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Center for Nonlinear Studies |
The long-time confinement of charged particles by nonuniform magnetic fields is based on the existence of a hierarchy of adiabatic invariants that begins with the guiding-center (gc) magnetic moment μgc (or the gyro-action Jgc = μgc B/Ω). The guiding-center phase-space transformation decouples the guiding-center gyro-angle ζgc (canonically-conjugate to Jgc) from the reduced guiding-enter Hamiltonian dynamics that involves the guiding-center position Xgc and the guiding-center parallel kinetic momentum p||gc, where the magnetic moment μgc appear as an invariant. This lecture summarizes the material presented in the review paper by Cary and Brizard [Reviews of Modern Physics 81, 693 (2009)]. The guiding-center polarization and magnetization are also calculated directly from the guiding-center transformation.