On the Lagrangian 1-Form Structure of the Hyperbolic Calogero-Moser System

TL;DR

This paper explores the Lagrangian 1-form structure of the hyperbolic Calogero-Moser system in both discrete and continuous time, deriving closure relations from compatibility conditions and continuum limits.

Contribution

It provides a new example of the Lagrangian 1-form structure for the hyperbolic Calogero-Moser system, connecting discrete and continuous formulations via pole reduction and continuum limits.

Findings

Discrete-time hyperbolic Calogero-Moser system derived from pole-reduction of semi-discrete KP equation

Discrete-time closure relation obtained from compatibility of Lax matrices

Continuous-time hierarchy derived through successive continuum limits

Abstract

In this work, we present another example of the Lagrangian 1-form structure for the hy- perbolic Calogero-Moser system both in discrete-time level and continuous-time level. The discrete-time hyperbolic Calogero-Moser system is obtained by considering pole-reduction of the semi-discrete Kadomtsev-Petviashvili (KP) equation. The key relation called the discrete-time closure relation is directly obtained from the compatibility between the temporal Lax matrices. The continuous-time hierarchy of the hyperbolic Calogero-Moser system is obtained through two successive continuum limits. The continuous-time closure relation, which is a consequence of continuum limits on the discrete-time one, is also shown to hold.