How can I let the length of a vector in VectorPlot be represented by it´s norm?

Question

I would like to draw a vector field and let the magnitude of the vectors change over time (t) using the Manipulate function. The direction of the vectors will be static.

Manipulate[
 VectorPlot[{Cos[t]*(-y), Cos[t]*x}, {x, -1, 1}, {y, -1, 1}]
 , {t, 0, 2 Pi}]

You can see that Cos[t] is the factor i use to scale the vectors. By default the plotted vectors all have the same length and a color function is used to indicate magnitude. However, i would like to have the magnitude represented by the length of the vectors, not by color. If i use VectorScaling, the vectors are scaled independently for every frame so if i play the animation there will be no change visible until the orientation switches. I would like to see the length of the vectors change over time. Is there an easy way to do this?

Any help will be greatly appreciated.

Greetings,

Fabian.

Answer 1

For version V12.1. I couldn't figure out a way to get VectorScaling to do anything except the limited options listed in the docs. (The old VectorScale seemed simultaneously more versatile and inscrutable.)

interp[p_, q_, s_] := ( p (1 - s) + q (1 + s))/2;
Manipulate[
 VectorPlot[{Cos[t]*(-y), Cos[t]*x}, {x, -1, 1}, {y, -1, 1},
   VectorScaling -> Automatic] /. 
  Arrow[{p_, q_}] :> 
   Arrow[{interp[p, q, -Abs@Cos[t]], interp[p, q, Abs[Cos[t]]]}],
 {t, 0, 2 Pi}]

Note that the rescaling by VectorPlot still makes the scaling by Cos[t] irrelevant, except for the sign of Cos[t]. One could omit the Cos[t] from the vector field and the Abs from the interpolation and get the same plot.