Observation of Periodic Orbits on Curved Two - dimensional Geometries

TL;DR

This paper demonstrates how periodic geodesic orbits influence the spectra of thin shells on curved geometries, linking classical orbits with quantum-like spectral features through a semiclassical trace formula.

Contribution

It introduces a semiclassical trace formula for elastomechanical spectra on curved shells, connecting periodic geodesic orbits with spectral properties in two-dimensional curved geometries.

Findings

Spectra are well described by a trace formula involving geodesic orbits.

Periodic orbits influence spectral clustering, especially in hemispherical shells.

Experimental spectra match predictions from the semiclassical model.

Abstract

We measure elastomechanical spectra for a family of thin shells. We show that these spectra can be described by a "semiclassical" trace formula comprising periodic orbits on geodesics, with the periods of these orbits consistent with those extracted from experiment. The influence of periodic orbits on spectra in the case of two-dimensional curved geometries is thereby demonstrated, where the parameter corresponding to Planck's constant in quantum systems involves the wave number and the curvature radius. We use these findings to explain the marked clustering of levels when the shell is hemispherical.