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Preface to the special issue dedicated to James Montaldi
Relative equilibria of the 3-body problem in $ \mathbb{R}^4 $
1. | IMCCE, UMR8028, Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France |
2. | School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia |
The classical equations of the Newtonian 3-body problem do not only define the familiar 3-dimensional motions. The dimension of the motion may also be 4, and cannot be higher. We prove that in dimension 4, for three arbitrary positive masses, and for an arbitrary value (of rank 4) of the angular momentum, the energy possesses a minimum, which corresponds to a motion of relative equilibrium which is Lyapunov stable when considered as an equilibrium of the reduced problem. The nearby motions are nonsingular and bounded for all time. We also describe the full family of relative equilibria, and show that its image by the energy-momentum map presents cusps and other interesting features.
References:
[1] |
A. Albouy, Mutual distances in celestial mechanics, Lectures at Nankai institute, Tianjin, China, preprint, (2004). Google Scholar |
[2] |
A. Albouy, H. E. Cabral and A. A. Santos,
Some problems on the classical $n$-body problem, Celestial Mechanics and Dynamical Astronomy, 113 (2012), 369-375.
doi: 10.1007/s10569-012-9431-1. |
[3] |
A. Albouy and A. Chenciner,
Le problème des $n$ corps et les distances mutuelles, Invent. Math., 131 (1998), 151-184.
doi: 10.1007/s002220050200. |
[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] |
A. Chenciner and H. Jiménez-Pérez, Angular momentum and Horn's problem, Mosc. Math. J., 13 (2013), 621–630,737.
doi: 10.17323/1609-4514-2013-13-4-621-630. |
[6] |
H. R. Dullin,
The Lie-Poisson structure of the reduced $n$-body problem, Nonlinearity, 26 (2013), 1565-1579.
doi: 10.1088/0951-7715/26/6/1565. |
[7] |
H. R. Dullin and J. Scheurle, Symmetry reduction of the 3-body problem in $R^4$, J. Geom. Mech., 12 2020, 377-394.
doi: 10.3934/jgm.2020011. |
[8] |
M. Herman,
Some open problems in dynamical systems, Doc. Math., 2 (1998), 797-808.
|
[9] |
J. L. Lagrange, Méchanique Analitique, Paris, 1788. Google Scholar |
[10] |
R. Moeckel,
Minimal energy configurations of gravitationally interacting rigid bodies, Celestial Mechanics and Dynamical Astronomy, 128 (2017), 3-18.
doi: 10.1007/s10569-016-9743-7. |
[11] |
D. J. Scheeres,
Minimum energy configurations in the $N$-body problem and the celestial mechanics of granular systems, Celestial Mechanics and Dynamical Astronomy, 113 (2012), 291-320.
doi: 10.1007/s10569-012-9416-0. |
[12] |
K. F. Sundman,
Mémoire sur le problème des trois corps, Acta mathematica, 36 (1913), 105-179.
doi: 10.1007/BF02422379. |
[13] |
A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, 5. Princeton University Press, Princeton, N. J., 1941.
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show all references
References:
[1] |
A. Albouy, Mutual distances in celestial mechanics, Lectures at Nankai institute, Tianjin, China, preprint, (2004). Google Scholar |
[2] |
A. Albouy, H. E. Cabral and A. A. Santos,
Some problems on the classical $n$-body problem, Celestial Mechanics and Dynamical Astronomy, 113 (2012), 369-375.
doi: 10.1007/s10569-012-9431-1. |
[3] |
A. Albouy and A. Chenciner,
Le problème des $n$ corps et les distances mutuelles, Invent. Math., 131 (1998), 151-184.
doi: 10.1007/s002220050200. |
[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] |
A. Chenciner and H. Jiménez-Pérez, Angular momentum and Horn's problem, Mosc. Math. J., 13 (2013), 621–630,737.
doi: 10.17323/1609-4514-2013-13-4-621-630. |
[6] |
H. R. Dullin,
The Lie-Poisson structure of the reduced $n$-body problem, Nonlinearity, 26 (2013), 1565-1579.
doi: 10.1088/0951-7715/26/6/1565. |
[7] |
H. R. Dullin and J. Scheurle, Symmetry reduction of the 3-body problem in $R^4$, J. Geom. Mech., 12 2020, 377-394.
doi: 10.3934/jgm.2020011. |
[8] |
M. Herman,
Some open problems in dynamical systems, Doc. Math., 2 (1998), 797-808.
|
[9] |
J. L. Lagrange, Méchanique Analitique, Paris, 1788. Google Scholar |
[10] |
R. Moeckel,
Minimal energy configurations of gravitationally interacting rigid bodies, Celestial Mechanics and Dynamical Astronomy, 128 (2017), 3-18.
doi: 10.1007/s10569-016-9743-7. |
[11] |
D. J. Scheeres,
Minimum energy configurations in the $N$-body problem and the celestial mechanics of granular systems, Celestial Mechanics and Dynamical Astronomy, 113 (2012), 291-320.
doi: 10.1007/s10569-012-9416-0. |
[12] |
K. F. Sundman,
Mémoire sur le problème des trois corps, Acta mathematica, 36 (1913), 105-179.
doi: 10.1007/BF02422379. |
[13] |
A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, 5. Princeton University Press, Princeton, N. J., 1941.
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