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Canonical transformations in three-dimensional phase space

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.