Domains of possible motion in the general three body problem

V. B. Titov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXIV, Tome 517 (2022), pp. 225-249

Résumé

The regions of possible motion of the general planar three-body problem are constructed in the form space, the factor-space of the configuration space of the problem by transfer and rotation. Such a space is a space of congruent triangles, and the sphere in this space is – similar triangles. The integral of energy in the form space gives the equation of the zero velocity surface. These surfaces separate regions of possible motion from regions where movement is impossible. These surfaces can also be obtained based on the Sundman inequality. Without loss of generality, we assume that the constant energy is equal $-1/2$ and the desired surfaces depend only on the value 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, the 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 region of possible motion separates into three disjoint regions, the motion is possible only inside them. Examples of the corresponding surfaces are given for each of the five types, their sections in the plane $xy$ and in the plane $xz$ and the surfaces themselves are constructed, their properties are studied. Orbits suffering collisions are studied separately: collinear and isosceles orbits. Collisions occurring in such orbits require regularization. In the form space, it is natural to use Lemaitre regularization. In the regularized space, the figure-“eight” orbit is also considered.

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