On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system

We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed.

💡 Research Summary

The paper provides a geometric construction of the bi‑Hamiltonian structure of the rational Calogero‑Moser system by means of two successive reductions of a very simple Poisson pair defined on the cotangent bundle of the matrix algebra $\mathfrak{gl}(n,\mathbb{R})$.

First, the authors recall the general theory: on any smooth manifold $Q$ a torsion‑free (1,1) tensor $L$ induces, via the deformed Liouville 1‑form $\theta_L$, a second Poisson tensor $P_1$ on $T^*Q$ that is compatible with the canonical Poisson tensor $P_0$. The recursion operator $N=P_1P_0^{-1}$ inherits the vanishing Nijenhuis torsion of $L$, and the functions $H_k=\frac{1}{k}\operatorname{tr}L^k$ generate a bi‑Hamiltonian hierarchy $P_1 dH_k=P_0 dH_{k+1}$.

Specialising to $Q=\mathfrak{gl}(n)$, the tensor $L_A(V)=AV$ is trivially torsion‑free. Identifying $T^*\mathfrak{gl}(n)$ with $\mathfrak{gl}(n)\times\mathfrak{gl}(n)$ via the trace pairing, the canonical Poisson tensor $P_0$ corresponds to the symplectic form $\omega_0=\operatorname{tr}(dB\wedge dA)$. The deformed form $\theta_L=\operatorname{tr}(BA,dA)$ yields a second Poisson tensor $P_1$ which can be written as a Lie‑Poisson structure on the semidirect product $\mathfrak{gl}(n)\ltimes\mathbb{R}^{n^2}$. The recursion operator $N$ takes the block form \