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.
"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)...
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.
Nice, I liked how I could see the relative attractiveness change as the bodies moved.
What was the shading on the bodies for? At first I thought the bright side might be pointed towards the centre of mass of the other two, but it seems not.
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?
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.
> 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?
Thanks. I'm not a physicist, I was merely drawing pretty pictures based on real physicists' work. The examples in the paper are all planar (and have zero angular momentum), but it would be easy to simulate arbitrary orbits if any are published.
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.
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.
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.
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
A pretty idea, but planetary orbits in a multi star system are pretty hard. You either have to orbit in close enough to one of the stars that the other star doesn't destabilize your orbit, or you have to orbit far enough out that you orbit the center of mass for all of the stars involved.
As a side note... While not quite the same we have observed some pretty cool orbits around the super massive black hole at the center of our galaxy. https://en.wikipedia.org/wiki/File:Galactic_centre_orbits.sv... (I doubt any of those stars have planets either. Each time they go wizzing by each other or the black hole the planets would be flung off. Pretty chaotic mess in there :D)
One could argue that given the number of stars out there, there must be some in fairly interesting orbits. I was trying to figure out what these would look like edge on if we were observing them from earth. In some scenarios (like the box) they would look kind of like a binary star system where the spectrum of the stars changed periodically (depending in which of the stars were in 'front' toward our point of view)
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.
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?"
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.
In Larry Niven's Ringworld we learn that the Puppeteer's fleet of planets are arranged in a perfect pentagon, fleeing the inevitable eruption of the galactic core.
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.
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.
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.
