One-parameter discrete-time Calogero–Moser system

We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero–Moser system. At the discrete level, the Lax pairs with a parameter are introduced and the discrete-time equations of motion are obtained as together with the corresponding discrete-time Lagrangian. The integrability property of this new system can be expressed in terms of the discrete Lagrangian closure relation by using a connection with the temporal Lax matrices of the discrete-time Ruijsenaars–Schneider system, an exact solution, and the existence of a classical \(r\) -matrix. As the parameter tends to zero, the standard Calogero–Moser system is recovered in both discrete-time and continuous-time forms.