Abstract
One may wonder why we introduce this chapter, since second-order systems of differential equations are reducible to the earlier discussed first-order systems of differential equations. The reason has at least two main sources. First of all, they appear in a natural traditional way beginning with the works of D’Alembert, Fermat, Maupertuis, Jean Bernoulli, Hamilton and Lagrange in that period, when mathematics and mechanics have inspired each other very strongly. The second reason is that the second-order differential equations are obtained from Newton’s second law or from Lagrange’s equations and they have a direct physical meaning. In addition, there exist some direct methods to deal with the second-order differential equations without their reduction to a set of first-order equations [59, 130, 160, 242].
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