论文标题
经典三体和三旋转问题的不稳定性和混乱
Instabilities and chaos in the classical three-body and three-rotor problems
论文作者
论文摘要
本论文以三体问题的平面3体问题以及不稳定性，混乱和怪异性来研究不稳定性和奇异性。 平面3体问题的轨迹表示为jacobi-maupertuis（JM）度量的大地测量，在配置空间$ C^3 $。翻译，旋转和缩放式异构体导致$ c^3 $的商的动力学减少，该动态编码有关完整动态的信息。里曼尼亚浸入订单用于查找商指标，并证明Geodesic配方将$ 1/r^2 $的碰撞定期，但不能以$ 1/r $ $的潜力。为了扩展蒙哥马利的工作，我们显示了标态曲率在质量配置空间的中心以及相等质量和零能量的某些商的负面性。还发现截面曲率在很大程度上表明广泛的地质不稳定性。 在三连杆问题中，3个相等的质量在一个受吸引人的余粒子间电位的圆上移动。这个问题是耦合约瑟夫森连接模型的经典限制。能量E充当控制参数。 We find analogues of the Euler-Lagrange family of periodic solutions: pendula and breathers at all E and choreographies up to moderate E. The model displays order-chaos-order behavior and undergoes a fairly sharp transition to chaos at a critical energy E$_c$ with several manifestations: (a) a dramatic rise in the fraction of Poincaré surfaces occupied by chaotic sections, (b) spontaneous破坏离散对称性，（c）板块中稳定性转变的几何级联和（d）JM曲率符号的变化。 Poincaré部分表明在略高于E $ _C $的一系列能量中，我们提供了有关liouville措施的良好性和混合的证据，并研究了复发时间的统计数据。
This thesis studies instabilities and singularities in a geometrical approach to the planar 3-body problem as well as instabilities, chaos and ergodicity in the 3-rotor problem. Trajectories of the planar 3-body problem are expressed as geodesics of the Jacobi-Maupertuis (JM) metric on the configuration space $C^3$. Translation, rotation and scaling isometries lead to reduced dynamics on quotients of $C^3$ that encode information on the full dynamics. Riemannian submersions are used to find the quotient metrics and to show that the geodesic formulation regularizes collisions for the $1/r^2$ but not for the $1/r$ potential. Extending work of Montgomery, we show the negativity of the scalar curvature on the center of mass configuration space and certain quotients for equal masses and zero energy. Sectional curvatures are also found to be largely negative indicating widespread geodesic instabilities. In the 3-rotor problem, 3 equal masses move on a circle subject to attractive cosine inter-particle potentials. This problem arises as the classical limit of a model of coupled Josephson junctions. The energy E serves as a control parameter. We find analogues of the Euler-Lagrange family of periodic solutions: pendula and breathers at all E and choreographies up to moderate E. The model displays order-chaos-order behavior and undergoes a fairly sharp transition to chaos at a critical energy E$_c$ with several manifestations: (a) a dramatic rise in the fraction of Poincaré surfaces occupied by chaotic sections, (b) spontaneous breaking of discrete symmetries, (c) a geometric cascade of stability transitions in pendula and (d) a change in the sign of the JM curvature. Poincaré sections indicate global chaos in a band of energies slightly above E$_c$ where we provide evidence for ergodicity and mixing with respect to the Liouville measure and study the statistics of recurrence times.
