The number of relative equilibria of the Newtonian 4-body problem is finite

You need to know: Euclidean plane ${\mathbb R}^2$ , rotations, translations, and dilations in the plane, (second) derivative of a function $x:{\mathbb R}\to {\mathbb R}^2$ , uniform rotation, angular velocity.

Background: Let n point particles with masses $m_i > 0$ and positions $x_i \in {\mathbb R}^2$ are moving according to Newton’s laws of motion: $m_j \frac{d^2 x_j}{dt^2} = \sum\limits_{i \neq j} \frac{m_i m_j(x_i - x j)}{r_{ij}^3}, \, 1 \leq j \leq n$ , where $r_{ij}$ is the distance between $x_i$ and $x_j$ . A relative equilibrium motion is a solution of this system of the form $x_i(t) = R(t)x_i(0)$ where $R(t)$ is a uniform rotation with constant angular velocity $v\neq 0$ around some point $c \in {\mathbb R}^2$ . Two relative equilibria are equivalent if they are related by rotations, translations, and dilations in the plane.

The Theorem: On 12th July 2004, Marshall Hampton and Richard Meockel submitted to Inventiones mathematicae a paper in which they proved that, for $n=4$ , there is only a finite number of equivalence classes of relative equilibria, for any positive masses $m_1, m_2, m_3, m_4$ .

Short context: The problem of describing motion of n bodies under gravitation (n-body problem) in space or plane is a fundamental problem in physics and mathematics. In general, the motion can be very complicated even for $n=3$ , but can we at least classify “nice” relative equilibrium motions on the plane? This problem is solved for $n=3$ : in this case, there are always exactly five relative equilibria, up to equivalence. However, for $n\geq 4$ , the question whether the number of relative equilibria is finite is a major open problem, which was included as problem 6 into Smale’s list of problems for the 21st century. The Theorem solves this problem for $n=4$ .