This Bachelor Thesis deals with the problem of variations. This is an
infinite-dimensional optimization problem, which is of great importance in
mechanics. The main results of this thesis state that a solution of the
problem of variations necessarily satisfies the Euler-Lagrange equation and
the Hamilton equation. The problem of variations is considered as a
generalization of a finite-dimensional optimization problem. A necessary
condition a solution of the problem of variations has to fulfill is the
vanishing of its so called first variation. This condition together with
the fundamental lemma of variations is used to show the Euler-Lagrange
equation. Using Legendre transformation the equivalence of the
Euler-Lagrange equation and the Hamilton equation is shown. Another part of
this thesis is dedicated to finding the solution of the catenary problem, a
classic problem of the calculus of variations. To solve the catenary
problem the Euler-Lagrange equation and the related Dubois-Reymond equation
are used.