Canonical transformations in three-dimensional phase space

TL;DR

This paper explores canonical transformations in three-dimensional phase space using Nambu brackets, defining them through canonoid transformations, and discusses their generating functions, infinitesimal forms, and decomposition methods.

Contribution

It extends the concept of canonical transformations to three-dimensional phase space with Nambu brackets, providing a comprehensive framework including generating functions and decomposition techniques.

Findings

Canonical transformations can be characterized via Pfaffian differential equations.

Multiple types of generating functions are identified and classified.

Decomposition of transformations is possible similarly to two-dimensional cases.

Abstract

Canonical transformation in a three-dimensional phase space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed as based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them is listed. Infinitesimal canonical transformations are also discussed. Finally, we show that decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.