https://doi.org/10.1140/epjc/s10052-019-6569-y
Regular Article - Theoretical Physics
Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants
1
Service de Physique de l’Univers, Champs et Gravitation, Research Institute for Complex Systems, Université de Mons-UMONS, Place du Parc 20, 7000, Mons, Belgium
2
Forme and Fonctionnement Humain Lab, Department of Physical Therapy, CERISIC, Haute Ecole Louvain en Hainaut, 136 rue Trieu Kaisin, 6061, Montignies sur Sambre, Belgium
3
Service de Physique Nucléaire et Subnucléaire, UMONS Research Institute for Complex Systems, Université de Mons, 20 Place du Parc, 7000, Mons, Belgium
4
Faculté des Sciences de la Motricité, Université catholique de Louvain, 1 Place Pierre de Coubertin, 1348, Louvain-la-Neuve, Belgium
5
Université de Bourgogne INSERM-U1093 Cognition, Action, and Sensorimotor Plasticity, Campus Universitaire, BP 27877, 21078, Dijon, France
* e-mail: buisseretf@helha.be
Received:
12
November
2018
Accepted:
6
January
2019
Published online:
28
January
2019
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.
© The Author(s), 2019
