Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems

Abstract

We introduce a noncanonical ("new-time") transformation which exchanges the roles of a coupling constant and the energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable systems and, for example, explain the observed duality between the H\'enon-Heiles and Holt models. It is shown that the transformation can sometimes connect weak- and full-Painlev\'e Hamiltonians. We also discuss quantum integrability and find the origin of the deformation $\ensuremath{-}\frac{5}{72}{\ensuremath{\hbar}}^{2}{x}^{\ensuremath{-}2}$.