Gyrosymmetry: Global Considerations

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Referee this paper: arXiv:1312.3975 by J. W. Burby, H. Qin

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

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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.