# $N$-body system subject to charge and gravity: explicit formulas for equilibrium?

Consider a system of $N$ classical particles $x_1, \ldots, x_N$ in $\mathbb{R}^2$ of the same mass exerting a repulsive "charge" force against one another (where all particles have the same charge strength). Assume further that the particle $x_1$ exerts an attractive "gravitational" force against the remaining particles, but that no other particle exerts such a force against any other.

From numerical experiments (and intuition), one expects a fairly simple equilibrium for this system: the particles $x_2, \ldots, x_N$ organize themselves into circular shells around $x_1$, with an angular offset between consecutive shells. This equilibrium has a lot of symmetry, so my question is:

Is there an explicit formula for the positions of the $N$ particles in this equilibrium?

The purpose of this question is to simplify some numerical simulations in which this configuration appears frequently.

• Unless you're including dissipative forces, then in general there should not be a stationary final configuration. Are you looking at the statistical distribution of very large $N$? – DilithiumMatrix Dec 7 '16 at 15:55
• What do you think? What is your attempt to find such a formula? ... What evidence do you have that the particles organise themselves into shells around $x_1$? ... Are you looking for solutions in which the particles are stationary or moving? – sammy gerbil Dec 7 '16 at 17:11

Actually the problem is not that simple. It depends a lot on the velocity of the individual particles, for which there's no information. To give you an idea, if you switch off the electric force and the velocity dispersion of the particles is constant with radius the volume mass density behaves as $\rho\sim r^{-2}$. On the other hand, if you have low velocity dispersion $\sigma\sim e^{-R/2R_0}$and high angular momentum, then you end up with a rotating disk with surface density $\Sigma \sim e^{-R/R_0}$
• That's a fair point. In my simulations I start the particles at rest with random initial positions within a rectangle of fixed size, and I end up with basically the same configuration every time (up to rotation about $x_1$ with constant angular velocity). This is the configuration for which I would like explicit formulas. – Paul Siegel Dec 7 '16 at 12:03
• @PaulSiegel What is the configuration you end up with? Again a box? If you start with a box large enough, then the force field is almost uniform, and the system will remain the same, this is specially the case if $F_{\rm electric} > F_{\rm gravitational}$. If on the other hand, the interaction is dominated by gravity then anisotropies will quickly develop and the density will cluster up – caverac Dec 7 '16 at 12:57