Summary: These notes are a brief introductory course to second order differential equations on manifolds and to some problems regarding forced oscillations of motion equations of constrained mechanical systems. The intention is to give a comprehensive exposition to the mathematicians, mainly analysts, that are not particularly familiar with the formalism of differential geometry. The material is divided into five sections. The background needed to understand the subject matter contained in the first three sections is mainly advanced calculus and linear algebra. The fourth and the fifth sections require some knowledge of degree theory and functional analysis.
We begin with a review of some of the most significant results in advanced calculus, such as the inverse function theorem and the implicit function theorem, and we proceed with the notions of smooth map and diffeomorphism between arbitrary subsets of Euclidean spaces. The second section is entirely devoted to differentiable manifolds embedded in Euclidean spaces and tangent bundles. In the third section, dedicated to differential equations on manifolds, a special attempt has been made to introduce the notion of second order differential equation in a very natural way, with a formalism familiar to any analyst. Section four concerns the concept of degree of a tangent vector field on a manifold and the Euler-Poincaré characteristic. Finally, in the last section, we deal with forced oscillations for constrained mechanical systems and bifurcation problems. Some recent results and open problems are presented.
For the entire collection see [Zbl 0829.00024].