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

Dedicated to James Montaldi

• 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 three 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.

Mathematics Subject Classification: 37N05, 70F10, 70F15, 70H33, 53D20.

 Citation: • • Figure 1.  Scaled energy-momentum diagram of the isosceles family of relative equilibria (or balanced configuration) in the 3-body problem in dimension 4 for two different mass ratios. These relative equilibria are minima of the Hamiltonian for sufficiently large negative scaled energy $h$, which occurs for small $b$ corresponding to small $\mu_2$

Figure 2.  Parameter space $n = m_1/m > 0$ and shape parameter $t \in (0, 1)$ of the isosceles equilibrium. The curves divide the positive quadrant into 6 regions. The horizontal line $t = 2 - \sqrt{3}$ corresponds to the equilateral triangles. The parabola-shaped curve $P_1(n, t) = 0$ indicates a vanishing of the determinant of the $(q_2, q_3)$-block. The curve $P_2(n, t) = 0$ starting at the origin indicates a vanishing of the determinant of the $(q_2, q_3)$-block and an infinity in the determinant of the $(p_2, p_3)$-block. In the region adjacent to the $n$-axis all eigenvalues are positive and the isosceles solution is a minimum of the 3-body problem in $\mathbb{R}^4$

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