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

## TL;DR

This paper performs a comprehensive symmetry reduction of the 3-body problem in four-dimensional space, revealing stable configurations and bounded initial conditions through Hamiltonian analysis.

## Contribution

It provides the first full symplectic reduction of the 3-body problem in $\

## Key findings

- Existence of Lyapunov stable relative equilibria for small $\
- demonstration of bounded initial condition sets without unbounded orbits
- Reduced Hamiltonian with 8-dimensional phase space and parameters related to angular momentum

## Abstract

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 $\mu_1 > \mu_2 \ge 0$, related to the conserved angular momentum. The limit $\mu_2 \to 0$ corresponds to the 3-dimensional limit. We show that the reduced Hamiltonian has relative equilibria that are local minima and hence Lyapunov stable when $\mu_2$ is sufficiently small. This proves the existence of balls of initial conditions of full dimension that do not contain any orbits that are unbounded.