Deformed Hamiltonian vector fields and Lagrangian fibrations

TL;DR

This paper introduces deformed Hamiltonian vector fields on Lagrangian fibrations, extending dissipative systems to symplectic manifolds, with implications for symplectic geometry and physics-inspired mathematical problems.

Contribution

It develops a new class of deformed Hamiltonian vector fields on Lagrangian fibrations, generalizing certain dissipative systems to symplectic geometry.

Findings

Properties of deformed Hamiltonian vector fields are characterized.

Connections to symplectic geometry are established.

Potential applications to physics-inspired mathematical problems are discussed.

Abstract

Certain dissipative physical systems closely resemble Hamiltonian systems in $\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in each conjugate pair rescaled by a real parameter. To generalise these dynamical systems to symplectic manifolds in this paper we introduce and study the properties of deformed Hamiltonian vector fields on Lagrangian fibrations. We describe why these objects have some interesting applications to symplectic geometry and discuss how their physical interpretation motivates new problems in mathematics.