A fibred manifold for calculus of variations in mechanics is defined as a triple \((Y,\,\pi ,\,X)\) with one-dimensional differentiable manifold X, \((m+1)\) -dimensional differentiable manifold Y and a surjective submersion \(\pi \) called the projection. Charts (coordinate systems on a fibred manifold) are introduced as well. Additional structures on the fibred manifold are sections (roughly speaking graphs of trajectories of mechanical systems). With help of so called sections equivalent up to the needed order of derivatives one can define jets of sections and the jet prolongations of a fibred manifold \((Y,\,\pi ,\,X)\) . For illustration: the first prolongation of \((Y,\,\pi ,\,X)\) can be understood as the phase space of the studied mechanical system, the second prolongation includes accelerations, etc. The first prolongation of a section represents the phase trajectory of this system. For the use of following chapters, homomorphisms and isomorphisms of fibred manifolds or a fibred manifold itself are defined and studied, as well its prolongations to jets of the underlying fibred manifold. The explanation is accompanied by illustrative examples and pictures.