The contact property for magnetic flows on surfaces
type: CreativeWork
description: This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the …
abstract: This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the …
author: Benedetti, Gabriele
language: en
keywords: Magnetic flows, Symplectic geometry, Periodic orbits
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/44772696-f4cf-4810-b9fb-4393ebef813d/download
license: https://www.rioxx.net/licenses/all-rights-reserved/
date published: 2015-01-06
date modified: 2026-07-22T22:24:11Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.16235