\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Dedicated to James Montaldi

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 37N05, 70F10, 70F15, 70H33, 53D20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • 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 $

    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 $

  • [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.
    [10] S. Smale, Topology and mechanics. I, Inv. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.
    [11] E. T. WhittakerA 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. 
  • 加载中

Figures(2)

SHARE

Article Metrics

HTML views(1927) PDF downloads(394) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return