I want to understand what causes this motion on contrails (Cirricular motion) and it’s name if possible. What effects it? How and why is it formed?
I will assume that what you mean by the circular motion is the motion of the contrail around its long axis.
As originally formed, the contrail in your picture does not have any circular motion in it because it was created by the hot steam in the exhaust streams shooting out of the tailpipes of the plane's engines.
However, because the wings of the airplane do produce circular motion in the air moving over them, that circular motion will then (weakly) disturb the exhaust plumes and stir them into circular motion as well.
The exhaust plumes therefore trace out the vortexes created by the wings. For a plane like a DC-9 or MD-11 that has its engines attached to its tail, rotation of the exhaust plumes by wing vortexes will be weak and hard to see. In the case of a B747 or the big Airbuses with 2 engines mounted on each wing, you can sometimes see the vortexes twisting the exhaust plumes around.