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.
KeywordsCharacteristic Frequency Equilibrium Position Inverted Pendulum Small Oscillation Lagrangian System
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- 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
- 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
- 43.It is useful to think of the case n = 3, k = 2.Google Scholar
- 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
- 46.In the case a(t) = cos t, Equation (4) is called Mathieu’s equation.Google Scholar
- 47.Cf., for example, the problem analyzed below.Google Scholar