On the superintegrability of the rational Ruijsenaars-Schneider model

The rational and hyperbolic Ruijsenaars-Schneider models and their non-relativistic limits are maximally superintegrable since they admit action variables with globally well-defined canonical conjugates. In the case of the rational Ruijsenaars-Schneider model we present an alternative proof of the superintegrability by explicitly exhibiting extra conserved quantities relying on a generalization of the construction of Wojciechowski for the rational Calogero model.

💡 Research Summary

The paper investigates the superintegrability of the rational Ruijsenaars‑Schneider (RS) model, a relativistic many‑body system that reduces to the Calogero‑Moser model in the non‑relativistic limit. While it is known that both the rational and hyperbolic RS models are maximally superintegrable because they admit globally defined action variables together with canonical conjugates, the authors provide a completely explicit construction of the additional integrals of motion for the rational case.

The authors begin by recalling the Lax representation of the rational RS model. The Lax matrix (L) is given by
(L_{ij}=p_i\delta_{ij}+i\frac{g}{x_i-x_j}(1-\delta_{ij})),
where (p_i) and (x_i) are the momenta and positions of the (N) particles and (g) is the coupling constant. A companion matrix (M) is chosen so that the Lax equation (\dot L=