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

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

Holger R. Dullin 1, and  Jürgen Scheurle 2,

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  September 2020 Early access  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.

Keywords: 3-body problem, symplectic symmetry reduction, Lyapunov stability.
Mathematics Subject Classification: 37N05, 70F10, 70F15, 70H33, 53D20.
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
References:
[1]

A. Albouy, Integral manifolds of the $N$-body problem, Invent. Math., 114 (1993), 463-488.  doi: 10.1007/BF01232677.  Google Scholar

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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

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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

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T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

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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

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T. Schmah and C. Stoica, On the n-body problem in $R^4$, arXiv: 1907.08746. Google Scholar

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S. Smale, Topology and mechanics. I, Inv. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar

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show all references

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
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 $
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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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