# Can we infer the existence of periodic solutions to the three-body problem from numerical evidence?

I recently found out about the discovery of 13 beautiful periodic solutions to the three-body problem, described in the paper

Three Classes of Newtonian Three-Body Planar Periodic Orbits. Milovan Šuvakov and V. Dmitrašinović. Phys. Rev. Lett. 110 no. 11, 114301 (2013). arXiv:1303.0181.

I am particularly impressed by how elaborate the solutions are, and I'm struck by the tantalizing hint of an infinity of other distinct orbits given by the analogy with a free group. The solutions can be viewed in the Three-Body Galery, which has animations of the new orbits in real space and in something called the 'shape sphere', which is described in the paper.

I was aware already of the figure-of-eight solution, which is described nicely in

A new solution to the three body problem - and more. Bill Casselman. AMS Feature Column.

and which was discovered numerically by Christopher Moore (Phys. Rev. Lett. 70, 3675 (1993)). I understand that the figure-of-eight solution has been proven to actually exist as a solution of the ODE problem, in

A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses. Alain Chenciner and Richard Montgomery. Ann. Math 152 no. 3 (2000), pp. 881-901.

There is also a large class of solutions called $$N$$-body choreographies by Carles Simó, in which a number of bodies - possibly more than three - all follow the same curve. Simó found a large class of them in 2000 (DOI/pdf), though this nice review (DOI) seems to imply that formal theorematic proof that they exist as periodic solutions of the ODE problem is still lacking.

So, this brings me to my actual question. For the numerical simulations, however well you do them, in the end you will only have a finite-precision approximation to a solution of a differential equation that is propagated for a finite time. Additionally, you might do a numerical stability analysis that strongly suggests (or rigorously proves?) that you are (or are not) in a stable orbit. However, this is quite a ways away from a rigorous existence theorem for a periodic orbit with that symmetry.

With this in mind, then, in what spirit are these simulations done? Is it purely a numerical approach, in the hope that good numerics do indicate existence, but with a rigorous proof left to the mathematicians through whatever other means they can manage? Or is there some overarching theorem that indicates the existence of a truly periodic solution after a given threshold? What tools are there for proving existence theorems of periodic solutions?

It seems like they were able to rigorously prove the existence of N-body choreographies by using interval Krawczyk method to show that a minimum exist to the variational problem solved in the subspace of the full phase space satisfying some symmetry conditions.

Following the links given I found this paper where they explain the method. It's not exactly a light reading material but on page 6 they say: "If all these conditions all fulfilled, then from Theorem 4.5 we are sure that in the set $Z \times \{c_0\}$ there is an initial condition for the choreography. Moreover, as the set Z is usually very small, the shape of the proved choreography is very similar to our first approximation."

It sounds like starting with "an initial guess", they are able to show that there exist an "exact solution" very close to this initial guess. And one can probably obtain a curve that is arbitrarily close to the actual solution by doing more and more precise calculation. But the existence of the choreography is established rigorously with the assist of their numerical method.

Note that in the beginning of the paper, they mention the solutions obtained by the usual numerical methods as "solutions produced in a non-rigorous numerical way."

• Thank you for your great answer. There is indeed a formal existence result which provides a rigorous proof of the existence of a periodic solution. Have a bounty! I was hoping that the bounty period would make people upvote this but for some reason they didn't :(. – Emilio Pisanty Apr 10 '14 at 15:14

Not the answer you want to, but... I did readings from some sources above. And had my eyes on some N-body problems.

What can I say - non-symplectic approach at $[0,\infty]$ is unstable by default. Runge-Kutta, any quantification methods - unstable. Absence of stability is the general issue. It holds for many $[0,\infty]$ problems.

Searching for periodic $[0,T]$ is almost the same as solution of $[0,\tau]$ with some nuances.

But basically, there is no answer to the next question (which is basically ergodic, but more deep and sophisticated as $3n+1$ problem):

How much time does it require for body to get free velocity and quit the system as a function of given initial conditions?

The symplectic approach can create virtual "real" solution to which you are moving with some series or another expansion technique. Then, you can extract series over series and turn problem into symbolic chaos. There is field of current research too.

App1. What is this so called "choreography"?

1. They find the $F([0,T])$ solution. It is well-known approach tested for many equations. T is very limited parameter. What if there is enumerable amount of orbits. This findings are fun, but are they useful? Symmetries does not help when you want free body solution.
2. When they are cheating with $1/r$ part of potential - it is very bad, because if they are forced to do this, they do all the thing wrong from the beginning. Quantization of time Runge-Kutta and other similar "Taylor fixed order" methods are wrong in their inside at approaching the initial $[0,\infty]$ problem. But they are working in iterative error-correction mode for $[0,\tau]$. So basically this guys are doing $[0,\tau]$, when they are lucky they stuck into the $[0,T]$ solution for small $T$, and call it "correct". But basically it is work of level of homework for some numerical university course about Taylor ODE methods.

They have solution of numerical problem, but no proof that this numerical scheme is absolutely integrable into real orbit. For integration into orbit, one must describe mechanism with which adding more computation, allows to have $$\Delta(\text{solution at any time})<\epsilon$$ for any given $\epsilon$ and any given time. Every existing technique with quantized time integration leads to $$\Delta(\text{solution})\sim \exp (t).$$

So it could not be absolutely integrable, you are not free at choosing $\epsilon$.

• Even if $\Delta$ grows as $\exp(t)$ periodic solution implies finite time interval so the $\Delta$ is still bounded, so one could obtain any desired accuracy. – user23660 Feb 8 '14 at 16:46