抛物型三涡运动的临界解

Errata and Addenda to: "Hydrodynamic Vortex on Surfaces" and "The motion of a vortex on a closed surface of constant negative curvature"

The two papers in the title contain some equations that are not complete. The missing terms, which are of topological origin, were recently unveiled by Bjorn Gustafsson. In this note we present the equations of Gustafsson in the case of a single vortex in a compact boundaryless surface, and show that many conclusions we have reached with the incomplete equations remain valid. It seems that the the extra-terms in Gustafsson's equations can be explicitly written in terms of elementary functions only for the two-torus, and this is done at the end of this note.

半平面上可积的两层点涡运动

Integrable two layer point vortex motion on the half plane

本文导出了上半平面上两层点涡运动的运动方程。我们研究了不变量使用对称性，包括哈密顿量，并表明，两个涡问题是可积的。我们表征所有两个涡运动的情况下，涡强度都是相等的，$Γ_{1}=Γ_{2}=1$和当它们是相反的$Γ_{1}=-Γ_{2}=1$。证明了当$Γ_{1}=-Γ_{2}=1$ .当$Γ_{1}=Γ_{2}=1$和涡处于不同的层中时，只有一个相对平衡位形。我们也观察到了两个涡旋在半平面中的有限时间坍缩。然后，我们比较了这两种情况下（运动的半平面）与整个平面上的双层涡问题的情况下的运动制度。我们还研究了几类流线拓扑的两个涡在不同的层。最后，我们通过研究可积对称组态，对半平面上可积两层三涡运动进行了哈密顿研究，并提供了丰富的新的相对平衡态。

Whiskered parabolic tori in the planar $(n+1)$-body problem

The planar $(n+1)$-body problem models the motion of $n+1$ bodies in the plane under their mutual Newtonian gravitational attraction forces. When $n\ge 3$, the question about final motions, that is, what are the possible limit motions in the planar $(n+1)$-body problem when $t\to \infty$, ceases to be completely meaningful due to the existence of non-collision singularities. In this paper we prove the existence of solutions of the planar $(n+1)$-body problem which are defined for all forward time and tend to a parabolic motion, that is, that one of the bodies reaches infinity with zero velocit…

Oscillatory Motions and Parabolic Manifolds at Infinity in the Planar Circular Restricted Three Body Problem

Consider the Restricted Planar Circular 3 Body Problem with both realistic mass ratio and Jacobi constant for the Sun-Jupiter pair. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on the method of correctly aligned windows and is computer assisted.

抛物芽的全纯运动、Fatou线性化和拟共形刚性

Holomorphic Motions, Fatou Linearization, and Quasiconformal Rigidity for Parabolic Germs

通过应用全纯运动，证明了抛物芽是拟共形刚性的，即任意两个拓扑共轭的抛物芽是拟共形共轭的，并且只要我们考虑这些芽定义在越来越小的邻域上，其共轭性就可以越来越接近共形.在证明这个定理之前，我们使用全纯运动的思想来给出Fatou线性化定理的概念性证明。作为一个副产品，我们还证明了任何有限数量的解析芽在不同的点在黎曼球面可以扩展到一个拟共形同胚，可以越来越近的共形，只要我们认为这些芽定义在这些点的越来越小的邻域。

牛顿N体问题中的全局极小抛物运动

Globally minimizing parabolic motions in the Newtonian N-body problem

考虑具有牛顿位势$1/r$的$\mathbb{R}^d$中的$N$ -体问题.我们证明了对于每一个初始组态$x_i$和每一个极小化的归一化中心组态$x_0$，存在一个从$x_i$开始并渐近于$x_0$的无碰撞抛物解.该解在每个时间区间都是极小元。证明利用变分结构的问题，它包括在寻找一个收敛的子序列在一个家庭的最小化轨迹。最难的部分是证明这个解是抛物线的，并且渐近于$x_0$。