Theory of Small Oscillations

  • Vladimir I. Arnold
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 3)


The study of the oscillations of a system in the neighborhood of an equilibrium position or a periodic motion usual begins with linearization. The linearized system can be integrated. After this is done, the main properties of the oscillations in the original system can frequently be determined using the theory of normal forms of Poincaré-Birkhoff. This theory is an analog of perturbation theory (Ch. 5, §2). Here the linearized system plays the role of the unperturbed system with respect to the original one. In this chapter we describe the basic elements of such an approach.


Normal Form Hamiltonian System Equilibrium Position Phase Portrait Small Oscillation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Vladimir I. Arnold
    • 1
  1. 1.Steklov Mathematical InstituteMoscowUSSR

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