Rotating Eights: I. The three Γi families

  title={Rotating Eights: I. The three $\Gamma$i families},
  author={Alain Chenciner and Jacques F{\'e}joz and Richard Montgomery},
  pages={1407 - 1424}
We show that three families of relative periodic solutions bifurcate out of the Eight solution of the equal-mass three-body problem: the planar Hénon family, the spatial Marchal P12 family and a new spatial family. The Eight, considered as a spatial curve, is invariant under the action of the 24-element group D6 × Z2. The three families correspond to symmetry breakings where the invariance group becomes isomorphic to D6, the three D6s being embedded in the larger group in different ways. The… 

Figures from this paper


From a normal form analysis near the Lagrange equilateral relative equilibrium, we deduce that, up to the action of similarities and time shifts, the only relative periodic solutions which bifurcate

Unchained polygons and the N-body problem

We study both theoretically and numerically the Lyapunov families which bifurcate in the vertical direction from a horizontal relative equilibrium in ℝ3. As explained in [1], very symmetric relative

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits and subsequently bifurcating families. The Lyapunov families arise

On the stability of the three classes of Newtonian three-body planar periodic orbits

Currently, the fifteen new periodic orbits of Newtonian three-body problem with equal mass were found by Šuvakov and Dmitra šinović [Phys Rev Lett, 2013, 110: 114301] using the gradient descent

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this

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

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

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680s. In the 300 years since this “three-body problem” was first recognized, only three families of periodic solutions had

Continuity and stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the figure eight orbit

Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit

In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent

On Action-Minimizing Retrograde and Prograde Orbits of the Three-Body Problem

A retrograde orbit of the planar three-body problem is a relative periodic solution with two adjacent masses revolving around each other in one direction while their mass center revolves around the



The Family P12 of the Three-body Problem – The Simplest Family of Periodic Orbits, with Twelve Symmetries Per Period

A beautiful plane eight-shaped orbit has been found by Alain Chenciner, Richard Montgomery and Carles Simo through the minimisation of the action between suitable limit conditions. The three masses

A remarkable periodic solution of the three-body problem in the case of equal masses

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry

Families of periodic orbits in the three-body problem

We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two

Some facts and more questions about the Eight

I discuss some properties of the “Eight” solution of the three-body problem, many of them conjectural. I describe in particular a simple approach to the P12 family, proposed by C. Marchal, which is a

Braids in classical gravity

Point masses moving in 2+1 dimensions draw out braids in spacetime. If they move under the influence of some pairwise potential, what braid types are possible? By starting with fictional paths of the

Braids in classical dynamics.

  • Moore
  • Mathematics
    Physical review letters
  • 1993
This work proposes this kind of topological classification as a tool for extending the «symbolic dynamics» approach to many-body dynamics by exploring the braid types of potentials of the form V∞r a from a≤-2, where all braidtypes occur, to a=2,where the system is integrable.

Lectures on Morse theory, old and new

Morse Theory is a beautiful and natural extension of the minimum principle for a continuous function on a compact space. In these lectures I would like to discuss it in the context of two problems in

The existence of simple choreographies for the N-body problem—a computer-assisted proof

We consider the question of finding a periodic solution for the planar Newtonian N-body problem with equal masses, where each body is travelling along the same closed path. We provide a

Periodic Solutions of an N-Body Problem

This thesis develops methods to identify periodic solutions to the n-body problem by representing gravitational orbits with Fourier series. To find periodic orbits, a minimization function was