Lectures on Classical Mechanics -- A Didactical Approach to Higher Mathematics

TL;DR

This paper provides a formal introduction to classical mechanics, proves key theorems like Noether's, and offers a didactical framework for teaching advanced mathematics through cognitive and assessment techniques.

Contribution

It introduces a didactical approach to higher mathematics in classical mechanics, including new insights into orbit cylinders and stability.

Findings

Proved Hamiltonian and Lagrangian versions of Noether's Theorem.

Presented a new statement on orbit cylinders and stability.

Developed a didactical framework for teaching advanced mathematics.

Abstract

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.