Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants
Abstract 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 actio...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-01-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | http://link.springer.com/article/10.1140/epjc/s10052-019-6569-y |
| _version_ | 1831765849587843072 |
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| author | Nicolas Boulanger Fabien Buisseret Frédéric Dierick Olivier White |
| author_facet | Nicolas Boulanger Fabien Buisseret Frédéric Dierick Olivier White |
| author_sort | Nicolas Boulanger |
| collection | DOAJ |
| description | Abstract 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. |
| first_indexed | 2024-12-22T06:11:20Z |
| format | Article |
| id | doaj.art-82a84888e66545b7b92122bacb86eebd |
| institution | Directory Open Access Journal |
| issn | 1434-6044 1434-6052 |
| language | English |
| last_indexed | 2024-12-22T06:11:20Z |
| publishDate | 2019-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | European Physical Journal C: Particles and Fields |
| spelling | doaj.art-82a84888e66545b7b92122bacb86eebd2022-12-21T18:36:14ZengSpringerOpenEuropean Physical Journal C: Particles and Fields1434-60441434-60522019-01-017911810.1140/epjc/s10052-019-6569-yHigher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariantsNicolas Boulanger0Fabien Buisseret1Frédéric Dierick2Olivier White3Service de Physique de l’Univers, Champs et Gravitation, Research Institute for Complex Systems, Université de Mons-UMONSForme and Fonctionnement Humain Lab, Department of Physical Therapy, CERISIC, Haute Ecole Louvain en HainautForme and Fonctionnement Humain Lab, Department of Physical Therapy, CERISIC, Haute Ecole Louvain en HainautUniversité de Bourgogne INSERM-U1093 Cognition, Action, and Sensorimotor PlasticityAbstract 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.http://link.springer.com/article/10.1140/epjc/s10052-019-6569-y |
| spellingShingle | Nicolas Boulanger Fabien Buisseret Frédéric Dierick Olivier White Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants European Physical Journal C: Particles and Fields |
| title | Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants |
| title_full | Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants |
| title_fullStr | Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants |
| title_full_unstemmed | Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants |
| title_short | Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants |
| title_sort | higher derivative harmonic oscillators stability of classical dynamics and adiabatic invariants |
| url | http://link.springer.com/article/10.1140/epjc/s10052-019-6569-y |
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