Multiplicative dynamical systems in terms of the induced dynamics

We realize an example of induced dynamics using new multiplicative determinant relations whose roots give the particle positions. We present both a general scheme for describing completely integrable dynamical systems parameterized by an arbitrary $N\times N$ matrix of momenta and an explicit model that interpolates between the Calogero–Moser and Ruijsenaars–Schneider hyperbolic systems. We consider some special cases of this model in detail.