Linear stability analysis of the figure-eight orbit in the three-body problem

We show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2×2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge–Kutta–Fehlberg algorithm. From this, we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.