Canonical (and non-canonical) Transformations: A Differential Approach

TL;DR

This paper introduces a differential method for calculating canonical transformations in Hamiltonian mechanics, simplifying the process and clarifying the underlying structure compared to traditional generating function approaches.

Contribution

A new algebraic differential approach for canonical transformations that directly handles variables and clarifies the Hamiltonian least-action principle.

Findings

Simplifies the calculation of canonical transformations.

Provides correct equations of motion for non-canonical variables.

Offers a more systematic and algebraic method than traditional generating functions.

Abstract

The traditional method of teaching canonical transformations involves the introduction of generating functions of various types. This method obscures the underlying structure of the Hamiltonian least-action principle, and can make a straightforward concept seem arcane. In this article, I present a method for calculating canonical changes of variable in Hamiltonian mechanics using a differential approach which is much more straightforward. This method handles canonical variables directly, but also returns the correct equations of motion for non-canonical variables. It is also much more algebraic than generating functions, making it easier to present in a systematic manner.