论文标题
三个身体问题中的双曲动力和振荡动作
Hyperbolic dynamics and oscillatory motions in the 3 Body Problem
论文作者
论文摘要
考虑质量$ M_0，M_1，M_2> 0 $的平面3身体问题。在本文中，我们解决了两个基本问题：振荡动作的存在和混乱的双曲线集。 1922年，Chazy归类了这三个物体的最终动作，即当时间趋于无穷大时，身体可能具有的行为。可能的行为之一是振荡动作，即三个身体问题的解决方案，使得身体的位置$ q_0，q_1，q_2 $ cassecty \ [\ liminf_ {\ liminf_ {t \ to \ pm \ pm \ pm \ pm \ pm \ pm \ pm \ pm \ sup_ \ quad \ text {and} \ quad \ limsup_ {t \ to \ pm \ pm \ infty} \ sup_ {i，j = 0,1,2，i \ neq j} \ | q_i-q_i-q_j \ | =+\ | =+\ iffty。 \]假设所有三个质量$ m_0，m_1，m_2> 0 $都不相等。然后，我们证明存在这样的动议。我们还证明，可以构建三个身体问题的解决方案，这些问题的前进和后向最终动作具有不同的类型。 该结果依赖于构建不变集的集合，其动力学与（无限符号）伯诺利（Bernouilli）移动相结合。这些集合对于符合直接减少的平面3身体问题是双曲线。结果，我们获得了混沌运动的存在，无限数量的周期轨道和3个身体问题的阳性拓扑熵。
Consider the planar 3 Body Problem with masses $m_0,m_1,m_2>0$. In this paper we address two fundamental questions: the existence of oscillatory motions and of chaotic hyperbolic sets. In 1922, Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of the possible behaviors are oscillatory motions, that is, solutions of the 3 Body Problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy \[ \liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad \limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty. \] Assume that all three masses $m_0,m_1,m_2>0$ are not equal. Then, we prove that such motions exists. We also prove that one can construct solutions of the three body problem whose forward and backward final motions are of different type. This result relies on constructing invariant sets whose dynamics is conjugated to the (infinite symbols) Bernouilli shift. These sets are hyperbolic for the symplectically reduced planar 3 Body Problem. As a consequence, we obtain the existence of chaotic motions, an infinite number of periodic orbits and positive topological entropy for the 3 Body Problem.
