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Three body problem; accumulation of phase transitions; transition to chaos; global chaos; choreographies.

We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum N-rotor problem has been used to model chains of coupled Josephson junctions and also arises via a partial continuum limit of the Wick-rotated XY model. In units of the coupling, the energy serves as a control parameter. We find periodic ‘pendulum’ and ‘breather’ orbits at all energies and choreographies at relatively low energies. They furnish analogs of the Euler–Lagrange and figure-8 solutions of the planar three body problem. Integrability at very low energies gives way to a rather marked transition to chaos at $E_{c} \approx 4$, followed by a gradual return to regularity as $E \rightarrow \infty$. We find four signatures of this transition: (a) the fraction of the areaof Poincare surfaces occupied by chaotic sections rises sharply at $E_{c}$, (b) discrete symmetries are spontaneously broken at $E_{c}$, (c) $E = 4$ is an accumulation point of stable to unstable transitions in pendulum solutions and (d) the Jacobi–Maupertuis curvature goes from being positive to having both signs above $E = 4$. Moreover, Poincare plots also reveal a regime of global chaos slightly above $E_{c}$.

GOVIND S KRISHNASWAMI¹ HIMALAYA SENAPATI¹

1. Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, India

Published

16 September 2019