Calogero-Moser Integrable System

The Calogero-Moser system of N particles on a line with pair potential g²/(xᵢ-xⱼ)² plus harmonic confinement ω²xᵢ²/2 is exactly integrable: it has N conserved quantities in involution. Remarkably, the particles never pass through each other, and the eigenvalues of a Lax matrix L encode all conserved charges.

Parameters

N particles: 5 Coupling g²: 1.00 Harmonic ω²: 0.50 Initial spread: 2.00

Total energy: —

Energy drift: —

Min separation: —

H = Σ pᵢ²/2 + ω²xᵢ²/2 + Σᵢ<ⱼ g²/(xᵢ-xⱼ)²

Calogero (1971): harmonic Calogero
Moser (1975): rational (no confinement)

Lax pair L, M: dL/dt = [L,M]
→ eigenvalues of L are conserved

Connection to random matrices (GUE level repulsion ↔ g=1)