Discrete-time Calogero-Moser system and Lagrangian 1-form structure

TL;DR

This paper explores the Lagrangian 1-form structure of the discrete and continuous Calogero-Moser system, demonstrating integrability, compatibility conditions, and a novel least-action principle through KP equation connections.

Contribution

It establishes the Lagrangian 1-form framework for the Calogero-Moser system in both discrete and continuous settings, linking it to KP solutions and integrability.

Findings

Discrete-time CM system arises from pole reduction of semi-discrete KP

Lax matrix compatibility leads to Lagrangian closure relation

Continuum limits recover Lagrange 1-form structure for CM model

Abstract

We study the Lagrange formalism of the (rational) Calogero-Moser (CM) system, both in discrete time as well as in continuous time, as a first example of a Lagrange 1-form structure in the sense of the recent paper [19]. The discrete-time model of the CM system was established some time ago arising as a pole-reduction of a semi-discrete version of the KP equation, and was shown to lead to an exactly integrable correspondence (multivalued map). In this paper we present the full KP solution based on the commutativity of the discrete-time flows in the two discrete KP variables. The compatibility of the corresponding Lax matrices is shown to lead directly to the relevant closure relation on the level of the Lagrangians. Performing successive continuum limits on both the level of the KP equation as well as of the CM system, we establish the proper Lagrange 1-form structure for the continuum case of the CM model. We use the example of the three-particle case to elucidate the implementation of the novel least-action principle, which was presented in [19], for the simpler case of Lagrange 1-forms.