I have been reading "variational principles in classical mechanics" by Douglas Cline and the following page rather confused me. It states that for a cylinder rolling without slipping, the velocity of rolling point of contact is not zero. What am I missing here?
The contact point is stationary when the wheel (or cylinder) is rolling. Otherwise it wouldn't be rolling, but sliding. The "without slipping" part is emphasizing this - if the two surfaces slide over one another, then we have slipping. If they don't, then we have no slipping.
And if they don't slide over one another, well, then they are stationary relative to each other.
But only for a moment.
To roll, the wheel therefor isn't sliding over the surface. Rather, the point-of-contact on the wheel is "placed on" the surface and the rest of the wheel moves over this point. Then this point is "lifted off" from the surface. A new point takes over right away, so that there is basically always contact. This contact point is stationary in each moment, but is constantly replaced by a new point on the wheel.