Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed

TL;DR

This paper explores conditions under which all orbits of certain flows on contact manifolds are closed with identical periods, providing examples and applications to quantum mechanics.

Contribution

It adapts geodesic circle foliation results to Reeb and Hamiltonian flows, showing all orbits are closed with equal periods on connected contact manifolds.

Findings

All orbits are closed with identical periods on connected contact manifolds.

Concrete examples include harmonic oscillators and Lotka-Volterra systems.

Application to semiclassical Schrödinger operators and spectral analysis.

Abstract

Our purpose here is to adapt the results of Geodesic circle foliations for Reeb flows or Hamiltonian flows on contact manifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. We also present concrete examples of periodic flows, all of whose orbits are closed, such as Harmonic oscillators, Lotka-Volterra systems, and others. Lotka-Volterra systems, Reeb flows, and some geodesic flows have non-trivial periods, whereas the periods of Harmonic oscillators and similar systems can be easily obtained through direct calculations. As an application to quantum mechanics, we examine the spectrum of semiclassical Shr\"odinger operators. Then we have one of the semiclassical analogies of the Helton-Guillemin theorem.