Gyro-gauge-independent formulation of the guiding-center reduction to arbitrary order in the Larmor radius

Abstract

The guiding-center reduction is studied using gyro-gauge-independent coordinates. The Lagrangian 1-form of charged particle dynamics is Lie transformed without introducing a gyro-gauge, but using directly the unit vector of the component of the velocity perpendicular to the magnetic field as the coordinate corresponding to Larmor gyration.The reduction is shown to provide a maximal reduction for the Lagrangian and to work to all orders in the Larmor radius, following exactly the same procedure as when working with the standard gauge-dependent coordinate.The gauge-dependence is removed from the coordinate system by using a constrained variable for the gyro-angle. The closed 1-form dθ is replaced by a more general non-closed 1-form, which is equal to dθ in the gauge-dependent case. The gauge vector is replaced by a more general connection in the definition of the gradient, which behaves as a covariant derivative, in perfect agreement with the circle-bundle picture. This explains some results of previous works, whose gauge-independent expressions did not correspond to a gauge fixing but indeed correspond to a connection fixing.In addition, some general results are obtained for the guiding-center reduction. The expansion is polynomial in the cotangent of the pitch-angle as an effect of the structure of the Lagrangian, preserved by Lie derivatives. The induction for the reduction is shown to rely on the inversion of a matrix which is the same for all orders higher than three. It is inverted and explicit induction relations are obtained to go to arbitrary order in the perturbation expansion. The Hamiltonian and symplectic representations of the guiding-center reduction are recovered, but conditions for the symplectic representation at each order are emphasized.

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

“…A first illustration is provided by the guiding-center transformation. For the gyro-angle c, the transformation is connection-independent 18,19 . For the coordinate θ, the transformationθ = ··e G 2 e G 1 θ is gauge dependent, but in such a way as to make the induced transformationc = ··e G 2 e G 1 c for c gauge independent, with G n the vector field generating the n-th order transformation.…”

“…The first of them considers the properties of the basic 1-forms (19) of the theory. The previous investigations showed that the basic differential form for the gyro-angle is δΘ, and that it is not closed.…”

“…It was shown in Ref. [19] that this intrinsic coordinate can be used also in the standard procedure for the guiding-center reduction, which provides a Hamiltonian structure for the guidingcenter dynamics and a maximal reduction for the guiding-center Lagrangian. All the results of the literature can thus be obtained using this physical but constrained coordinate.…”

“…Indeed, the guiding-center derivations in Refs. [17][18][19] showed that, in the gauge-independent approach, all the features of the standard approach seemed to be present, including a kind of generalized gauge vector. This makes it necessary to clarify whether the gauge-independent coordinate actually resolves the questions or only transfers them into other questions.…”

“…shows thatφ is also singular in p sin ϕ = 0. At higher orders, this singularity is expected to affect all the coordinates, because it is related to the order in the cotangent of the pitch-angle compared to the order in the Larmor radius, as is emphasized in Refs [17][18][19]…”