• V. I. Arnold
Part of the Graduate Texts in Mathematics book series (GTM, volume 60)


Because linear equations are easy to solve and study, the theory of linear oscillations is the most highly developed area of mechanics. In many nonlinear problems, linearization produces a satisfactory approximate solution. Even when this is not the case, the study of the linear part of a problem is often a first step, to be followed by the study of the relation between motions in a nonlinear system and in its linear model.


Characteristic Frequency Equilibrium Position Inverted Pendulum Small Oscillation Lagrangian System 
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  1. 41.
    If the equilibrium position is unstable, we will talk about “unstable small oscillations” even though these motions may not have an oscillatory character.Google Scholar
  2. 42.
    If one wants to, one can introduce a euclidean structure by taking the first form as the scalar product, and then reducing the second form to the principal axes by a transformation which is orthogonal with respect to this euclidean structure.Google Scholar
  3. 43.
    It is useful to think of the case n = 3, k = 2.Google Scholar
  4. 45.
    The distance between two linear systems with periodic coefficients, (math), is defined as the maximum over t of the distance between the operators B 1(t) and B 2(t).Google Scholar
  5. 46.
    In the case a(t) = cos t, Equation (4) is called Mathieus equation.Google Scholar
  6. 47.
    Cf., for example, the problem analyzed below.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • V. I. Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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