On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top

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.

💡 Research Summary

The paper investigates the celebrated Hirota‑Kimura (HK) discretization of the so(3) Euler top and establishes a full Hamiltonian framework for the map. The Euler top is a classical integrable system whose continuous dynamics are governed by the Lie‑Poisson bracket ({x_i,x_j}=\varepsilon_{ijk}x_k) and a quadratic Hamiltonian (H=\frac12\sum a_i x_i^2). Hirota and Kimura introduced an explicit, birational time‑step map that preserves two independent integrals and admits solutions expressed through elliptic functions, thereby demonstrating integrability at the level of first integrals. However, a Hamiltonian description—essential for Liouville‑Arnold integrability—had remained missing.

The authors begin by restating the HK map in a compact form: \