[hintymad]

This introduction must assume that the reader already understands physics deeply, right? For instance, the page on Hamiltonian mechanics stated that force is the derivative of momentum with respect to time. I can't imagine how one will understand the intuition behind the definition without having already learned at least college-level physics.

[abnry]

If you know F=ma, p=mv, and by calculus that a=v', all of which is high school level, then the time derivative of momentum is clearly force.

[hintymad]

This is derivation, which is different from using momentum to define force. That definition actually has many benefits in Hamiltonian mechanics. The key challenge is to intuitively understand why such definition makes intuitive sense.