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The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps
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 |
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.
References:
[1] |
A. Albouy,
Integral manifolds of the $N$-body problem, Invent. Math., 114 (1993), 463-488.
doi: 10.1007/BF01232677. |
[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. |
[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. |
[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. |
[5] |
M. Herman,
Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Doc. Math., 2 (1998), 797-808.
|
[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. |
[7] |
T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. |
[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. |
[9] |
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. |
[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.
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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. |
[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. |
[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. |
[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. |
[5] |
M. Herman,
Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Doc. Math., 2 (1998), 797-808.
|
[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. |
[7] |
T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. |
[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. |
[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. |
[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.
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