Coordinate Transformations and Invariance Properties

Abstract

This chapter completes the illustration of the basic principles of Analytical Mechanics started in the previous one. The purpose of the two chapters is to introduce a number of concepts that are not only useful per se, but also constitute a basis for the concepts of Quantum Mechanics that are introduced in later chapters. The first subject is that of the canonical transformations, followed by the definition and properties of the Hamilton characteristic function and of the phase velocity. Then, the invariance properties that derive from the symmetries of the Lagrangian function are discussed. The chapter continues with a short description of the Maupertuis principle and of the expression of the angular momentum in spherical coordinates. The last paragraphs deal with the linear motion and the action-angle variables.