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Instabilities and chaos in the classical three-body and three-rotor problems
Chennai Mathematical Institute.
In this thesis, we study instabilities and singularities in a geometrical approach to the planar three-body problem as well as instabilities, chaos and ergodicity in the three-rotor problem.
Trajectories of the three-body problem are expressed as geodesics of the Jacobi-Maupertuis (JM) metric on the configuration space. Isometries lead to reduced dynamics on quotients of the configuration space, which encode information on the full dynamics. Riemannian submersions are used to find quotient metrics and to show that the geodesic formulation regularizes collisions for the inverse-square, but not for the Newtonian potential. Extending work of Montgomery, we show the negativity of the scalar curvature on the center of mass configuration space and certain quotients. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities.
In the three-rotor problem, three 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 serves as a control parameter. We find analogues of the Euler-Lagrange family of periodic solutions: pendula, breathers and choreographies. The model displays order-chaos-order behavior and undergoes a fairly sharp transition to chaos at a critical energy with several novel manifestations. Poincaré sections indicate global chaos in a band of energies slightly above this transition where we provide numerical evidence for ergodicity and mixing with respect to the Liouville measure and study the statistics of recurrence times.