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 quasiperiodic and a KAM 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 respect to most frames, we call the phenomenon pattern evocation.

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

“…The resonant structures that one sees in the rotating frames of interest in the present paper appear similar to what one sees in the phenomenon of pattern evocation ͑see Marsden and Scheurle, 27 Marsden, Scheurle and Wendlandt 28 ͒ when rotationally symmetric systems are viewed from the point of view of an appropriate rotating frame. Of course, for the restricted three-body problem there is a simple and natural choice of a rotating frame.…”

Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

“…This point of view is closely related to recent developments of the geometric phase in the context of control theory, which is discussed in [22]. Finally, Marsden and Scheurle [29] use the geometric phase idea on mechanical systems with symmetries to show how symmetric patterns in the phase space of the system can be brought out that would not otherwise be seen (see also [30]). By now, there are several sources where one can get an overview of the various interpretations and applications of the geometric phase in various contexts.…”

“…Its integral around the circular vortex orbit gives the Hannay-Berry phase (29). Hence, the geometric phase in this problem is given by…”

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

“…The Double Spherical Pendulum. The simulation in this section is motivated by our work on pattern evocation for the double spherical pendulum given in Marsden and Scheurle [1995] and Marsden, Scheurle and Wendlandt [1996]. The double spherical pendulum consists of two constrained point masses.…”

“…Here, the linear space is, V = R 3 × R 3 , and the constraint manifold, Q ⊂ V , is S 2 × S 2 . This example is motivated by our work on pattern evocation (Marsden and Scheurle [1995] and Marsden, Scheurle and Wendlandt [1996].…”