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 Henon-Heiles and Holt models. It is shown that the transformation can sometimes connect weak-and full-Painleve Hamiltonians. We also discuss quantum integrability and find the origin of the deformation -^-# 2 x~2.PACS numbers: 03.20. + i, 02.30. +gThe search for (and the discovery of) integrable dynamical systems is a most fascinating branch of nonlinear physics, one which has been the center of intensive activity in the past decade. 1 Integrable systems are quite rare and still only a few examples are known. In this paper we will present a novel transformation that relates integrable Hamiltonian systems.Several methods have been devised for the investigation of integrability. One method that has met with particular success in the last few years is singularity analysis, which associates integrability with the Painleve property, i.e., a movable polelike singularity (t-t 0 )~" in the solution of the equations of motion. It was used a century ago by Kowalevskaya, 2 who identified with it the last integrable configuration of the heavy top. The method was resurrected by Ablowitz, Ramani, and Segur 3 and by now several works which have combined Painleve analysis with explicit construction of constants of motion have yielded new integrable systems (see, e.

Superintegrability in a two-dimensional space of nonconstant curvature

“…In particular we note that each of the three superintegrable systems we have examined are such that when we write out the classical equation H = E and factor out the denominator we recover a variant of a superintegrable system corresponding to flat space [4]. This is an example of what is called coupling-constant metamorphosis [20]. It has been proven in [12] that all of the superintegrable systems in the plane are such that the bound states energies can be calculated algebraically.…”

Hidden symmetries of dynamics in classical and quantum physics

“…We know specific examples but there is not yet a complete theory. Even deeper relations among Hamiltonian systems, such as for example the coupling constant metamorphosis discussed in (Hietarinta et al, 1984) and following works, are not related to canonical transformations and represent an open window on a more complete understanding.…”

Quantum superintegrability and exact solvability in n dimensions

“…We see here a phenomenon which has been called "metamorphosis" 34 or "migration" 22 of the coupling constant. In equation (33) the energy E plays the role of the frequency of a harmonic oscillator whereas the Coulomb coupling constant γ plays the role of an eigenvalue of Q 0 .…”