[Paper Review] On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top

Matteo Petrera, Yuri B. Suris|ArXiv.org|Jul 30, 2007

Nonlinear Waves and Solitons8 references3 citations

TL;DR

This paper establishes the Hamiltonian structure of the Hirota-Kimura discretization of the Euler top, proving its integrability in the Liouville-Arnold sense by constructing a bi-Hamiltonian formulation. The authors provide a streamlined derivation of existing conserved quantities and elliptic function solutions, and identify an invariant volume form and a family of compatible Poisson tensors, completing the standard integrability framework for this explicit, birational map on R³.

ABSTRACT

This paper deals with a remarkable integrable discretization of the so(3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a bi-Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville-Arnold sense.

Motivation & Objective

• To provide a Hamiltonian formulation for the Hirota-Kimura discretization of the Euler top, which had not been previously established.
• To streamline and simplify the presentation of prior results on conserved quantities and elliptic function solutions from Hirota and Kimura.
• To establish the map’s integrability in the Liouville-Arnold sense by constructing a bi-Hamiltonian structure.
• To lay the foundation for understanding a broader class of integrable discretizations in classical mechanics.

Proposed method

• Derive the explicit birational map from the Hirota-Kimura discretization equations: $\tilde{x}_i - x_i = \delta_i(\tilde{x}_j x_k + x_j \tilde{x}_k)$ with $\delta_i = \epsilon \alpha_i / 2$.
• Re-derive the two independent integrals of motion and the solution in terms of Jacobi elliptic functions using a simplified approach.
• Identify an invariant volume form for the map, confirming preservation of phase space volume.
• Construct a family of compatible Poisson tensors to establish a bi-Hamiltonian structure.
• Verify that the Poisson tensors are invariant under the map and that the integrals of motion are in involution.
• Use the bi-Hamiltonian structure to confirm integrability in the Liouville-Arnold sense.

Experimental results

Research questions

• RQ1 Does the Hirota-Kimura discretization of the Euler top admit a Hamiltonian formulation?
• RQ2 Can a bi-Hamiltonian structure be constructed for this explicit, birational map on R³?
• RQ3 Are the known conserved quantities in involution with respect to the Poisson brackets of the bi-Hamiltonian structure?
• RQ4 Does the existence of a bi-Hamiltonian structure confirm integrability in the Liouville-Arnold sense?
• RQ5 What is the role of the invariant volume form and compatible Poisson tensors in the integrability of this map?

Key findings

• The Hirota-Kimura discretization admits a bi-Hamiltonian structure, confirming its integrability in the Liouville-Arnold sense.
• Two independent integrals of motion are explicitly derived and shown to be in involution with respect to the Poisson brackets of the bi-Hamiltonian structure.
• An invariant volume form is identified, which is preserved under the map, supporting the system’s phase space structure.
• The solution to the map is expressed in terms of Jacobi elliptic functions, with amplitudes and modulus determined by the integrals of motion.
• The bi-Hamiltonian formulation is constructed via a family of compatible Poisson tensors, with explicit expressions derived for the Poisson structures.
• The parameterization of the solution depends on the region of phase space, with two distinct forms (26) and (27) corresponding to different inequalities on the integrals of motion.