The Lagrangian structure of Calogero's goldfish model

From a Lax pair ansatz, we obtain the discrete-time rational Calogero goldfish system. The discrete-time Lagrangians of the system have a discrete-time $1$-form structure similar to the Lagrangians in the discrete-time Calogero–Moser system and the discrete-time Ruijsenaars–Schneider system. We obtain the Lagrangian hierarchy for the system as a result of a two-step passage to the continuum limit. As expected, the continuous-time Lagrangian preserves the $1$-form structure. We establish a connection with the Kadomtsev–Petviashvili lattice systems.