# American Institute of Mathematical Sciences

September  2020, 12(3): 377-394. doi: 10.3934/jgm.2020011

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

 1 School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia 2 Zentrum Mathematik, M8, TU München, Boltzmannstraße 3, D-85748 Garching bei München, Germany

Dedicated to James Montaldi

Received  August 2019 Revised  October 2019 Published  March 2020

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.

Citation: Holger R. Dullin, Jürgen Scheurle. Symmetry reduction of the 3-body problem in $\mathbb{R}^4$. Journal of Geometric Mechanics, 2020, 12 (3) : 377-394. doi: 10.3934/jgm.2020011
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##### References:
 [1] A. Albouy, Integral manifolds of the $N$-body problem, Invent. Math., 114 (1993), 463-488.  doi: 10.1007/BF01232677.  Google Scholar [2] A. Albouy and A. Chenciner, Le problème des $n$ corps et les distances mutuelles, Invent. Math., 131 (1998), 151-184.  doi: 10.1007/s002220050200.  Google Scholar [3] A. Albouy and H. R. Dullin, Relative equilibra of the 3-body problem in $R^4$, J. Geom. Mech., 12, 2020, 323-341. doi: 10.3934/jgm.2020012.  Google Scholar [4] A. Chenciner, The angular momentum of a relative equilibrium, Discrete Contin. Dyn. Syst., 33 (2013), 1033-1047.  doi: 10.3934/dcds.2013.33.1033.  Google Scholar [5] M. Herman, Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Doc. Math., 2 (1998), 797-808.   Google Scholar [6] C. G. J. Jacobi, Sur l'élimination des noeuds dans le problème des trois corps, J. Reine Angew. Math., 26 (1843), 115-131.  doi: 10.1515/crll.1843.26.115.  Google Scholar [7] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [8] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. on Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar [9] T. Schmah and C. Stoica, On the n-body problem in $R^4$, arXiv: 1907.08746. Google Scholar [10] S. Smale, Topology and mechanics. I, Inv. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar [11] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, 4th edition, Cambridge University Press, New York, 1959.   Google Scholar
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 $\mu_2$
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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