[Paper Review] Hidden Nambu mechanics - A variant formulation of Hamiltonian systems -

A. Horikoshi, Yoshiharu Kawamura|arXiv (Cornell University)|Apr 8, 2013

Control and Dynamics of Mobile Robots1 references52 citations

TL;DR

This paper proposes a novel formulation of Hamiltonian systems using Nambu mechanics with redundant variables, showing that any Hamiltonian system can be recast as a Nambu system with generalized Nambu brackets involving both original Hamiltonians and induced constraints. The key contribution is a unified framework where extended phase spaces with extra degrees of freedom yield Nambu dynamics, applicable even to systems with first-class constraints and with associated partition functions derived.

ABSTRACT

We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.

Motivation & Objective

• To develop a variant formulation of Hamiltonian systems using variables with redundant degrees of freedom.
• To show that such systems can be described by Nambu dynamics with generalized master equations involving multiple Hamiltonians and constraints.
• To extend the formulation to many-body systems and systems with first-class constraints.
• To derive partition functions for the extended dynamics in many degrees of freedom.
• To establish a bridge between standard Hamiltonian mechanics and Nambu mechanics via variable redefinition.

Proposed method

• Introduce N ≥ 3 variables (x₁,…,xₙ) as fundamental degrees of freedom, functionally related to canonical (q,p) variables.
• Derive induced constraints from the functional dependence of the xᵢ on (q,p), treating them as additional conserved quantities.
• Formulate time evolution using generalized Nambu equations with N−1 Hamiltonians, including the original H and the constraints as generalized Hamiltonians.
• Use the Nambu bracket defined via N-dimensional Jacobian determinants to express the dynamics.
• Apply Dirac's constraint formalism to handle first-class constraints in the extended phase space.
• Construct partition functions for the extended Nambu systems by generalizing the path integral approach to Nambu dynamics.

Experimental results

Research questions

• RQ1 Can Hamiltonian systems be reformulated using Nambu mechanics with redundant variables?
• RQ2 How do constraints arising from variable redundancy affect the Nambu dynamics?
• RQ3 Can the Nambu formalism be generalized to include both original Hamiltonians and induced constraints as dynamical generators?
• RQ4 What is the structure of the partition function in the extended Nambu formulation?
• RQ5 How does the Nambu formulation handle systems with first-class constraints?

Key findings

• Any Hamiltonian system with canonical variables (q,p) can be recast as a Nambu system with N ≥ 3 variables (x₁,…,xₙ), where the dynamics are governed by generalized Nambu equations.
• The induced constraints from the variable redundancy are shown to behave like additional Hamiltonians in the Nambu framework, leading to a consistent Nambu equation with N−1 Hamiltonians.
• The time evolution of any function ƒ(x₁,…,xₙ) is expressed via the Nambu bracket with the generalized Hamiltonians, including both the original H and the constraint functions G_c.
• The formulation is extended to many degrees of freedom, where partition functions are explicitly constructed for the extended Nambu dynamics.
• For systems with first-class constraints, the Nambu formulation naturally incorporates the constraints as conserved quantities, and the dynamics remain consistent under the generalized Nambu bracket.
• The paper demonstrates that Nambu mechanics is not limited to special systems but is a hidden structure underlying general Hamiltonian dynamics when redundant variables are used.