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(summary thanks to JoshBurby, author)

This paper is concerned with addressing a mathematical problem that arises when studying the dynamics of charged particles in a strong magnetic field.

Under the assumption of a strong, slowly-varying magnetic field, the Lorentz force vector field on the position-velocity phase space, \(X=\frac{q}{m}v\times B\cdot\frac{\partial}{\partial v }+v\cdot\frac{\partial}{\partial x}\)is nearly invariant under the \(S^1\)action \(\Phi_\theta(x,v)=(x,v_\theta)\), where \(v_\theta\)is the velocity vector rotated about the magnetic field vector \(B(x)\) by \(\theta\)radians. This approximate symmetry can be exploited to greatly simplify the dynamical description of strongly-magnetized charged particle dynamics; see, e.g. this paper on guiding center theory.

This circle action endows the position-velocity space with the structure of a principal circle bundle. The submitted paper determines when this circle bundle admits a global section. It also addresses the question of whether or not the usual guiding center theory (as outlined in the earlier link) is valid when a global section does not exist.