Functional Analysis in Mechanics

Long ago it was traditional to apply mathematics only to mechanics and physics. Now it is almost impossible to find an area of knowledge in which mathematics is not used as a tool to create new models and to simulate them. This is due mainly to the fantastic ability of computers to process models having thousands of parameters.

Consider a set of particles P _i , i = 1,..., n. To locate these particles in the space E³, we need a reference system. Let the Cartesian coordinates of Pi be (ξ _i , η _i , ζ _i ) for each i. Identifying (ξ _i , η _i , ζ _i ) with (x ₁, x ₂, x ₃), (ξ _i , η _i , ζ _i ) with (x ₄, x ₅, x ₆), and so on, we obtain a vector x of the Euclidean space ℝ³ⁿ with coordinates (x ₁, x ₂,..., x _3n ). This vector determines the positions of all particles in the set.

This chapter aims to present in more detail some results of the theory of linear operators. We cannot pretend to give a full treatment of this vast area, and shall select only those parts which are useful in the applications under consideration. Of course, we are forced to give some general theoretical background.

From the viewpoint of functional analysis, nonlinear problems of mechanics are more complicated than linear problems; as in mechanics, they require new techniques for their study. Many of them, such as nonlinear elasticity in the general case, provide a wide field of investigation for mathematicians (see Antman [2]) ; the problem of existence of solutions in nonlinear elasticity in general is still open.