Relative equilibria of the four-body problem

Finiteness of Relative Equilibria of the Four-body Problem

We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried out by computer.

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

We study the relative equilibria of the limit case of the planar Newtonian 4–body problem when three masses tend to zero, the so-called (1 + 3)–body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1 + ...

Relative Equilibria of a Gyrostat in the Three Body Problem

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the three body problem. By means of geometric-mechanics methods we will study the approximate dynamics that arises when we develop the potential in series of Legendre and truncate this in an arbitrary order. Working in the reduced problem, we will study the existence of relative equilibria that we will denominate of Euler and L...

Central configurations and relative equilibria in the n–body problem

Relative Equilibria in the Spherical, Finite Density Three-Body Problem

The relative equilibria for the spherical, finite density three-body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical five relative equilibria for the point-mass three-body problem. None of the identified relative equilibria exist or are stable over all values of angular momentum. The stability and bifurcation pathways o...