A criterion of integrability for perturbed nonresonant harmonic oscillators. “Wick ordering” of the perturbations in classical mechanics and invariance of the frequency spectrum

@article{Gallavotti1982ACO,
  title={A criterion of integrability for perturbed nonresonant harmonic oscillators. “Wick ordering” of the perturbations in classical mechanics and invariance of the frequency spectrum},
  author={G. Gallavotti},
  journal={Communications in Mathematical Physics},
  year={1982},
  volume={87},
  pages={365-383}
}
  • G. Gallavotti
  • Published 1982
  • Mathematics
  • Communications in Mathematical Physics
We introduce an analogue to the renormalization theory (of quantum fields) into classical mechanics. We also find an integrability criterion guaranteeing the convergence of Birkhoff's series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi-periodic motions. We point out its possible relevance to the transition to chaos. 
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References

SHOWING 1-10 OF 17 REFERENCES
Perturbations of geodesic flows on surface of constant negative curvature and their mixing properties
We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for theExpand
PROOF OF A THEOREM OF A.?N.?KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN
CONTENTS § 1. Introduction § 2. Formulation of the theorems § 3. Proofs § 4. Technical lemmas § 5. Appendix. The rotatory motion of a heavy asymmetric rigid bodyReferences
Critical behavior of a KAM surface: I. Empirical results
Kolmogorov-Arnol'd-Moser (KAM) surfaces are studied in the context of a perturbed two-dimensional twist map. In particular, we ask how a KAM surface can disappear as the perturbation parameter isExpand
Smooth prime integrals for quasi-integrable Hamiltonian systems
SummaryA Hamiltonian withN degrees of freedom, analytic perturbation of a canonically integrable strictly nonisochronous analytic Hamiltonian, is considered. We show the existence ofN functions onExpand
Lectures on Hamiltonian systems . Rigorous and formal stability of orbits about an oblate planet
Idealized point mass motion in axisymmetric gravitational field, discussing orbital stability about oblate planet
The spectral class of the quantum-mechanical harmonic oscillator
AbstractThe purpose of this paper is to study the so-calledspectral classQ of anharmonic oscillatorsQ=−D2+q having the same spectrum λn=2n (n≧0) as the harmonic oscillatorQ0=−D2+x2−1. ThenormingExpand
Critical behaviour of a KAM surface: I
  • Empirical Results. J. Stat. Phys. 27, 631 (1982); Shenker, S.: Scaling behaviour in a map of a circle onto itself. Empirical results. Physica D (to appear); Feigenbaum, M., Kadanoff, L., Shenker, S.: Quasi periodicity in dissipative systems. A renormalization group analysis. Preprint, Los Alamos, 19
  • 1982
Integrability of hamiltonian systems on cantor sets
private communication Communicated by A
  • Jaffe Received July
  • 1982
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