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Variational principle of action and group theory for bifurcation of figure-eight solutions

Toshiaki Fujiwara, Hiroshi Fukuda, Hiroshi Ozaki

Abstract

Figure-eight solutions are solutions to planar equal mass three-body problem under homogeneous or inhomogeneous potentials. They are known to be invariant under the transformation group $D_6$: the dihedral group of regular hexagons. Numerical investigation shows that each figure-eight solution has some bifurcation points. Six bifurcation patterns are known with respect to the symmetry of the bifurcated solution. In this paper we will show the followings. The variational principle of action and group theory show that the bifurcations of every figure-eight solution are determined by the irreducible representations of $D_6$. Each irreducible representation has one to one correspondence to each bifurcation. This explains numerically observed six bifurcation patterns. In general, in Lagrangian mechanics, bifurcations of a periodic solution is determined by irreducible representations of the transformation group that leaves this solution invariant.

Variational principle of action and group theory for bifurcation of figure-eight solutions

Abstract

Figure-eight solutions are solutions to planar equal mass three-body problem under homogeneous or inhomogeneous potentials. They are known to be invariant under the transformation group : the dihedral group of regular hexagons. Numerical investigation shows that each figure-eight solution has some bifurcation points. Six bifurcation patterns are known with respect to the symmetry of the bifurcated solution. In this paper we will show the followings. The variational principle of action and group theory show that the bifurcations of every figure-eight solution are determined by the irreducible representations of . Each irreducible representation has one to one correspondence to each bifurcation. This explains numerically observed six bifurcation patterns. In general, in Lagrangian mechanics, bifurcations of a periodic solution is determined by irreducible representations of the transformation group that leaves this solution invariant.

Paper Structure

This paper contains 46 sections, 4 theorems, 180 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Figure-eight solutions
  3. Symmetry group for figure-eight solutions: $D_6$
  4. Bifurcations of figure-eight solutions
  5. Variational principle of action for bifurcation
  6. Higher derivatives of action and definition of Hessian
  7. Necessary condition for bifurcation
  8. Expression for higher derivatives in terms of eigenfunctions of $\mathcal{H}$
  9. Lyapunov-Schmidt reduction
  10. Bifurcations for non-degenerate case
  11. Bifurcations for degenerate case
  12. Projection operator
  13. Inheritance of symmetry
  14. Bifurcations of $D_6$
  15. Representation I: $\mathcal{P_C}'=\mathcal{M}'=\mathcal{S}'=1$
  16. ...and 31 more sections

Key Result

Theorem \oldthetheorem

If a group $G$ is a symmetry group for $q_o$ and the action, a stationary point in subspace $\mathcal{P}'_G=1$ is a stationary point in whole space, namely, a solution of the equations of motion.

Figures (4)

  • Figure 1: Bifurcations and symmetry breaking of $D_6$ and $D_2$. Bifurcations of $D_6$ and $D_2$ are represented by thick arrows and dashed thick arrows respectively. Each vertex represents the solution of the equations of motion. The symbol "$G:\mathcal{P}$" at each vertex represents symmetry group $G$ and projection operator $\mathcal{P}$ for the solution. The symbol "No:$\slashed{\mathcal{O}}$" on the arrows represents the number of irreducible representation in table \ref{['table6RepsOfD6']} and broken symmetry $\mathcal{O}$. The fork in VI shows that this bifurcation yields two bifurcated solutions, one with symmetry group $D_1$ and the other with $D'_1$. The bifurcation I in table \ref{['table6RepsOfD6']} is bifurcation of $D_6 \to D_6$ which is omitted in this figure.
  • Figure 2: Reduced action $S_{LS}$ of representation V for $A_3, A_4>0$. Upper: Contour plot of $S_{LS}$ in orthogonal coordinates $(x,y)$. Lower: $S_{LS}$ for $y=0$. From left to right $\kappa<0, \kappa=0$ and $\kappa>0$. Gray circle and black circles represent $q_o$ and $q_b, \mathcal{C}q_b, \mathcal{C}^2 q_b$ respectively. For sufficiently small $\kappa$ and short range of $r$, terms of $O(r^4)$ have no effect.
  • Figure 3: Reduced action $S_{LS}$ of representation VI for $A_4(0)<0, A_{6+}>A_{6-}>0$. Left: Contour plot of $S_{LS}$ for $\kappa>0$ in orthogonal coordinates $(x,y)$. Gray circle at the centre is $q_0$ and black circles and black stars are $\mathcal{B}^k q_{b+}$ and $\mathcal{B}^k q_{b-}$ with $k=0,1,2,\dots,5$ respectively. For this assignment of $A_4(0)$ and $A_{6\pm}$, $\mathcal{B}^k q_{b+}$ are local maximum and $\mathcal{B}^k q_{b-}$ are saddle. Right: $S_{LS}$ for $y=0$. Three curves represent $\kappa$ negative (dashed), zero (dotted) and positive(solid curve) respectively. For $A_4(0)<0$, bifurcated solutions $q_{b+}$ and $-q_{b+}$ (black circles) exist for $\kappa>0$. Gray circle at the origin is $q_0$.
  • Figure 4: The figure-eight (left) and bifurcated solution for IV (right) under $U_h$ at $a=-0.2$. Points represent position of particles at $t=-T/12$ (solid circles), $0$ (black stars), $T/12$ (hollow circles), and $2 T/12$ (grey stars).

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • Theorem \oldthetheorem
  • proof
  • proof