When I strike a ball with my foot, look at the exact contact point (and pretend the ball doesn’t squish across my foot—even though it does, because my pegas are so poderosas). There’s a
tangent space at that
contact point and the exact particulars of my posture, shape of my foot|shoe, and so on determine the force vector that I use to bend the ball like Beckham.
f=ma corrected Aristotle’s theory of motion
Vacuum isn’t possible: Vacuum doesn’t occur, but hypothetically, terrestrial motion in a vacuum would be indefinitely fast.
So we know we only need that one strike and then inertia minus drag convolved with lay-of-the-land will determine a full path for the ball: till my teammate’s head contacts it—another tangent space, another vector—and we score.
That’s football for simple tangent spaces—the Aristotle-v-Newton scenario. Now onto the most poetic sport, where two holonomic dynamical systems dance within each other’s gyre, moving in and out of each other’s movements and—occasionally—osculating a tangent that links one part of me with one part of you.
All of the logic about connecting vectors head-to-tail with parallelograms is only to reason about one single strike. The whole linear algebra on force vectors is a complete examination of the possibilities of the variations and the ways to strike at that exact same point.
To talk about striking different points requires a connection (parallel transport). [And remember if you hook my cheek versus jab my forehead, those arenotparallel so even more maths is involved. Also hard tissue (bone) versus soft tissue (cheek).]
Striking the ball on the side (to spin it) or under (to chip it) would be a
connection on the S² manifold of the ball — moving the point of tangency — which is a different algebra’s worth of logic. Landing a punch on a different part of me is also a connection (that would be parallel transport of the strike vector on the surface manifold (with boundary) of my face/torso/armpit/whatever).
Keeping my torso over the ball is even yet another calculus—one that I suspect is much, much more complicated. Even though Julian James Faraway and others work on these questions of whole-body mechanics, I don’t think ∃ a complete mathematical theory of how all of the fixed
holonomic parts of bodies that have very similar morphogenetic shapes (similar ratios of forearm to humerus length, etc) interact—how soft tissue and hard tissue in the usual places interact and specific angles can make a judoist with excellent technique and little strength able to whirl the body of a weight-lifter around her fulcrum.
Or how this guy can deliver a lot of force with proper dynamical posture (“shape”) when he’s clearly weak and fat. I can start to imagine the beginnings of something like that but it doesn’t obviously fit into the tangent space points & vectors story, except in a very complicated way of vectors connected to vectors connected to vectors, with each connection (not the same as the parallel-transport connection sense of the word I used above! Back to the base namespace) holonomically constrained or even "soft-constrained" in the case of soft tissue. Same with a blow landing parallel-transported different places on the surface of my body. The transport takes care of the initial strike vector but not how those forces travel and twist through my skeleton, soft tissues, down to the floor (either through my feet or through my *rse if the blow knocks me down). And sure, you could FEM-approximate my interior with a 3-D mesh and a deep geometric logic—and even make the ribs more rigid than the liver—but that’s just going to make you want higher maths even more as you grapple with the army of ε’s you’ll convoke with all the approximations,
dx size, parameters, and so on.
Or why it’s better to swing the bat this way, or teach your body to do its swimming strokes in that motion, or various other dynamical or static particulars of human body shape and internal anatomical facts.
BTW, an amateur ultimate fighter once told me that the order of importance for winning a fight is:
- technique (brasilian jiu jitsu technique)
He didn’t mention tolerance for getting punched in the head but he seemed to put up with it remarkably. PS: judo guys have sexy bodies.