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

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", $\tilde{T}(\vec{p},\vec{q})$ 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": $\tilde{T}(\vec{p},\vec{q})=T(\vec{q},\vec{p})$. 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.

LaTeX, 16 pages (acknowledgements improved)

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• Duality in elliptic Ruijsenaars system and elliptic symmetric functions
• On complete solution of the quantum Dell system
• Challenges of Matrix Models
• On the status of DELL systems
• Characteristic determinant and Manakov triple for the double elliptic integrable system
• pq-Duality: a set of simple examples
• On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta