

An Old Galactic Result - tcoppi
http://rjlipton.wordpress.com/2014/07/25/an-old-galactic-result/

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antognini
I work on other problems in three body dynamics, but I'm very pleased to see
this article show up here on HN!

One thing I could expand on a little is these singularities in Sundman's
series. These singularities are nothing more than collisions! It's these
collisions and near-collisions that can make the three body problem so hard.
Gravity is a 1/r^2 force, so the forces get very strong and change very
rapidly when the two objects come close to each other. This is not only hard
to handle analytically, but numerically as well!

In this field there is a well known transformation known as the Kustaanheimo-
Stiefel transformation (although the full name is not so well known---everyone
just refers to it as the KS transformation or KS regularization). What this
transformation does is it moves you from a regular Cartesian or polar
coordinate system (which has the force diverge when r --> 0) to a coordinate
system in which this singularity is moved to infinity. After a KS
transformation a computer can integrate even extremely elliptical orbits or
collisional orbits. The only downside to this is the algorithmic complexity.
In an ordinary Cartesian system, if you were to perform some numerical
integration the complexity goes like O(N^2), with N being the number of
objects in your system. The complexity of computing an orbit after KS
regularization is O(N^3), however. (My understanding of this is that KS
regularization works by turning position vectors into quaternions---which then
can basically be treated as matrices. Numerically evolving the system then
amounts to performing matrix multiplication, which goes as O(N^3). My
intuition may be flawed, however!) KS regularization therefore can't help if
you have more than ~10 objects. In practice in my research I haven't found it
useful even at N=3.

One last thing to mention is that even though these orbits can in principle be
solved numerically, they can't in practice. Three body systems are chaotic and
the machine precision of your computer leads to round off errors that cause
your calculation to diverge from the true solution remarkably quickly. There
are special-purpose programs that can calculate orbits to arbitrary precision,
but they run really, really slowly (as you might imagine), so they're not
useful for more general studies of three body systems. Fortunately, back in
1991, Quinlan & Tremaine showed that there exist so-called "shadow orbits"
that track the orbit numerically computed by your computer. These shadow
orbits are real orbits for some initial conditions that come arbitrarily close
to the numerically calculated orbits. This leads some credence to numerical
few body studies. Unfortunately, what's still unknown is whether these shadow
orbits have the same statistical properties as the general set of orbits. Not
enough research has been done on this question because it's still open and the
accuracy of my research depends on this property being true! But ultimately,
this question is probably linked to the ergodic hypothesis which is central to
the foundations of statistical mechanics---so it's an important question for
all physicists, not just me!

~~~
Sharlin
> One last thing to mention is that even though these orbits can in principle
> be solved numerically, they can't in practice.

This is true in the general case, but there are numerous examples of N-body
systems that are both of real-world interest and remarkably stable. The Solar
System, for instance. The reason is ultimately anthropic, of course - if it
weren't stable, we probably wouldn't be here observing it.

~~~
antognini
Many systems (including the Solar System) are stable over short time periods,
but the stability of the Solar System is actually unknown over longer time
periods. As an example, one of the numerical experiments Scott Tremaine has
done has been to simulate the solar system, but to perturb the initial
conditions of the planets by 1mm (that's one _millimeter_!). Over ~100 million
years, in ~1% of systems Mercury actually crashes into Venus! [1]

[1] [http://www.ias.edu/about/publications/ias-
letter/articles/20...](http://www.ias.edu/about/publications/ias-
letter/articles/2011-summer/solar-system-tremaine)

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troymc
If you're curious about the historical details of the various "solutions" of
the three-body problem (prior to 1996), then there's an excellent book for the
layman, titled _Poincare and the Three Body Problem_ by June Barrow-Green.

[http://www.worldcat.org/search?qt=worldcat_org_bks&q=Poincar...](http://www.worldcat.org/search?qt=worldcat_org_bks&q=Poincare+and+the+Three+Body+Problem&fq=dt%3Abks)

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RBerenguel
Fond memories of when I took a course on celestial mechanics. Weirdly enough,
the theoretical framework was taught by a more "applied" researcher, whereas
the "problem solving" classes were taught by one local leading researcher in
the field. I still remember determining degrees of singularities on collisions
and regularisations near collisions as an "easy problem to get up to speed"!

