Topological characterizations of Hamiltonian flows with finitely many singular points on unbounded surfaces

TL;DR

This paper provides a topological classification of Hamiltonian flows with finitely many singular points on unbounded surfaces, establishing invariants and embedding results under certain conditions.

Contribution

It introduces a topological framework for Hamiltonian flows on unbounded surfaces and constructs complete invariants under regularity assumptions.

Findings

Topological invariants for Hamiltonian flows on unbounded surfaces are constructed.

Hamiltonian flows on unbounded surfaces can be embedded into flows on compact surfaces under finite volume.

The paper distinguishes Hamiltonian flows from non-Hamiltonian flows on the plane based on topological properties.

Abstract

Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems. Though various fluid phenomena are modeled as flows on the plane, it is not obvious to determine if the flows are Hamiltonian, even the singular point set is totally disconnected and every orbit is contained in a straight line parallel to the $x$-axis. In fact, there are such non-Hamiltonian flows on the plane. On the other hand, this paper topologically characterizes Hamiltonian flows on unbounded surfaces and constructs their complete invariant under a regularity condition for singular points. In addition, under finite volume assumption, Hamiltonian flows on unbounded surfaces can be embedded into those on compact surfaces.