Equations modelling a snowboarder/skier on a jump

Question

I am currently a highschool student elaborating an internal assessment for my mathematics AA HL course (a sort of research paper we must elaborate for the IB) and was looking into the relationship between mathematics and snowboarding. My current idea for research was investigating the feasibility of a quintuple cork given some initial conditions of the jump (angle, length, height, slope angle, etc.) and the rider (height, mass, etc.).

However, I've encountered some difficulties in finding appropriate differential equations to model a snowboarder's motion during a jump. Most existing models are either too complex, relying on matrix methods and other mathematics that are beyond my current level of math, or too simplistic, relying on basic kinematic equations that wouldn't allow for a meaningful investigation.

I would greatly appreciate your help in finding or developing differential equations that can accurately model a snowboarder's motion during a jump. I believe that with the right equations, I can utilize various numerical methods such as Euler and Runge-Kutta to analyze the feasibility of a quintuple cork on a specific jump. Whether you can suggest equations, recommend textbooks or research papers, or provide insights into the application of calculus and differential equations in this context, I would greatly appreciate your input.

Let me make it clear I do not want my work done for me, since it would be against guidelines, rather some pointers to help me get started

Answer 1

Difficult!!

The "easy" part is the motion of the centre of mass of the snowboarder. Ignoring air drag that is just projectile motion, no matter how they spin. So we can set that part aside.

The difficult part is the spinning. If you make enough simplifying assumptions you can make it "easy". You seem to be looking for a sweet spot between the ridiculously complicated, fully realistic treatment, vs. the sorts of simplistic situations that might be suitable for a 1st or 2nd year undergraduate physics student. Here are some benchmarks for levels of complexity based on simplifying assumptions:

1. Assume the snowboarder has cylindrical symmetry (so, no arms, no snowboard...) and rotates purely around their long axis. In this case they rotate at a fixed rate and there is no differential equation needed. I think you are looking for something a bit juicier than this.

2. Assume the snowboarder has cylindrical symmetry (no arms, no snowboard...) and they rotate around their long axis, but also around some other axis. Now you will need Euler's Equations. There are lots of things called "Euler's equations". I mean these ones: https://en.wikipedia.org/wiki/Euler%27s_equations_(rigid_body_dynamics) . The rotation will now not be at a fixed rate, around a fixed axis. There will be precession and libration. This might be the sweet spot you are looking for.

3. As 2. above, but drop the assumption of cylindrical symmetry. This will still be solvable with Euler's equations, but finding the moment of inertia will be harder and the behaviour will be richer. If you manage to do 2. you might try this as a stretch goal.

4. The snowboarder isn't a rigid body. Their arms/legs probably move, they may bend their torso, etc. Achieving a quintuple cork might require that they do this. Modeling it is very hard, and is at the edge of academic research into classical dynamics (e.g. people interested in rotation of space probes sometimes need to grapple with this, and you can find recent papers dealing with the problem). This is way beyond what is realistic for you to try. It is basically graduate school level stuff.