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

Dedicated to James Montaldi

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  • 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:

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  • 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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