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September  2020, 12(3): 323-341. doi: 10.3934/jgm.2020012

Relative equilibria of the 3-body problem in $ \mathbb{R}^4 $

Alain Albouy 1, and  Holger R. Dullin 2,

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

Dedicated to James Montaldi

Received  September 2019 Revised  December 2019 Published  September 2020 Early access  March 2020

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.

Keywords: 3-body problem, symplectic symmetry reduction, Lyapunov stability.
Mathematics Subject Classification: 37N05, 70F10, 70F15, 70H33, 53D20.
Citation: Alain Albouy, Holger R. Dullin. Relative equilibria of the 3-body problem in $ \mathbb{R}^4 $. Journal of Geometric Mechanics, 2020, 12 (3) : 323-341. doi: 10.3934/jgm.2020012
References:
[1]

A. Albouy, Mutual distances in celestial mechanics, Lectures at Nankai institute, Tianjin, China, preprint, (2004).

[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.

[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. 

show all references

References:
[1]

A. Albouy, Mutual distances in celestial mechanics, Lectures at Nankai institute, Tianjin, China, preprint, (2004).

[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.

[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. 
Figure 2.  Three distinct masses $ (m_1, m_2, m_3) = (3, 2, 1)/6 $. The Lagrange equilateral family is the vertical line, not extending all the way to $ k = 1/4 $. The three non-equilateral families emerge from Euler's collinear configurations at $ k = 0 $ and for $ h \to -\infty $ approach collision configurations. The two short families have a cusp each. The long family emerges at the Euler collinear configuration with the smallest energy, then touches the endpoint of the equilateral family, and then is tangent to the maximum at $ k = 1/4 $. Past this tangency it corresponds to minimal energy at fixed $ k $ and hence is non-linearly stable
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Figure 3.  Equal mass case. The Lagrange equilateral family is the vertical line, in this case extending all the way to $ k = 1/4 $. The three isosceles families coincide and emerge at the collinear Euler solution at $ k = 0 $. Isosceles triangles with $ \rho < 1 $ are non-linearly stable because they are minima in the energy for fixed $ k $
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Figure 4.  Two equal masses, third mass smaller ($ \mu = 1/2 $). The Lagrange equilateral family is the vertical line, not extending all the way to $ k = 1/4 $. At the endpoint it meets the isosceles long family, which later touches $ k = 1/4 $. Both short families of asymmetric triangles emerge from a collinear Euler configuration at $ k = 0 $ and have a cusp beyond which $ h $ approaches $ -\infty $. The isosceles configurations to the left of the tangency with $ k = 1/4 $ are the absolute minimum of the energy and hence are non-linearly stable
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Figure 5.  Two equal masses, third mass bigger ($ \mu = 2 $). The Lagrange equilateral family is the vertical line, not extending all the way to $ k = 1/4 $. At the endpoint it meets the long isosceles family (red), which later touches $ k = 1/4 $. One short family (green) of asymmetric triangles emerges from a collinear Euler configuration at $ k = 0 $, has a cusp tangent to the long family and retraces itself back down. The other short family (blue) of asymmetric triangles starts and finishes at the collision where $ h \to -\infty $, and has a cusp tangent to the long family. These asymmetric triangles of absolute minimal energy are non-linearly stable. There is a tiny part of the long family of symmetric isosceles triangles which have absolute minimal energy and hence are non-linearly stable
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Figure 6.  Three distinct masses, somewhat close to the two isosceles cases. Left: masses $ (12, 5, 4)/21 $, Right: masses $ (6, 5, 2)/13 $. The left figure illustrates that there is no continuity in the balanced families when perturbing from the case with two equal masses and the third mass larger than the equal ones, compare Fig. 5
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Figure 1.  Three smooth families of balanced configurations. Long family red, short families blue and green. Masses $ (m_1, m_2, m_3) = (3, 2, 1)/6 $. Isosceles shapes are shown as dashed blue lines. Left: $ a(b) $ for $ c = 1 $, the long family exists for all values of $ b $. Right: The extended triangle of shapes $ I = const $ with boundary black dashed where one side length vanishes. The thick black ellipse marks shapes with area $ A = 0 $ with contour lines of constant positive area inside. The other set of contour lines indicate $ V = const $. Special points are marked by their projective triple $ [a, b, c] $
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