You would think that after three hundred years the consequences of the law of gravity for “simple” systems would have been pretty well studied and nothing surprising would emerge. Amazingly there is such richness and complexity in the physics of three masses that new discoveries are still being made.
The opening of the door was the paper “Braids in Classical Gravity” by Christopher Moore 1. This did open the door but the paper seems to have missed its audience a bit since one of its results was rediscovered a few years later 2. What these papers discovered was new families of solutions to the gravitational N-body problem not just for N=3 but for much more complex systems. They involve a mathematical assumption that the N bodies follow the same path and were termed choreographies. The gem at the center of all these solutions is a stable figure-8 solution for three equal masses.
These solutions are hypnotizing and as soon as I learned of them I had to get them into the N-body app. The solutions have very special initial conditions and when I first put them in they did not evolve as expected. The version of N-body at the time had a very simple numerical integrator, known as Leapfrog with a fixed time step 3. The choreographies required a more careful approach. Even though N-body was close to being ready to go I could just not give up on these choreographies. So I put in a 4th order Hermite integrator with some adaptive time step capabilities. This was enough to get the simple chain choreographies running. (I then found some more cool choreographies on arXiv, in fact new classes of choreographies. These will need more integrator tuning but look for them in a future update to N-body. In addition the Java gallery by Vanderbei has some solutions I’d like to include in the next release).
I’d love to understand how these solutions are found, but I’m just not there yet. Scholarpedia N-Body Choreographies is great place to start. A useful expository paper that also finds even more solutions (including some new non-choreography solutions) is Vanderbei’s New orbits for the N-body problem. His paper provides code to calculate choreographies in the specialized language AMPL. This makes the details of the search quite clear, even if you (like me) have never seen AMPL before.
Based on what I’ve absorbed thus far it seems the general idea involves using a path optimization approach using variational principles. This comes from an idea which is in the very core of physics, the notion of finding the path along which a specific quantity (the action) is an extreme value. I learned how to do this when I was in graduate school but I have always been a bit baffled by how the mathematical machinery is searching over paths. There are such a huge number of possible paths! The key mathematics here is the calculus of variations. I’ve gotten as far a buying a copy of the classic staring point Calculus of Variations (aka Gelfand and Fomin)
Notes:
- Phys.Rev.Lett 70:3675-3679. This was in turn anticipated by M. H´enon (1976) “A family of periodic solutions to the planar threebody problem, and their stability.” Celestial Mechanics 13 267–285. ↩
- A. Chenciner and R. Montgomery (2001) “A remarkable periodic solution of the three body problem in the case of equal masses.” Annals of Mathematics 152 881–901. ↩
- This is a good place to acknowledge the places I read about N-body evolution codes. Firstly, Moving Stars Around which is written in a quirky Socratic-dialog style. As part of the learning arc some of the code shown has deliberate errors that are then explained, so be careful. Everything you ever wanted to know about the subject is in the comprehensive Gravitational N-Body Simulations: Tools and Algorithms (Cambridge Monographs on Mathematical Physics)
. For a more concise summary of N-body codes and their pros and cons the section in Galactic Dynamics: (Second Edition) (Princeton Series in Astrophysics)
is very to the point. ↩