## Introduction:

In my previous blog post I explained that the gravitational force between a pair of masses may be linearised in terms of the mass-distance ratio using the logarithm. It’s easy to show that for more than two masses such a linearisation is no longer possible even if all the masses are the same as the logarithm isn’t a linear operator.

However, in the case of $N > 2$ equal point masses we may consider questions that are mathematically interesting. Here I’ll share a few.

## Gravitational N-body problems:

1. Is there a topologically interesting family of N-body choreographies $\forall N \in \mathbb{N}$?

a. It is known for example that for any $N$ we can construct a simple choreography of point masses chasing each other around the circle.

b. Note that I don’t say anything about stability here.

2. Are all KAM stable orbits smooth?

a. It’s important to note that there are classical dynamical systems that aren’t smooth and yet stable with respect to small perturbations. A desk that is abruptly perturbed for example.

b. The intuition behind this question rests on the fact that all N-body choreographies are smooth.

3. What is the statistical behaviour of a macroscopic number of point masses in a universe with a spherical boundary as $N \rightarrow \infty$?

a. What if we drop the assumption that we are dealing with point masses and allow the diameter $d$ of these planets to be an experimental variable?

b. If the diameter of the universe is given by $D$, is the ratio $\frac{D}{d}$ a sufficient statistic for the condensation phase of the system?

## Discussion:

Of all the questions above, I find the last two to be the most interesting and I think I will attempt to answer the last question first. I am curious about whether the asymptotic statistical behaviour of these point masses is independent of the initial conditions and whether the resulting system will behave like a gas.