Summary of the phase-space coordinate transform

The coordinate transform from particle coordinates (x,v) to guiding-center coordinates (X,𝜀,μ,α,σ) is given by

( e∥(x) |||| X(x,v) = x+2 v × Ω(x) ||{ 𝜀(x,v ) = |v|2-+ qΦ0m(x) μ(x,v) = |v−v⋅e∥(x)|2 |||| 2B0(x) ||( α(x,v) = (angle between v⊥ and locale⊥) σ(x,v) = sign(v ⋅e∥(x))

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As mentioned above, the dependence of the distribution function on σ will not be explicitly indicated in the following.