Abstract
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.
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If the equilibrium position is unstable, we will talk about “unstable small oscillations” even though these motions may not have an oscillatory character.
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.
It is useful to think of the case n = 3, k = 2.
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).
In the case a(t) = cos t, Equation (4) is called Mathieu’s equation.
Cf., for example, the problem analyzed below.
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© 1978 Springer Science+Business Media New York
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Arnold, V.I. (1978). Oscillations. In: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1693-1_5
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DOI: https://doi.org/10.1007/978-1-4757-1693-1_5
Publisher Name: Springer, New York, NY
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