Symmetry reduction of the 3-body problem in $\mathbb{R}^4$

The 3-body problem in $\mathbb{R}^4$ 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 $μ_1 > μ_2 \ge 0$, related to the conserved angular momentum. The limit $μ_2 \to 0$ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has relative equilibria that are local minima and hence Lyapunov stable when $μ_2$ is sufficiently small. This proves the existence of balls of initial conditions of full dimension that do not contain any orbits that are unbounded.

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