Lagrangian 1-form structure of Calogero-Moser type systems

TL;DR

This paper develops a variational principle based on Lagrangian 1-forms for Calogero-Moser type systems, deriving integrability conditions and Hamiltonian structures for elliptic models like Calogero-Moser, Ruijsenaars-Schneider, and Goldfish.

Contribution

It introduces a multiform variational framework for these systems, deriving their elliptic forms and associated Hamiltonian structures without specifying potentials.

Findings

Derived elliptic forms of CM, RS, and Goldfish systems from integrability criteria.

Established a multiform variational principle ensuring multidimensional consistency.

Extended the analysis to Hamiltonian formalism with commuting flows.

Abstract

We consider the variational principle for the Lagrangian 1-form structure for long-range models of Calogero-Moser (CM) type. The multiform variational principle involves variations with respect to both the field variables as well as the independent variables corresponding to deformations of the time-curves in a multi-time space. The ensuing generalised Euler-Lagrange (gEL) equations comprise a system of multi-time EL equations, as well as constraints from so-called `alien derivatives' and `corner equations' arising from how variations on different coordinate curves match up. The closure relation, i.e. closedness of the Lagrange 1-form on solutions of the EL system, guarantees the stationarity of the action functional under deformation of the time-curves, and hence the multidimensional consistency of the corresponding gEL system. Using this as an integrability criterion on the Lagrangian level, we apply the system to some ans\"atze on the kinetic form of the Lagrangian components, associated with models of CM type without specifying the potentials. We show that from this integrability criterion the general elliptic form of the three systems, Calogero-Moser, Ruijsenaars-Schneider, and Goldfish systems, can be derived. We extend the analysis to an associated Hamiltonian formalism, via Noether's theorem and by applying Legendre transformations. Thus, the multiform variational principle leads to a system of generalised Hamilton equations describing Hamiltonian commuting flows for the mentioned elliptic models.