[Paper Review] Nambu dynamics and its noncanonical Hamiltonian representation in many degrees of freedom systems

A. Horikoshi|arXiv (Cornell University)|Mar 1, 2021

Advanced Topics in Algebra16 references5 citations

TL;DR

This paper demonstrates that noncanonical Hamiltonian dynamics can preserve consistent time evolution even when Nambu dynamics fails due to violation of the fundamental identity in many-degrees-of-freedom systems. By constructing noncanonical Poisson brackets from the Nambu bracket using one Hamiltonian, the authors show that the Jacobi identity—essential for consistent dynamics—can still hold if the Poisson matrices are independent across degrees of freedom, as verified in a semiclassical two-oscillator model.

ABSTRACT

Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example, we evaluate these identities for a semiclassical system of two coupled oscillators.

Motivation & Objective

• To investigate whether consistent time evolution can be restored in noncanonical Hamiltonian dynamics when Nambu dynamics violates the fundamental identity in many-degrees-of-freedom systems.
• To derive the mathematical condition under which the Jacobi identity holds for the noncanonical Poisson bracket derived from the Nambu bracket.
• To demonstrate this restoration of consistency using a semiclassical model of two coupled oscillators with a hidden Nambu structure.
• To clarify the distinction between the roles of the two Nambu Hamiltonians in determining whether the Jacobi identity is satisfied in the noncanonical formulation.

Proposed method

• Define the Nambu bracket for N=3 variables in a system of n coupled degrees of freedom using the 3D Levi-Civita symbol and Jacobian determinant.
• Construct noncanonical Poisson brackets via Poisson matrices derived from one of the Nambu Hamiltonians (G or H), using Jij = εijk ∂G/∂xk.
• Derive the condition for the Jacobi identity to hold: ∂Jα_ij /∂xβ_k = 0 for α ≠ β, ensuring Poisson matrices of different degrees of freedom are independent.
• Apply the formalism to a semiclassical two-oscillator system with Hamiltonians H (with coupling) and G (decoupled), using expectation values of quantum operators.
• Verify the fundamental identity violation in Nambu dynamics by computing {{A,B,C},D,E} − [sum of cyclic terms] for specific functions, yielding non-zero results.
• Check the Jacobi identity in both noncanonical formulations: one using G (satisfies condition, identity holds), the other using H (violates condition, identity fails).

Experimental results

Research questions

• RQ1 Can consistent time evolution be restored in noncanonical Hamiltonian dynamics even when the Nambu dynamics violates the fundamental identity in many-degrees-of-freedom systems?
• RQ2 What condition on the Poisson matrices ensures the Jacobi identity holds in the noncanonical representation of Nambu dynamics?
• RQ3 Does the choice of Hamiltonian used to define the Poisson matrix affect the validity of the Jacobi identity in the noncanonical formulation?
• RQ4 In a system with coupled degrees of freedom, can the noncanonical Poisson bracket still satisfy the Jacobi identity if the Poisson matrix is constructed from a decoupled Hamiltonian?
• RQ5 How does the violation of the fundamental identity in Nambu dynamics affect the actual time evolution, and can this be corrected via noncanonical representation?

Key findings

• The fundamental identity of Nambu dynamics is violated in the two-oscillator system: for (A,B,C) = (x²₁, x²₂, x¹₂) and (D,E) = (H,G), the left-hand side of Eq. (3) is zero while the right-hand side is −λ, indicating broken consistency.
• The noncanonical Poisson bracket defined via the decoupled Hamiltonian G satisfies the Jacobi identity because the Poisson matrices Jα_ij depend only on their respective degrees of freedom, fulfilling the condition ∂Jα_ij /∂xβ_k = 0 for α ≠ β.
• In contrast, the noncanonical Poisson bracket defined via the coupled Hamiltonian H violates the Jacobi identity, as the Poisson matrices ˜Jα_ij depend on variables from other degrees of freedom, breaking the independence condition.
• For the choice (A,B) = (x¹₂, x²₂) and C = H, the left-hand side of the Jacobi identity is zero, but the right-hand side is −λm₁ω²₁x²₁, confirming the violation.
• The Liouville theorem holds in both noncanonical formulations, preserving phase space volume despite the violation of the Jacobi identity in one case.
• The study shows that consistent time evolution can be restored in the noncanonical representation even when the original Nambu dynamics is inconsistent, provided the Poisson matrix is constructed from a decoupled Hamiltonian.