Central configuration

Remarks

The constant $\lambda$ is always positive. Central configurations are counted up to translation, rotation and scaling, since these maps send central configurations to central configurations. Equivalently, they are the critical points of the potential $U = \sum_{i < j} G m_i m_j / \norm{q_i - q_j}$ restricted to configurations with a fixed moment of inertia about the centre of mass.

Example: three equal masses $m = 1$, with $G = 1$, at the vertices of an equilateral triangle of side $s$. Each body feels two pulls of size $1/s^2$ with an angle of $60^\circ$ between them, so the total pull has size $\sqrt{3}/s^2$ and points to the centre. The distance to the centre is $s/\sqrt{3}$, so the condition holds with $\lambda = 3/s^3$. By a result of Lagrange the equilateral triangle is central for any three positive masses.

Non-example: three unit masses on a line at $0$, $1$ and $3$, with $G = 1$, so $c = 4/3$. The body at $0$ has acceleration $1 + 1/9 = 10/9$, which forces $\lambda = 5/6$. The body at $1$ has acceleration $-1 + 1/4 = -3/4$, which forces $\lambda = -9/4$. No single $\lambda$ works, so the configuration is not central.

These configurations give solutions of the n-body problem in which the shape stays similar to itself. In the plane, a rigid rotation with angular speed $\sqrt{\lambda}$ about $c$ is a solution, called a relative equilibrium.