  The aim of this paper is twofold: First, we give a formal introduction to the
basics of the mathematical framework of classical mechanics. Along the way, we
prove a Hamiltonian and a Lagrangian version of Noether's Theorem, an important
result concerning continuous symmetries of physical systems. At the end, we
prove a new statement about orbit cylinders on homotopies of stable regular
energy hypersurfaces. The main question we answer is what is the dynamical
meaning of stability? The second aim is to provide a didactical framework for
introducing advanced mathematics, which can also be of used in other topics. A
broad range of established methods belonging to the realm of cognitive
activation as well as formative assessment techniques are used.
