Pattern evocation and geometric phases in mechanical systems with symmetry

Abstract

This paper is concerned with the relation between the dynamics of a given Hamiltonian system with a given symmetry group and its reduced dynamics. We illustrate the process of visualization of reduced orbits using the double spherical pendulum. In this process of visualization, one sees certain patterns when the dynamics is viewed relative to rotating frames with certain critical angular velocities. By using the reduced dynamics, we also explain these patterns. We show that if the motion on the phase space reduced by a continuous symmetry group at a given momentum level is periodic, then there is a uniformly rotating frame, that is, a one-parameter group motion, relative to which the unreduced trajectory is periodic with the same period. If the continuous symmetry group of the system is Abelian, which corresponds to the system having cyclic variables, we derive an explicit expression for the required angular velocity in terms of the dynamic phase (an average of the mechanical connection) and the geometric phase (the holonomy of the mechanical connection). We show that one can also find such a frame if the reduced orbit is quasi-periodic and a KAM (Kolmogorov-Arnold-Moser) condition is satisfied The almost periodic case is also discussed. An important aspect of this procedure is how to use it in the presence of discrete symmetries. We show that, under appropriate conditions, the visualized orbit has, relative to a suitable uniformly rotating frame, the same temporal behavior and discrete symmetries as the reduced orbit. Since these spatio-temporal patterns are not apparent with repect to most frames, we call the phenomenon pattern evocation.