Toshiaki Fujiwara, Richard Montgomery
The figure eight is a remarkable solution to the Newtonian three-body problem in which the three equal masses chase each around a planar curve having the qualitative shape and symmetries of a figure eight. Here we prove that each lobe of this eight is convex.
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We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones type, $1/r^{12}-1/r^6$ where $r$ is a distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no $1/r^{12}$ term in the potential. The rest are very different from them and have several points with large curvatures in their figure-eight orbits, at which particles are repelled. Here figure-eight choreographies are the periodic m…
We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local minimizer. We then apply the theory to circular orbits of the Kepler problem with potentials of type $1/r^α, \, α> 0$. By using numerical computations, we show that circular solutions are strong local minimizers for $α> 1$, while they are saddle points for $α\in (0,1)$. Moreover, we show that for $α\in (1,2)$ the global minimizer of the action over periodic cu…
Numerical solutions are presented for a family of finite angular momentum orbits with three equal masses which connects the classical circular Lagrange orbit with the recently discovered planar figure eight orbit. Each member of this family is a periodic orbit in a frame rotating around the horizontal symmetry axis of the figure eight orbit. Numerical evidence is given that this family is a continuous function of the angular rotation frequency. Similar numerical solutions are also found for n>3 equal masses, where n is an odd integer. Finite angular momentum orbits also have been obtained for …
In this paper, we first describe how we can arrange any bodies on Figure-Eight without collision in a dense subset of $[0,T]$ after showing that the self-intersections of Figure-Eight will not happen in this subset. Then it is reasonable for us to consider the existence of generalized solutions and non-collision solutions with Mixed-symmetries or with Double-Eight constraints, arising from Figure-Eight, for N-body problem. All of the orbits we found numerically in Section \ref{se7} have not been obtained by other authors as far as we know. To prove the existence of these new periodic solutions…
In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the eight figure choreography, and we explore some symmetric periodic orbits of this system which turns out to be non autonomous. We demonstrate that certain families of involutions are reversing symmetries of certain N-body problem. We consider a particular case of such families of reversing symmetries and we use it to study both theoretically and numerically certain type of symmetric periodic orbits of this system. The symmetric perio…
From the Lagrange Triangle to the Figure Eight Choreography: Proof of Marchal's Conjecture
本文证明了在等质量三体问题中，拉格朗日等边三角形解最对称的延续类包含摩尔发现的八边形编队，解决了马歇尔1999年提出的猜想。
The present work studies the continuation class of the regular $n$-gon solution of the $n$-body problem. For odd numbers of bodies between $n = 3$ and $n = 15$ we apply one parameter numerical continuation algorithms to the energy/frequency variable, and find that the figure eight choreography can be reached starting from the regular $n$-gon. The continuation leaves the plane of the $n$-gon, and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the $n$-gon solution is complicated by the fact that the kernel of the linearization th…
Lorenz equations and the figure eight knot
本文研究 Lorenz 方程在第二个 T 点参数处的动力学，通过建立几何模型证明系统存在无穷多个周期轨道，且这些轨道作为纽结均为正、素且纤维化的纽结。
Self orbit equivalences on the Anosov flows on Dehn surgeries on the figure-eight knot
本文证明在八字结上大次数Dehn手术所得流形上，标准Anosov流的自轨道等价群同构于$\mathbb{Z}_2\oplus\mathbb{Z}_2$，且映射类群每个元素均可由自轨道等价实现。
The path closure of figure eight knot complement space, $N_0$, supports a natural DA (derived from Anosov) expanding attractor. Using this attractor, Franks-Williams constructed the first example of non-transitive Anosov flow on the manifold $M_0$ obtained by gluing two copies of $N_0$ through identity map along their boundaries, named by Franks-Williams manifold. In this paper, our main goal is to classify expanding attractors on $N_0$ and non-transitive Anosov flows on $M_0$. We prove that, up to orbit-equivalence, the DA expanding attractor is the unique expanding attractor supported by $N_…
The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov f…
Time-periodic perturbations of an asymmetric Duffing-Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincare map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computat…
The interplay between 1D traveling pulses with oscillatory tails (TPO) and heterogeneities of bump type is studied for a generalized three-component FitzHugh-Nagumo equation. First, we present that stationary pulses with oscillatory tails (SPO) form a ``snaky'' structure in homogeneous spaces, after which TPO branches form a ``figure-eight-like stack of isolas'' located adjacent to the snaky structure of SPO. Herein, we adopted input resources as a bifurcation parameter. A drift bifurcation from SPO to TPO was observed by introducing another parameter at which these two solution sheets merged.…
We revisit the three-body problem in the framework of general relativity. The Newtonian N-body problem admits choreographic solutions, where a solution is called choreographic if every massive particles move periodically in a single closed orbit. One is a stable figure-eight orbit for a three-body system, which was found first by Moore (1993) and re-discovered with its existence proof by Chenciner and Montgomery (2000). In general relativity, however, the periastron shift prohibits a binary system from orbiting in a single closed curve. Therefore, it is unclear whether general relativistic eff…
Periodic orbits near collision in a restricted four-body problem for the eight choreography
本文研究三体沿八字形 choreography 运动时无质量粒子在碰撞附近的周期轨道，通过正则化坐标和反转对称性结合边值问题求解对称周期轨道。
After the existence proof of the first remarkably stable simple choreographic motion-- the figure eight of the planar three-body problem by Chenciner and Montgomery in 2000, a great number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs and none of them are stable. Most important to astronomy are stable periodic solutions which might actually be seen in some stellar system. A question for simple choreographic solutions on $n$-body problems naturally arises: Are there any other stable simple choreographic solutions except th…
We consider a 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 computer assisted proof for the following facts: local uniqueness and convexity of Chenciner and Montgomery Eight, an existence (and local uniqueness) for Gerver's SuperEight for 4-bodies and a doubly symmetric linear chain for 6-bodies.
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