Choreographies in the $n$-body problem

Renato Calleja, IIMAS, UNAM (Mexico)
Eusebius Doedel, Department of Computer Science, Concordia University (Canada)
Carlos García-Azpeitia, IIMAS, UNAM (Mexico)

This page presents numerical and computer-assisted results on choreographies in the Newtonian $n$-body problem. A choreography is a periodic motion in which all bodies have equal masses and traverse the same closed curve, separated only by a constant phase shift in time.

The examples below are obtained by numerically continuing global Lyapunov families emanating from polygonal relative equilibria in a rotating frame. When the frequency of a Lyapunov orbit and the frequency of the rotating frame satisfy an appropriate rational relation, the orbit is periodic in the inertial frame. A dense set of the resonant Lyapunov orbits obtained in this way are choreographies. The computations displayed here include planar and spatial choreographies for $n=4,\ldots,9$, as well as torus-knot choreographies.

Interactive examples

Spatial choreography with 3 bodies

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Spatial choreography with 5 bodies

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Spatial choreography with 9 bodies

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Background

The study of $n$ equal masses following the same path has attracted considerable attention since the numerical discovery of the figure-eight solution. Moore found a three-body choreography in which the bodies follow one another around a figure eight by minimizing the action among symmetric paths. Chenciner and Montgomery later gave a rigorous proof of the existence of this orbit by minimizing the action over paths connecting a collinear configuration and an isosceles configuration.

These results initiated the systematic use of variational methods to study choreographies. In this approach, one seeks critical points of the classical action for Newton's equations of motion. The main difficulties are the possible occurrence of collisions and the lack of compactness of the action. Terracini and Ferrero used the principle of least action systematically over symmetric paths to avoid collisions, building on ideas introduced by Marchal. For a discussion of these and related variational approaches, see the references in the bibliography.

Continuation from polygonal relative equilibria

Another way to obtain choreographies is by continuation. Chenciner and Féjoz pointed out that choreographies occur in dense sets along vertical Lyapunov families arising from $n$ bodies rotating in a polygon. The local existence of the vertical Lyapunov families follows from Weinstein--Moser theory. The computations on this page continue these families far from the polygonal equilibrium and detect resonant orbits that become choreographies in the inertial frame.

Let $(u_j,z_j)\in\mathbb{C}\times\mathbb{R}$ denote the position of the $j$-th body in a rotating frame, where all bodies have equal unit mass. The angular velocity of the rotating frame can be chosen so that the unit polygon

u_j=e^{ij\zeta},\qquad z_j=0

is a relative equilibrium of Newton's equations. The Lyapunov families emanate with frequencies equal to the natural oscillation modes of the polygonal equilibrium. García-Azpeitia and Ize proved the global existence of the bifurcations of planar and vertical Lyapunov families using equivariant degree theory.

Planar Lyapunov families

For $n\geq 7$ and each integer $k$ satisfying

2\leq k\leq n-2,

the polygonal relative equilibrium has a global bifurcation of planar periodic solutions with symmetries

u_j(t)=e^{ij\zeta}u_n(t+jk\zeta),\qquad u_n(t)=\overline{u_n}(-t).\qquad \mathrm{(PS)}

Spatial Lyapunov families

For $n\geq 3$ and each integer $k$ satisfying

1\leq k\leq n/2,

the polygonal relative equilibrium has a global bifurcation of spatial periodic solutions satisfying the symmetries (PS) together with

z_j(t)=z_n(t+jk\zeta).\qquad \mathrm{(SS)} \] \[ u_n(t)=u_n(t+\pi),\qquad z_n(t)=-z_n(t+\pi).\qquad \mathrm{(8)}

In the case $k=n/2$, with $n$ even, one has $k\zeta=\pi$ and the symmetries imply

u_j(t)=e^{ij\zeta}u_n(t+j\pi)=e^{ij\zeta}u_n(t),\qquad z_j(t)=z_n(t+jk\zeta)=(-1)^j z_n(t).

Solutions with these symmetries are known as Hip-Hop solutions.

Resonant Lyapunov orbits and torus knots

The planar and spatial Lyapunov solutions with symmetries (PS) and (SS) are traveling waves: each body follows the same path up to a rotation and a time shift. These symmetries allow one to identify a dense set of solutions along the families that are choreographies in the inertial frame.

More precisely, a planar or spatial Lyapunov orbit is called $\ell:m$ resonant if its period and frequency satisfy

T=\frac{2\pi}{\sqrt{s_1}}\left(\frac{\ell}{m}\right),\qquad \nu=\sqrt{s_1}\frac{m}{\ell},

where $\ell$ and $m$ are relatively prime integers such that

\frac{k\ell-m}{n}\in\mathbb{Z}.

The $\ell:m$ resonant Lyapunov orbits are choreographies in the inertial frame. The integers $k$, $\ell$, and $m$ determine the geometry of the choreography: the projection of the curve onto the $xy$-plane has winding number $\ell$ around a center, it is invariant under the $\mathbb{Z}_m$ group of rotations by $2\pi/m$, and the $n$ bodies form groups of $h$-polygons, where $h=\gcd(k,n)$. Some of these curves wind around a toroidal surface with winding numbers $\ell$ and $m$; in that case, the choreography is a $(\ell,m)$-torus knot.

Related direction: Marchal's conjecture

A closely related problem in three-body choreographies is Marchal's conjecture, which concerns a continuous family of periodic orbits connecting the Lagrange equilateral triangle solution to the figure-eight choreography. This family is often denoted by $P_{12}$. In joint work with Olivier Hénot, Carlos García-Azpeitia, Jean-Philippe Lessard and Jason D. Mireles James, this conjecture was proved using computer-assisted methods, validated continuation and interval arithmetic.

Navigation

Use the menu above to browse the galleries of planar choreographies, spatial choreographies, unchained choreographies, Lyapunov families and bibliography.

Page maintained by Renato Calleja. The animations rely on the existing local WebGL files in webgl_coreografias/.