On an intrinsic approach of the guiding-center anholonomy and gyro-gauge-arbitrariness

In guiding center theory, the standard gyro-angle coordinate is associated with gyro-gauge dependence, the global existence problem for unit vectors perpendicular to the magnetic field, and the notion of anholonomy, which is the failure of the gyro-angle to return to its original value after being transported around a loop in configuration space. We analyse these three intriguing topics through the lens of a recently proposed, global, gauge-independent gyro-angle. This coordinate is constrained, and therefore necessitates the use of a covariant derivative. It also highlights the intrinsic meaning and physical content of gyro-gauge freedom and anholonomy. There are, in fact, many possible covariant derivatives compatible with the intrinsic gyro-angle, and each possibility corresponds to a different notion of gyro-angle transport. This observation sheds new light on Littlejohn's notion of gyro-angle transport and suggests a new derivation of the recently-discovered global existence condition for unit vectors perpendicular to the magnetic field. We also discuss the relationship between Cartesian position-momentum coordinates and the intrinsic gyro-angle.