p,q-Duality and Hamiltonian Flows in the Space of Integrable Systems or Integrable Systems as Canonical Transforms of the Free Ones

TL;DR

This paper explores the duality and Hamiltonian flows in the space of integrable systems, showing how coupling constant variations correspond to canonical transformations and how dual systems relate through momentum-coordinate interchange.

Contribution

It introduces a framework connecting integrable systems' coupling constant variations to Hamiltonian flows and duality transformations, with explicit formulas for specific models.

Findings

Coupling constant variations act as canonical transformations.

Dual integrable systems are related by momentum-coordinate interchange.

Explicit formulas provided for harmonic oscillator and Calogero-Ruijsenaars-Dell systems.

Abstract

Variation of coupling constants of integrable system can be considered as canonical transformation or, infinitesimally, a Hamiltonian flow in the space of such systems. Any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system: this gives an idea of how many integrable systems there are in the space of coupling constants. Inverse flow is generated by a dual "Hamiltonian", $\widetilde T(\vec p, \vec q)$ associated with the dual integrable system. In vicinity of a self-dual point the duality transformation just interchanges momenta and coordinates in such a "Hamiltonian": $\widetilde T(\vec p, \vec q) = T(\vec q, \vec p)$. For integrable system with several coupling constants the corresponding "Hamiltonians" $T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: [ d/dg_a - T_a(p,q), d/dg_b - T_b(p,q) ] = 0. Some explicit formulas are given for harmonic oscillator and for Calogero-Ruijsenaars-Dell system.