On Globalization Problem of Multi-Hamiltonian Formalisms

TL;DR

This paper addresses the globalization problem for local Hamiltonian structures by developing a unified locally conformal framework for Poisson, Nambu–Poisson, and generalized Poisson geometries. It introduces locally conformal Nambu–Poisson and locally conformal generalized Poisson manifolds, showing they naturally induce Nambu–Jacobi and generalized Jacobi structures, respectively, and derives Hamiltonian-type dynamics, including locally conformal bi-Hamiltonian formulations for 3-Nambu–Poisson systems. The authors establish a contraction hierarchy linking higher-order NP and Jacobi-type structures, and demonstrate how LCNP dynamics lift to locally conformal higher-order generalized Poisson structures. The work provides a cohesive geometric and dynamical framework applicable to classical, Nambu–Poisson, and generalized Poisson settings within a locally conformal context, with implications for irreversible multi-Hamiltonian processes and potential time-dependent extensions.

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.

Theorems & Definitions (20)

• Theorem 2.1
• Proof 2.2
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Proof 3.4
• Corollary 3.5
• Theorem 3.6
• Theorem 4.1
• Proof 4.2