# Integrable many-body systems of Calogero-Ruijsenaars type

## TL;DR

This thesis advances the understanding of Calogero-Ruijsenaars integrable systems by deriving explicit formulas, exploring dualities, and constructing new deformations and Lax pairs for various models within this class.

## Contribution

It provides explicit spectral coordinates, explores dualities, and constructs deformations and Lax pairs for multiple Calogero-Ruijsenaars type systems, enhancing their mathematical understanding.

## Key findings

- Explicit spectral coordinates for rational Calogero-Moser system.
- Action-angle duality for BC(n) Sutherland system established.
- Poisson-Lie deformation of BC(n) Sutherland system derived.

## Abstract

This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle duality for the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 3. We derive a Poisson-Lie deformation of the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 4. We construct a Lax pair for the hyperbolic Ruijsenaars-Schneider system with two couplings. 5. We present an explicit construction of compactified trigonometric and elliptic Ruijsenaars-Schneider systems.

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Source: https://tomesphere.com/paper/1705.01333