Hyperbolic and Bi-hyperbolic solutions in the planar restricted (N+1)(N+1)-body problem

Consider the planar restricted $(N+1)$-body problem with trajectories of the $N(\ge 2)$ primaries forming a collision-free periodic solution of the $N$-body problem, for any positive energy $h$ and directions $\theta_{\pm} \in [0, 2\pi)$, we prove that starting from any initial position $x$ at any initial time $t_x$, there are hyperbolic solutions $\gamma^{\pm}|_{[t_x, \pm \infty)}$ satisfying $\gamma^{\pm}(t_x) =x$ and $$\lim_{t \to \pm \infty} \gamma^{\pm}(t) / |\gamma^{\pm}(t)| = e^{i \theta_{\pm} (\text{mod } 2\pi)}, \;\;\lim_{ t \to \pm \infty} \dot{\gamma}^{\pm}(t) = \pm \sqrt{2h} e^{i \theta_{\pm} (\text{mod } 2\pi)}.$$ Moreover we also prove the existence of a bi-hyperbolic solution $\gamma|_{\mathbb{R}}$ satisfying $$\lim_{t \to \pm \infty} \gamma(t) / |\gamma(t)| = e^{i \theta_{\pm} (\text{mod } 2\pi)}, \;\;\lim_{ t \to \pm \infty} \dot{\gamma}(t) = \pm \sqrt{2h} e^{i \theta_{\pm} (\text{mod } 2\pi)}.$$Comment: 37 pages, 4 figures; Comments are welcome

A symplectic dynamics approach to the spatial isosceles three-body problem

We study the spatial isosceles three-body problem from the perspective of Symplectic Dynamics. For certain choices of mass ratio, angular momentum, and energy, the dynamics on the energy surface is equivalent to a Reeb flow on the tight three-sphere. We find a Hopf link formed by the Euler orbit and a symmetric brake orbit, which spans an open book decomposition whose pages are annulus-like global surfaces of section. In the case of large mass ratios, the Hopf link is non-resonant, forcing the existence of infinitely many periodic orbits. The rotation number of the Euler orbit plays a fundamental role in the existence of periodic orbits and their symmetries. We explore such symmetries in the Hill region and show that the Euler orbit is negative hyperbolic for an open set of parameters while it can never be positive hyperbolic. Finally, we address convexity and determine for each parameter whether the energy surface is strictly convex, convex, or non-convex. Dynamical consequences of this fact are then discussed.Comment: 66 pages, 15 figure