Singular cotangent models and complexity in fluids with dissipation

TL;DR

This paper explores singular $b$-cotangent models in fluid dynamics with dissipation, providing geometric interpretations and extending Hamiltonian frameworks to non-conservative systems.

Contribution

It introduces $b$-cotangent models for dissipative fluids, linking singular symplectic geometry with physical interpretations and Hamiltonian formulations of non-conservative systems.

Findings

Models relate fluid complexity to Reynolds number.

Twisted $b$-cotangent models include dissipative fluids.

Extended Hamiltonian framework for non-conservative systems.

Abstract

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a $b$-cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in $b$-cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The first one models systems on manifolds with boundary and the twisted model represents Hamiltonian systems where the singularity of the system is in the fiber of the bundle. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We relate the complexity of the fluids in terms of the Reynolds number and the (non)-existence of cotangent lift dynamics. We also discuss more general physical interpretations of the twisted and non-twisted $b$-symplectic models. These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems.