Hannay Angle: Yet Another Symmetry Protected Topological Order Parameter in Classical Mechanics

Topological way of thinking now goes beyond conventional solid materials, and topological characterization of classical mechanical systems governed by Newton's equation of motion begins to attract much attention. To have a deeper insight on physical meaning of topological numbers in mechanical systems, we demonstrate the use of the Hannay angle, a classical counterpart of the Berry phase, as a symmetry protected topological order parameter. We first derive the Hannay angle using a canonical transformation that maps the Newton's equation to the Schr\"{o}dinger type equation. The Hannay angle is then used to characterize a simple spring-mass model topologically with a particular focus on the bulk-edge correspondence and new aspects of the symmetry in a classical system.