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 derivation of the well‑known bi‑Hamiltonian structure of the rational n‑particle Calogero‑Moser (CM) system. The authors start by recalling that the CM system is an archetype of integrable many‑body dynamics and that its integrability is usually exhibited through a Lax pair (L, M) together with two compatible Poisson brackets. Rather than taking the Lax representation as the primary object, the authors ask where the two Poisson structures themselves originate.

To answer this, they consider the Lie algebra gl(n,ℝ) and its cotangent bundle Tgl(n,ℝ) ≅ gl(n,ℝ) × gl(n,ℝ)⁎. On this space they introduce two globally defined 2‑forms. The first, ω₀ = tr(dX∧dY), is the canonical symplectic form of the cotangent bundle and yields the standard Poisson tensor P₀. The second, ω₁ = tr(dX·X∧dY), is built from the matrix product and encodes an “R‑matrix‑type” bracket; it gives a second Poisson tensor P₁. The pair (P₀,P₁) is compatible, i.e., any linear combination is again a Poisson tensor, thus forming a Poisson pair on Tgl(n,ℝ).

The next step is a double reduction (projection) that transports this Poisson pair onto the phase space of the CM system. The first reduction is a Marsden–Weinstein symplectic reduction with respect to the coadjoint action of GL(n,ℝ). The associated moment map is μ₀(X,Y) =