Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants

• Published in: The European physical journal. C, Particles and fields Vol. 79; no. 1; pp. 1 - 8
• Main Authors: Boulanger, Nicolas, Buisseret, Fabien, Dierick, Frédéric, White, Olivier
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2019; Springer; Springer Nature B.V; Springer Verlag (Germany); SpringerOpen
• Subjects: Adiabatic flow; Astronomy; Astrophysics and Cosmology; Classical mechanics; Dynamic stability; Elementary Particles; General Physics; Hadrons; Harmonic oscillators; Heavy Ions; High Energy Physics - Theory; Invariants; Measurement Science and Instrumentation; Nuclear Energy; Nuclear Physics; Oscillators; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Regular Article - Theoretical Physics; String Theory; Time dependence; oscillator: harmonic; perturbation: time dependence; derivative: high; Hamiltonian; mechanics: classical; stability
• ISSN: 1434-6044; 1434-6052
• DOI: 10.1140/epjc/s10052-019-6569-y

The status of classical stability in higher-derivative systems is still subject to discussions. In this note, we argue that, contrary to general belief, many higher-derivative systems are classically stable. The main tool to see this property are Nekhoroshev’s estimates relying on the action-angle formulation of classical mechanics. The latter formulation can be reached provided the Hamiltonian is separable, which is the case for higher-derivative harmonic oscillators. The Pais–Uhlenbeck oscillators appear to be the only type of higher-derivative harmonic oscillator with stable classical dynamics. A wide class of interaction potentials can even be added that preserve classical stability. Adiabatic invariants are built in the case of a Pais–Uhlenbeck oscillator slowly changing in time; it is shown indeed that the dynamical stability is not jeopardised by the time-dependent perturbation.