Solution to Solving More Difficult Partial Differential Equations (P. 4): The Vlasov Equation

The Problem again:

Show that:

Ñ x · v = 0   and:  Ñ v · (E  + v x B )  = 0

And write the resulting partial differential equation.

 Solutions:

Ñ x · v = 0 because x and v are independent phase variables.

 Ñ v · (E  + v x B )  = 0   because:

a)  E is a function of x only, independent of v so: Ñ v · E = 0

b) Ñ v · ( v x B )  = 

¶ (v y B _z- v z B _y) / ¶ v x  + ¶ (v z B_x- v x B _z) / ¶ v y  +

¶ (v x B_y - v y B_x) / ¶ v z    = 0 + 0 + 0 =  0

The resulting Vlasov partial DE can be written:

¶ f/ ¶ t + v  · Ñ x f +  q/m (E  + v x B ) · Ñ v f = 0

Or:

[¶ f/ ¶ t + v  · Ñ x  · q/m (E  + v x B ) · Ñ v ] f = 0