Instabilities and chaos in the classical three-body and three-rotor problems
Himalaya Senapati
备注：
PhD thesis, defended July 23, 2020, Chennai Mathematical Institute, 101 pages. Based on arXiv:1606.05091, arXiv:1901.07289, arXiv:1810.01317, arXiv:1811.05807 and arXiv:1910.04455
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.
Ergodicity, mixing and recurrence in the three rotor problem
在经典的三转子问题中，三个相等的点质量在一个圆周上运动，受到强度为g的吸引余弦势的作用。在质心系中，能量E是唯一已知的守恒量。在早期的工作中[Krishnaswami and Senapati，arXiv：1810.01317，Oct. 2018，arXiv：1811.05807，Nov. 2018]，在这个系统中发现了一个有序-混沌-有序的转变，以及一个5.33g < E < 5.6g的全局混沌带。在这里，我们提供了数值证据遍历性和混合在这个波段。沿一般轨迹的相对角和角动量的分布被示出为在时间上以幂律形式接近恒定能量超曲面（由刘维尔测度加权）上的相应分布。此外，从一个小体积发出的轨迹被证明是均匀分布在恒定的能量超曲面，这表明动力学是混合的。在这个带之外，遍历性和混合失败，虽然在恒定能量超曲面上的角动量的分布显示出有趣的相变从维格纳双峰随着能量的增加。最后，在全局混沌带，发现有限大小的细胞的复发时间的分布遵循指数规律，平均复发时间满足一个标度律，涉及一个指数一致的全局混沌和遍历性。
Intramolecular vibrational energy redistribution from a high frequency mode in the presence of an internal rotor: Classical thick-layer diffusion and quantum localization
