Certain aspects of the integrability/solvability of the
Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen
systems with rational and trigonometric potentials are reviewed. The
equilibrium positions of classical multi-particle systems and the
eigenfunctions of single-particle quantum mechanics are described by the same
orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson
and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum
mechanical systems have two remarkable properties, factorization and shape
invariance.Comment: 30 pages, 1 figure. Contribution to proceedings of RIMS workshop
"Elliptic Integrable Systems" (RIMS, Nov. 2004
