Patrick wrote:Is there a relation between rolling your belly, which is primarily done by controlling your breathing, and power output?
What we can see in these videos is that they can bounce something from the belly when they belly inflates quickly. So how does this transfer to the structure of the body?
First, it isn't "primarily done by controlling your breathing". Second, it isn't about "inflating" the belly, implying it is filling with air.
The "stuff" of the abdomen is connected to other stuff. When the "stuff" of the abdomen is, for example, contracted, it causes other stuff to contract or stretch. As Windwalker pointed out, changing one thing changes other things connected to it. Put another way, the manipulation of the abdomen that you see is the result of more than just the abdomen. This is very clear in the video of Chen Yu's student. The perineum (huiyin) is involved, the anus, the spine, muscles of the back, muscles of the chest and rib cage, the hips, legs ...
Movement of "the dan tian" is not an end. It is a means to connecting in action the parts of the body: when one part moves, all parts move. One way of looking at it is that when all of the parts of the body are appropriately connected, the movement of the abdomen is the result. Another way of looking at it is that when the abdomen does its thing, the other parts of the body are enlisted. A crude analogy is "the dan tian" (i.e. abdomen) is an input gear that drives the other gears (body parts) in the gear train. In Feng's teachings, there are 18 major "joints" that are conceptualized as being balls (e.g. ball bearings in a race of a bearing) in contact with one another: rotate one ball and the others rotate. This gives rise to his "18 ball silk reeling exercises".
Coordinated breathing can be used to augment the compression and contraction. At some stages, it can feel like that it is what drives the compression and contraction.