Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910588151529472 |
|---|---|
| author | Miyanishi, Yoshihisa |
| author_facet | Miyanishi, Yoshihisa |
| contents | 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 Shrödinger operators. Then we have one of the semiclassical analogies of the Helton-Guillemin theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed Miyanishi, Yoshihisa Mathematical Physics Differential Geometry Dynamical Systems Symplectic Geometry Spectral Theory 37J46, 35P20, 70H05 (primary), 37E35, 46N50, 81Q80, 58C40 (secondary) 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 Shrödinger operators. Then we have one of the semiclassical analogies of the Helton-Guillemin theorem. |
| title | Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed |
| topic | Mathematical Physics Differential Geometry Dynamical Systems Symplectic Geometry Spectral Theory 37J46, 35P20, 70H05 (primary), 37E35, 46N50, 81Q80, 58C40 (secondary) |
| url | https://arxiv.org/abs/2408.06056 |