Conformal bi-Hamiltonian structure and integrability of an interacting Pais-Uhlenbeck oscillator

TL;DR

This paper demonstrates that an interacting Pais-Uhlenbeck oscillator with specific interactions exhibits integrability, conformal bi-Hamiltonian structure, and explicit solutions, linking it to a well-known integrable system.

Contribution

It establishes the integrability and geometric structure of an interacting higher-derivative oscillator, connecting it explicitly to the integrable Hénon-Heiles system.

Findings

Existence of bounded, regular trajectories.

Construction of explicit solutions using elliptic functions.

Identification of a second conserved Hamiltonian.

Abstract

We investigate an interacting Pais-Uhlenbeck oscillator with a Landau-Ginzburg type interaction term and analyse its classical dynamics from a geometric and numerical point of view. We show that the resulting fourth-order equation of motion admits a conformal bi-Hamiltonian formulation, possesses a non-trivial set of Lie symmetries and we demonstrate the existence of bounded and regular trajectories in representative parameter regimes. By establishing an explicit correspondence with an integrable generalised H\'enon-Heiles system, we show that the interacting higher-derivative dynamics inherits the integrability properties of the latter. This connection allows us to construct a second conserved Hamiltonian, to clarify the geometric origin of separability, and to obtain explicit classical solutions in terms of elliptic functions. Our results provide a concrete example of an interacting higher-derivative system for which integrability and periodic classical solutions can be established in a fully explicit manner.