Hamiltonian theory of guiding centre motion revisited

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Abstract
The classical problem of the motion of a charged particle in a slowly varying electromagnetic field is reconsidered in the framework of ‘pseudo-canonical transformations’ in a Hamiltonian formalism. As compared with Littlejohn's important recent work, we develop a method which we believe to be more transparent. It consists, in essence, of exploiting directly the requirement that the Lie brackets of the guiding centre variables be independent of the (new) gyrophase. By using this method, we construct explicitly the complete set of functional relationships between the guiding centre variables and the particle variables, up to second order in the drift parameter є These expressions are valid for magnetic and electric fields of arbitrary geometry (within the drift approximation), which may be slowly varying in time.