On the Lagrangian structure of Calogero's Goldfish model

TL;DR

This paper explores the Lagrangian structure of Calogero's goldfish model, deriving discrete and continuous hierarchies, and connecting it to lattice KP systems, revealing its integrable structure.

Contribution

It introduces a Lagrangian hierarchy for the discrete-time Calogero's goldfish system and establishes its connection to lattice KP systems, extending understanding of its integrability.

Findings

Discrete-time Lagrangians have a 1-form structure.

Continuum limits produce a hierarchy of Lagrangians.

Connection to lattice KP systems is established.

Abstract

The discrete-time rational Calogero's goldfish system is obtained from the Ansatz Lax pair. The discrete-time Lagrangians of the system possess the discrete-time 1-form structure as those in the discrete-time Calogero-Moser system and discrete-time Ruijsenaars-Schneider system. Performing two steps of continuum limits, we obtain Lagrangian hierarchy for the system. Expectingly, the continuous-time Lagrange 1-form structure of the system holds. Furthermore, the connection to the lattice KP systems is also established.