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

TL;DR

This work presents a concrete interacting Pais-Uhlenbeck oscillator with a Landau-Ginzburg-type term and shows that its fourth-order equation of motion admits a conformal bi-Hamiltonian formulation. By establishing a precise map to a generalized Henon–Heiles system, the authors import integrability tools—two compatible Hamiltonians, Lie symmetries, and separation of variables—into the higher-derivative setting, enabling explicit elliptic-function solutions. They construct a second conserved quantity, clarify the geometric origin of separability, and demonstrate bounded classical trajectories in representative parameter regimes via direct numerics that align with the integrable structure. The findings provide a transparent, fully explicit example where integrability and periodic classical solutions persist in an interacting higher-derivative model, with potential implications for quantization and broader classes of higher-derivative dynamics.

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 Hénon-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.

Figures (3)

• Figure 1: $(q,\dot{q})$ and $(\ddot{q},\dddot{q})$ phase space, for the parameters $\alpha=5$, $\beta=4$, $E=-0.5$, initial values $q(t=0)=0.5$ , $\dot{q}(t=0) = \ddot{q}(t=0)=\dddot{q}(t=0) =0$ for different values of the coupling constant $g$. The time is running from $t_0=0$ to $t=250$ in panels (a)-(f) and from $t_0=0$ to $t=60$ in panels (g) and (h).
• Figure 2: Conformal factor in equation (\ref{['confact']}) as function of $t$ for the solutions from figure \ref{['phasespace']}.
• Figure 3: Periodic solution to the fourth order equation (\ref{['equmPUI']}) for the parameters $\tilde{\alpha} = 4.1$, $\tilde{\beta} = 2.35$, $E=E_1=-0.181609$, $E_2=- 8.15687$, $a_+ = 1.35991$, $a_- = 5.64009$ and $a_3=16.6$.