Calogero-Moser Integrable System

H = Σpᵢ²/2 + ω²Σqᵢ²/2 + g²Σᵢ≠ⱼ (qᵢ−qⱼ)⁻² · particles never cross

H = —
4
10
1.0
20
The Calogero-Moser system is exactly integrable: N particles on a line with harmonic confinement and inverse-square repulsion. It has N conserved quantities in involution — Lax pair L_ij = pᵢδᵢⱼ + ig(1−δᵢⱼ)/(qᵢ−qⱼ). Remarkably, the eigenvalues of L are conserved, and particles never cross — they approach, slow down (inverse-square repulsion diverges), and bounce back. The quantum version has wavefunctions with Vandermonde structure.