论文标题
平面限制$（n+1）$ - 身体问题的双曲线和双曲线解决方案
Hyperbolic and Bi-hyperbolic solutions in the planar restricted $(N+1)$-body problem
论文作者
论文摘要
考虑平面限制$（n+1）$ - 身体问题，具有$ n（\ ge 2）$的轨迹，$ n $ n $ body问题的原始解决方案，对于任何积极的能量$ h $和方向$θ_{\ pm} \ in [0,2oud）$，我们证明，在任何初始位置$ x $ x $ x $ t时$γ^{\ pm} | _ {[t_x，\ pm \ infty）} $满足$γ^{\ pm}（t_x）= x $和$$ \ liM_ { ＝ 2π）}。$$，我们还证明存在Bi-Hyperbolic解决方案$γ| _ {\ Mathbb {r}} $满足$ \ lim_ {t \ to \ pm \ pm \ infty}γ（t） / |γ（t） / |γ（t） / |γ（t）| ＝
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 $θ_{\pm} \in [0, 2π)$, we prove that starting from any initial position $x$ at any initial time $t_x$, there are hyperbolic solutions $γ^{\pm}|_{[t_x, \pm \infty)}$ satisfying $γ^{\pm}(t_x) =x$ and $$ \lim_{t \to \pm \infty} γ^{\pm}(t) / |γ^{\pm}(t)| = e^{i θ_{\pm} (\text{mod } 2π)}, \;\;\lim_{ t \to \pm \infty} \dotγ^{\pm}(t) = \pm \sqrt{2h} e^{i θ_{\pm} (\text{mod } 2π)}.$$ Moreover we also prove the existence of a bi-hyperbolic solution $γ|_{\mathbb{R}}$ satisfying $$ \lim_{t \to \pm \infty} γ(t) / |γ(t)| = e^{i θ_{\pm} (\text{mod } 2π)}, \;\;\lim_{ t \to \pm \infty} \dotγ(t) = \pm \sqrt{2h} e^{i θ_{\pm} (\text{mod } 2π)}.$$
