On the derivation of Darboux form for the action- angle dual of trigonometric BCn Sutherland system

Tamás F Görbe

doi:10.1088/1742-6596/563/1/012012

Abstract

Recently Fehér and the author have constructed the action-angle dual of the trigonometric BCn Sutherland system via Hamiltonian reduction. In this paper a reduction- based calculation is carried out to verify the canonical Poisson bracket relations on the phase space of this dual model. Hence the material serves complementary purposes whilst it can also be regarded as a suitable modification of the hyperbolic case previously sorted out by Pusztai.

Highlights

The integrable one-dimensional many-body systems of Calogero, Moser, and Sutherland and generalized versions of them have proven to be a fruitful source of both diverse physical applications and connections between seemingly distant areas of mathematics
With Darboux coordinates q, p and λ, θ, respectively, are said to be duals of each other if there is a global symplectomorphism R : M → Mof the phase spaces, which exchanges the canonical coordinates with the action-angle variables for the Hamiltonians
This means that H ◦ R−1 depends only on λ, while H ◦ R only on q

Summary

Introduction

The integrable one-dimensional many-body systems of Calogero, Moser, and Sutherland and generalized versions of them have proven to be a fruitful source of both diverse physical applications and connections between seemingly distant areas of mathematics. Two Liouville integrable many-body Hamiltonian systems (M, ω, H) and (M , ω, H ). With Darboux coordinates q, p and λ, θ, respectively, are said to be duals of each other if there is a global symplectomorphism R : M → Mof the phase spaces, which exchanges the canonical coordinates with the action-angle variables for the Hamiltonians. This means that H ◦ R−1 depends only on λ, while H ◦ R only on q. Q are the particle positions for H and action variables for H , and λ are the positions of particles modelled by the Hamiltonian Hand action variables for H

Objectives

Conclusion