Abstract

The zero-velocity surfaces of the general planar three-body problem are constructed in form space, i.e., the factor space of the configuration space by transfer and rotation. Such a space is the space of congruent triangles, and the sphere in this space is similar triangles. The integral of energy in form space gives the equation of a zero-velocity surface. These surfaces can also be obtained based on the Sundman inequality. Such surfaces separate areas of possible motion from areas where motion is impossible. Without loss of generality, we can assume that the constant energy is −1/2 and the sought surfaces depend only on the magnitude of the angular momentum of the problem, J. Depending on this value, five topologically different types of surfaces can be distinguished. For small J, the surface consists of two separate surfaces, internal and external ones, motion is possible only between them. With J increasing the inner surface increases, the outer surface decreases, the surfaces first have a common point at some value of J, with a further increase in J, their topological type changes and finally the zero-velocity surface splits into three nonintersecting surfaces, and motion is possible only inside them. Examples of the corresponding surfaces are given for each of these types, their cross sections in the plane xy and in the plane xz and the surfaces themselves are constructed, and their properties are studied.

中文翻译：

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一般三体问题中的零速度面

摘要

一般平面三体问题的零速度面是在形空间即位形空间的因子空间中通过传递和旋转构造的。这样的空间就是全等三角形空间，这个空间中的球面就是相似三角形。形式空间中的能量积分给出了零速度表面的方程。这些曲面也可以根据 Sundman 不等式获得。这些表面将可能运动的区域与不可能运动的区域分开。在不失一般性的情况下，我们可以假设恒定能量为 −1/2，并且所寻找的表面仅取决于问题的角动量J的大小。根据此值，可以区分五种拓扑不同类型的表面。对于小J，表面由两个独立的表面组成，内部表面和外部表面，运动只能在它们之间进行。随着J的增加，内表面增加，外表面减少，表面首先在某个J值处有一个公共点，随着J的进一步增加，它们的拓扑类型发生变化，最后零速度表面分裂成三个不相交的表面，只有在它们内部才有可能运动。对于这些类型中的每一种都给出了相应表面的示例，它们在平面xy和平面xz中的横截面以及表面本身被构造，并且它们的特性被研究。