Canonical transformations: from the coordinate based approach to the geometric one

TL;DR

This paper explores the theory of canonical transformations in classical mechanics, transitioning from coordinate-based methods to a geometric framework, highlighting the role of Hamiltonian vector fields and their flows.

Contribution

It provides a detailed comparison of geometric and coordinate approaches to canonical transformations, emphasizing the geometric interpretation and formalism.

Findings

Canonical transformations are identified with flows of Hamiltonian vector fields.

Infinitesimal generators of invariance transformations are characterized geometrically.

The connection between geometric and coordinate frameworks is illustrated with examples.

Abstract

In this paper the theory of time-dependent and time-independent canonical transformations is considered from a geometric perspective. Both the geometric formalism and the coordinate based approach are described in detail. In particular, one-parameter groups of canonical transformations are geometrically identified with flows of Hamiltonian vector fields which, in turn, are their infinitesimal generators. Likewise, infinitesimal generators of invariance transformations are geometrically characterized. The main results are established in the form of theorems and the connection between the geometric and the coordinate based frameworks is remarked using concrete examples.