On Globalization Problem of Multi-Hamiltonian Formalisms

TL;DR

This paper explores the extension of locally conformal geometric structures to multi-Hamiltonian systems, introducing new frameworks for modeling irreversible dynamics with rich geometric and Hamiltonian properties.

Contribution

It systematically extends locally conformal analysis to multi-Hamiltonian geometries, linking them to Nambu--Jacobi and generalized Jacobi manifolds, and constructs Hamiltonian dynamics for these structures.

Findings

Locally conformal Nambu--Poisson structures induce Nambu--Jacobi manifolds.

Locally conformal generalized Poisson structures induce generalized Jacobi manifolds.

Introduces locally conformal bi-Hamiltonian systems suitable for irreversible processes.

Abstract

The globalization problem arises when local tensor fields possess a given property (such as being symplectic or Poisson) but cannot be consistently extended to a global object due to incompatibilities on chart overlaps. A notable instance occurs in locally conformal analysis, where local representatives coincide only up to conformal factors. The locally conformal approach not only enables the definition of novel and rich geometric structures but also provides Hamiltonian formulations for irreversible systems, yielding physically meaningful dynamical consequences. While extensively studied for symplectic, cosymplectic, and Poisson geometries, its systematic extension to multi-Hamiltonian settings remains largely unexplored. In this work, we investigate locally conformally Nambu--Poisson and locally conformally generalized Poisson manifolds, showing that these structures naturally induce Nambu--Jacobi and generalized Jacobi manifolds, respectively. From a dynamical point of view, we construct Hamiltonian-type evolution equations, and for locally conformal $3$-Nambu structures, we introduce locally conformal bi-Hamiltonian systems. The resulting dynamics are particularly suitable for modeling irreversible multi-Hamiltonian processes, as they generally do not preserve the system's energy. Collectively, this work provides a unified framework for understanding both the geometric structures and Hamiltonian dynamics of classical, Nambu--Poisson, and generalized Poisson manifolds within a locally conformal context.