Instabilities and chaos in the classical three-body and three-rotor problems

TL;DR

This thesis explores the geometric and dynamical complexities of the classical three-body and three-rotor problems, revealing instabilities, chaos, and ergodic behavior through curvature analysis and numerical simulations.

Contribution

It extends geometric methods to analyze instabilities and chaos in these classical systems, including regularization of collisions and identification of chaos transition mechanisms.

Findings

Negative scalar curvature indicates geodesic instability.

Chaos emerges sharply at a critical energy in the 3-rotor problem.

Evidence of ergodicity and mixing near the chaos threshold.

Abstract

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 Poincar\'e 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. Poincar\'e 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.