SYMMETRY REDUCTION of the 3-BODY PROBLEM in R4

The 3-body problem in R4 has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space with 8 dimensions. The Hamiltonian depends on two parameters Âμ 1 Âμ 2 ≥ 0, related to the conserved angular momentum. The limit Âμ 2 ! 0 corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has three relative equilibria that are local minima and hence Lyapunov stable when Âμ 2 is sufficiently small. This proves the existence of balls of initial conditions of full dimension that do not contain any orbits that are unbounded.