Final form of Vlasov equation in guiding-center coordinates

Using the above results, the Vlasov equation in guiding-center coordinates is written

∂fg [∂fg q ∂fg] ∂t-+ v ⋅ ∂X-+ [λB1 + λB2]fg − m-E0-∂𝜀 ( ) + q(E + v ×B )⋅ I×-e∥⋅ ∂fg+ v ∂fg+ v⊥-∂fg + eα-∂fg m Ω ∂X ∂𝜀 B0 ∂μ v⊥ ∂α = 0 (32)

Using tensor identity a ⋅ I × b = a × b, equation (32) is written as

[ ] ∂fg + v⋅ ∂fg+ [λB1 + λB2 ]fg −-qE0 ∂fg ∂t ∂X m ∂𝜀 ( ) -q (e∥) ∂fg q- eα-∂fg + m (E + v× B )× Ω ⋅∂X + m (v × B) ⋅ v⊥ ∂α q ( ∂fg v ⊥∂fg eα ∂fg) + m-E ⋅ v ∂𝜀-+ -B--∂μ + v--∂α- 0 ⊥ = 0, (33)

This is the Vlasov equation in guiding-center coordinates.