New periodic orbits in the planar equal-mass three-body problem
It is known that in the planar equal-mass three-body problem, there exist two families of orbits: ``retrograde'' orbit and ``prograde'' orbit: ``A retrograde orbit is a relative periodic solution of the Newton's law of gravitation, with two adjacent masses revolving around each other in one direction while their mass centre revolves around the third mass in the other direction. The orbit is ``prograde'' if both revolutions follow the same direction''. In this paper, the existence of a new set of orbits in the planar equal-mass three-body problem is studied, for the case of masses $m_1=m_2=m_3=1$. A two-point free boundary value problem with topological constraints are considered. The first configuration $Q_S$ is on a horizontal line with the three collinear points, and the $Q_E$ configuration is an isosceles triangle with the point $q_1$ as the vertex, and the symmetry axis of $q_2$, $q_3$ is a counterclockwise $\theta$ rotation of the $x$-axis. For $\theta=\pi/10$, and for $\theta$ close to $\pi/10$, the test path is defined. For each $\theta \in [0.084\pi, 0.183\pi]$, there exists a nontrivial and collision-free minimizing path $P_{0,\theta} = P_{0,\theta} (t \in [0,1])$ connecting the two configuration sets $Q_S$ and $Q_E$, and there exists a new set of periodic orbits for a small interval of rotation angle $\theta$. Finally, in Appendix A, Data of 7 test paths are given.
scientific article; zbMATH DE number 6919331
scientific article