Impedance and parametric excitation of oscillators

B.Ya. Zel’dovich^a, b
^aUniversity Academic Division of Nonlinear Optics, Institute of Electro-Physics of the Ural Division of the Russian Academy of Sciences, pr. Lenina 76, Chelyabinsk, 454010, Russian Federation
^bCollege of Optics and Photonics/CREOL, University of Central Florida, 4000 Central Florida Blvd., Orlando, Florida, 32816-2700, USA

This article deals with the linear ordinary differential equations for one or several coupled oscillators. Emphasis is placed on the separate notions of the frequency matrix (as a kinematic entity) and the impedance matrix. The latter matrix is explicitly introduced in this article, and its time dependence is shown to be responsible for parametric excitation and for nonconservation of adiabatic invariants.

PACS: 05.45.Xt, 45.05.+x, 84.30.−r (all)
DOI: 10.1070/PU2008v051n05ABEH006498
URL: https://ufn.ru/en/articles/2008/5/d/
Web of Science

000259376200004
Scopus

2-s2.0-51549117543
ADS NASA

2008PhyU...51..465Z
Citation: Zel’dovich B Ya "Impedance and parametric excitation of oscillators" Phys. Usp. 51 465–484 (2008)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Зельдович Б Я «Импеданс и параметрическое возбуждение осцилляторов» УФН 178 489–510 (2008); DOI: 10.3367/UFNr.0178.200805d.0489

References (13) ↓ Cited by (7) Similar articles (12)

Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki 3-e izd. (M.: Nauka, 1989); Translated into English, Arnold V I Mathematical Methods Of Classical Mechanics 2nd ed. (Berlin: Springer-Verlag, 1989)
Arnol’d V I Obyknovennye Differentsial’nye Uravneniya 3-e izd. (M.: Nauka, 1984) p. 196; Translated into English, Arnold V I Ordinary Differential Equations (Berlin: Springer-Verlag, 1992)
Landau L D, Lifshits E M Mekhanika (M.: Nauka, 1982); Translated into English, Landau L D, Lifshitz E M Mechanics (Oxford: Pergamon Press, 1976)
Goldstein H Classical Mechanics (Cambridge, Mass.: Addison-Wesley, 1950); Goldstein G Klassicheskaya Mekhanika (M.: Nauka, 1975)
Sussman G J, Wisdom J Structure And Interpretation Of Classical Mechanics (Cambridge, Mass.: MIT Press, 2001)
José J V, Saletan E J Classical Dynamics : A Contemporary Approach (Cambridge: Cambridge Univ. Press, 1998)
Berndt R Einführung In Die Symplektische Geometrie (Braunschweig: Vieweg Verlag, 1998); Translated into English, Berndt R An Introduction To Symplectic Geometry (Providence, RI: American Mathematical Society, 2001) p. 74-75
Paradoksov P Usp. Fiz. Nauk 89 707 (1966); Paradoksov P Sov. Phys. Usp. 9 618 (1967)
Berestetskii V B, Lifshits E M, Pitaevskii L P Kvantovaya Elektrodinamika (M.: Nauka, 1980); Translated into English, Berestetskii V B, Lifshitz E M, Pitaevskii L P Quantum Electrodynamics (Oxford: Pergamon Press, 1982)
Glauber R J "Optical coherence and photon statistics" Optique Et électronique Quantiques. Quantum Optics And Electronics. Lectures, Hes Houches, 1964 Session Of The Summer School Of Theoretical Physics, Univ. Of Grenoble (Eds C DeWitt, A Blandin, C Cohen-Tannoudji) (New York: Gordon and Breach, 1965) p. 63; Glauber R Kvantovaya Optika i Kvantovaya Radiofizika. Lektsii v Letnei Shkole Teoreticheskoi Fiziki Grenobl’skogo Universiteta, Lezush, Frantsiya (M.: Mir, 1966)
Landau L D, Lifshits E M Statisticheskaya Fizika (M.: Nauka, 1976); Translated into English, Landau L D, Lifshitz E M Statistical Physics (Oxford: Pergamon Press, 1980)
Landau L D, Lifshits E M Teoriya Polya (M.: Nauka, 1988); Translated into English, Landau L D, Lifshitz E M The Classical Theory Of Fields (Oxford: Pergamon Press, 1983)
Gantmakher F R Teoriya Matrits (M.: Nauka, 1988); Translated into English, Gantmakher F R Applications Of The Theory Of Matrices (Mineola, NY: Dover Publ., 2005)