Bracket Structure for Gyrokinetic Theory

The Lagrangian for the gyrokinetic description of magnetised plasma

dynamics in canonical form has a structure very close to the one for

the kinetic plasma under Maxwell's equations. All of the basic theorems

result, especially energy and momentum conservation. In the gyrokinetic

case, the conserved momentum is toroidal angular momentum. Any

complication due to extra finite gyroradius or finite field amplitude is

automatically included in this description because there is no change to

the underlying structure. The equation for the gyrocenters is readily

put into the form of a four-dimensional phase space Poisson bracket.

The main complication is that the coefficients are not constant, though

a model geometry can be constructed under conventional tokamak ordering

which reduces to two two-dimensional brackets with constant coefficients.

The procedure by which a phase-space test function is used to construct

Lie-Poisson functional brackets is given, and the Jacobi identity is

proved using the same index manipulation by which the four-bracket is

constructed in the first place. Progress in further exploration of the

Hamiltonian field theory structure is given.