# Deriving the Vlasov equation in {$\vec r, v_{||}, \mu, \varphi$} coordinates

I'm reading some lecture notes on drift kinetics and I'm having trouble with one derivation. The general idea is changing phase space coordinates from {$$\vec r, \vec v$$} to {$$\vec r, v_{||} \text{ (parallel velocity)}, \mu \text{ (magnetic moment)}, \varphi \text{ (gyrophase)}$$} and writing the Vlasov equation (2.1):

in these new coordinates. The coordinate transform is done by using the chain rule term by term to obtain these simple relations:

Now, using these relations (2.1) should get the form:

,

and this is what I'm in trouble with. In the notes it says that this operator has been used to obtain (2.9):

,

but writing this using (2.6)-(2.8) simply gives me equation (2.1) just with a lot more terms because of the coordinate transformation. If I write it for $$f_s$$ using (2.6)-(2.8), do I substitute this into (2.1) (in place of the first term) and how can I discard the the third term (dot product with $$\nabla_v f_s)$$? So, to put it simply, how do I obtain (2.9), what do I do with (2.10)? This should be a relatively simple derivation, but I just can't wrap my head around it, so any help is greatly appreciated.

Equation (2.9) is a mathematical identity -- it is always true because (in the collisionless limit) $$f$$ satisfies

$$\frac{df}{dt}=0$$

Fundamental to the calculation in Felix's notes is that

$$\dot{\varphi} = \Omega_s + \text{small corrections}$$

It is not possible to make this derivation comprehensive with any simple tricks or explanations. The key points to make, however, are:

(1) The entirety of these complex notes is to carry out an asymptotic expansion in the limit $$\Omega_s \rightarrow \infty$$.

(2) Page 5 in the notes is a detailed discussion of the justification for $$\partial f/\partial \varphi \sim 0$$, meaning the distribution function is independent of gyroangle -- except for $$\tilde{f}_{s,1}$$ in (2.37).

The reason for employing this ordering, and for the notes, is to write a drift-kinetic equation (2.45) for the gyroangle-independent part of $$f$$.