provides three things: enumerated list
of URL for sprite graphics, starting conditions for those objects both simple
coordinates and maybe "hidden" variables, and the math equation(s) governing.
Not general purpose "here's mathematica in node.js" just dynamics.

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peterarmitage
A few weeks ago some new solutions to the three-body problem were discovered:
<https://news.ycombinator.com/item?id=5347412>

I made some simulations of them using three.js:
<http://www.funcmain.com/three_body_problem>

Mine are pretty much the same kind of examples as those posted here, except
specifically for the case N=3.

~~~
sengstrom
Nicely done - I like the option to spin around the group and get a 3D feel for
it, but it seems that all the examples you have are strictly planar. Are there
any true 3D solution to these choreographed N-body problems?

~~~
ISL
The problem is vastly easier with initial positions and velocities constrained
to the plane. If they start that way, they can't ever leave. Even if you have
a coulomb/newtonian potential, proving stability will be far more
straightforward.

I'd love to be surprised, but I suspect closed and collisionless solutions
with any out-of-plane components would be newsworthy and immediately
publishable.

~~~
csense
> closed and collisionless solutions with any out-of-plane components would be
> newsworthy and immediately publishable.

Trivial example: What about two tiny spaceships (they're small enough that
their effect on each other is negligble) orbiting a star, one orbiting in the
XY plane the other in the XZ plane?

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raverbashing
This is very interesting

As someone that did a planet trajectory simulation program (but much simpler
than that) this is very nice and interesting.

What would be interesting though is adding perturbation to the orbits to see
how stable are they, or how likely is this to happen in a real situation

It usually happens with one or two massive elements (stars) and less massive
elements (planets), all with different masses

So for two relatively equal stars and one big and one small planet with
different energies I bet there are several 'choreographies' available, maybe
with chaotic behaviour.

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arnarbi
Amazing. If we ever find stars orbiting like that, I might start believing in
intelligent design.

Imagine living on a planet in the 7-butterfly pattern. Twice a year (or
whatever) you'd see the other planets heading straight for you as you whizz
through the apex in the middle.

~~~
splat
It's still unknown whether or not these orbits are stable. It seems probable
that they're unstable, in which case you would never expect to find them in
the universe.

~~~
leephillips
They're probably all unstable except for the figure-8. Numerical experiments
suggest that there should be somewhere from one to 100 figure-8 systems in the
observable universe:
<http://www.scholarpedia.org/article/N-body_choreographies>

~~~
lholden
Also worth noting that the mass of each star involved in a figure-8 would have
to be pretty much identical.

~~~
leephillips
All these solutions are for bodies of equal mass.

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lmm
I'm reminded of an SF story (Reynolds?) in which a set of three identically-
sized bodies in orbit were the one indisputable relic of an alien civilization
- maybe a work of art, maybe a demonstration of power, maybe something else.

~~~
VLM
"maybe something else."

Alien contact sci-fi plots usually operate from human experience of conquest
or a couple other ideas that are way too serious. I've often thought an
interesting new plot idea would be a practical joke. Squaring a circle is so
crude but still kinda funny. We know ringworlds are unstable so for a peculiar
definition of funny, a stable apparent ringworld (perhaps built with stealthy
invisible stabilizing structures or something). An apparently long term stable
3 body equal mass orbit would count as a practical joke. I'm just saying a
story about an artifact that appeals to a stage magician with some sneakiness
might be a fun change of pace from an artifact designed to appeal to the
physicists and .mil generals and computer scientists.

I wonder what an alien would think of watching a really accomplished human
stage magician. "Holy F these guys have teleporters who couldda guessed?"
"Whoa some members of this species are also telepaths?" "These guys can
bioassemble/replicate a living rabbit using lab gear that fits in a modest top
hat?"

~~~
sesqu
There's Pratchett's book The Dark Side of the Sun, in which one of the alien
races, the Jokers, is known only by the artefacts they've left behind. These
include fossils from the future, monomolecular towers, and ring stars.

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RBerenguel
There are quite a few also here: <http://www.maia.ub.es/dsg/nbody.html> Don't
expect anything fancy: you need to untar the file and have gnuplot to make it
work.

Carles was one of the teachers in the PhD program I was (or am? I'm finishing
my PhD thesis so I don't know how the timing has to be considered) in.

------
est
If you turn off trajectory, you will discover amazing similar patterns with
this factorization animation:

[http://www.datapointed.net/visualizations/math/factorization...](http://www.datapointed.net/visualizations/math/factorization/animated-
diagrams/)

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kvb
Perhaps also of interest, here's a page that lets you draw an arbitrary curve
and attempts to find a choreography close to it: <http://gminton.org/#choreo>.

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mjcohen
I find it interesting that some things are visible if the trajectories are
turned off that I would never see with them turned on.

For example, if the last example (10 on an octagram) is viewed with
trajectories off, I see two rotating intersecting pentagons, and I missed this
completely with traj on.

Similarly, with 8 on a (9/4) enneagram, I see two non-intersecting rotating
squares.

Nice.

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obviouslygreen
Oh man... this reminds me of Gravitation:

<http://macintoshgarden.org/games/gravitation-ltd-50>

We played with this constantly when I was a kid.

~~~
lholden
I used to play with something similar back in the 80s. Wish I could remember
what it was!

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halodweller
With all the globular clusters out there, some formation like these must've
broken away somewhere...amazing! The non-symmetrical one was the coolest.

~~~
nine_k
Live in a world shaped like one of these? Enjoy your attempts to discover the
Newtonian laws of gravity :)

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sdoering
Never heard of n-Body choreographies before and beautifully lost 5 minutes of
my life. Thanks a lot for this. really great and quite interesting.

