This is all lots of fun, though I'm sure it leaves out forum members who think physics is evil or confusing.
I'll add a few sample ideas.
I was thinking a lot about the physics edges and creating and dumping angular momentum, in connection with blade shapes and skate sharpening. Remarkably small shifts in weight and blade shape can have tremendous effects on initiating and checking turns, spins and jumps, as well as preserving speed and glide.
It is unstable for a center of rotation (note that surface contact constrains the axis of rotation to be at right angles to the ice surface, which is not the effective vertical created by the vector sum of gravitational and various inertial forces) to be ahead of the center of lateral resistance.
Consider how unstable a weather vane is when it has the point of major resistance temporally upwind of the center of rotation is unstable. It will turn so that the part of the vane with the largest resistance is downwind.
Proper use of these principles helps initiate turns, and checks them. Like a skateboarder who pushes down one part of the board to the surface to initiate a turn. Or the classic bootlegger's turn, where you use the car's emergency brakes (which only brake the front wheels) to create and control a rapid turn, or various forms of wheely [sp?] on bikes. 3 turns and brackets follow similar principles.
It helps explains why a parabolic blades should in theory be better at both initiating and checking turns, spins and jumps, but only if you use them right. The ends of the blade have more resistance, so placing your weight and contact right makes it easier to control angular momentum. Tapered blades only make sense if you restrict yourself to forward-to-backward turns.
There are a lot of trade-offs on things like side honing - i.e., it should cause some things to work better, some worse.
When you edge a blade, you are tilting the rocker a bit into the horizontal plane, and thereby effectively creating a curved blade that want to turn. That also shortens the length in contact - though parabolic blades paradoxically lengthen the contact area.
It is also amazing how the tiny amount of lateral resistance on a blade created by friction on the side of the blade towards which you are edged can make 3-turns so much easier to initiate (but harder to check, because you have more angular momentum to rapidly dump) than brackets.
Edge shape actually has very complicated physics. Plus, there are a lot of trade-offs.
One thing that is very hard for me to understand is that some blade shape differences that are around 1/1000 inch make a huge difference in the blade/ice interaction and how I skate. It's fun to play with subtle sharpening changes.
But there is no obvious way to model anything physically, when people aren't certain how the underlying blade/ice physics works. People try to use intuitions derived from fluid dynamics (e.g., boats, aircraft), which probably don't apply very well.
Another physics issue is understanding the ways you transfer angular momentum from one body part to another. For example, people often swing legs or arms to temporarily hold the angular momentum being accumulated by an edge. When the swing stops, angular momentum is put back into the main body, and the rotation occurs.
For some reason, judges hate some forms of momentum transfer. If you swing your arms around to create a turn, test judges (especially for dance and moves) will mark you poorly. They like more subtle motions. But many of the best freestyle skaters begin some rotations with up stretched hands, transfer it to the arms, and then to the torso - and use this very unsubtle technique to win national and international competitions.
Again, tension without internal motion (e.g., pulling back a shoulder to initiate or check a turn) should have essentially no effect on the external motions of the body (standard physics theorem) - but it does. In part it reduces vibration and flopping around, which wastes energy. It seems to have huge effects it shouldn't have. Perhaps when we pull a shoulder back a few thousandths of an inch, it transfers a bit of angular momentum at the beginning and end of the pull, which is enough to initiate and check turns. Those displacements are so small, it seems implausible. Yet - what else could it be?
The physics of falling are also interesting. You have to dissipate the energy and momentum in a way that spreads it over a large portion of the body. This creates pressure and sheer waves, fluid motions, and compressions and tensions in many different body components. Some materials within the body are good at taking forces in very specific directions. Bone takes compression well (as long as it is altered slowly), sheer not so well, but fractures quite easily under tension, including that generated by torsion and bending. Muscles and ligaments take tension quite well along the fibers, not so well if it tends to split fibers apart, and have problems taking sheer and compression. People probably often destroy Achilles tendons precisely because the tendon isn't designed to take the compression and sheer forces the back of a freestyle boot imposes.
Body design is based and developmentally adapts and changes around the idea of compensating for the specific weaknesses and strengths of the component materials. It is not designed perfectly, in engineering terms - perhaps ligaments and muscles should pass through bones, so their tension is balanced against bone compression, for maximum strength. Plus, humans and animals don't take advantage of materials and structures that developed for other biological creatures (E.g., wood, Chitin, silicates), nor do they use steel or aluminum.
Consider the ways in which stiffness is created. If you want a lightweight composite material stiffness, you do the same thing - balance compression of something that mostly resists compression against tension of something that mostly resists stretching. For this to work, both materials must have substantial intra-laminar sheer strength, and they must be held together by an adhesive or sewn fibers that provide substantial inter-laminar sheer strength. This is used in composite material hockey and speed skates, as well as composite kayaks, canoes, and ultralight aircraft. The component layers do not have to be stiff or dense - they are only stiff when adhered or sewn together. In theory, you can create (non-isotropic) stiffness in selective directions, by controlling the directions in which the forces are balanced. E.g., it is possible to create boots that have a lot of sideways resistance, but are quite flexible for ankle point and flex, though I'm not sure anyone does that.
But stiffness in traditional leather boots is often achieved in a much less efficient fashion. The material is quite dense, and is saturated by adhesives that make the entire material rigid. This makes for a heavy boot, and it is very difficult to give it selective direction stiffness. I believe properly designed composite boots could have an overwhelming performance advantage over traditional leather boots.
Anyway, I hope you all enjoy playing with physics ideas.