Calogero-Moser and Ruijsenaars-Schneider Systems

Abstract

Undersystems of the Calogero-Moser typewe shall understand multi-particle systems with non-local pair-wise interactions. The original Calogero-Moser models enjoyed generalizations in various directions, so that the area became highly ramified. Already these original systems are known in three appearances: rational, hyperbolic (trigonometric), and elliptic.Calogero-Moser (CM) modelsare natural Hamiltonian systems on the canonical phase space $\mathbb{R}^{2N} \left( {x,p} \right)$ with the symplectic Poisson bracket (1.3.7) and the Hamilton functions

$${{H}_{{CM}}}\left( {x,p} \right) = \frac{1}{2}\sum\limits_{{k = 1}}^{N} {p_{k}^{2}} + \frac{1}{2}{{\mathop{\sum }\limits_{{k,j}} }^{\prime }}\mathcal{V}\left( {{{x}_{k}} - {{x}_{j}}} \right), $$

(27.1.1)