# Rotating Eights: I. The three Γi families

@article{Chenciner2005RotatingEI, title={Rotating Eights: I. The three $\Gamma$i families}, author={Alain Chenciner and Jacques F{\'e}joz and Richard Montgomery}, journal={Nonlinearity}, year={2005}, volume={18}, 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…

## 40 Citations

### THE FLOW OF THE EQUAL-MASS SPATIAL 3-BODY PROBLEM IN THE NEIGHBORHOOD OF THE EQUILATERAL RELATIVE EQUILIBRIUM

- Mathematics
- 2008

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

- Mathematics
- 2009

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

- MathematicsCelestial Mechanics and Dynamical Astronomy
- 2018

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

- Physics
- 2014

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

- Mathematics
- 2004

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

- Mathematics
- 2020

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

- Physics
- 2017

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

- Physics
- 2005

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

- Mathematics
- 2009

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

- Physics
- 2009

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…

## References

SHOWING 1-10 OF 31 REFERENCES

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

- Physics
- 2000

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

- Physics, Geology
- 2000

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

- Physics, Mathematics
- 1974

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

- Physics
- 2003

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

- Physics
- 1993

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.

- MathematicsPhysical 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

- Mathematics
- 1982

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

- Mathematics
- 2003

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

- Physics, Geology
- 2004

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…