In Hamiltonian mechanics, angular momentum is a certain momentum map and a component of the angular momentum is the generator function of the action of a one-parameter subgroup of the rotation group $SO(3)$.
In Newtonian physics, angular momentum is something that causes the precession of a bicycle wheel that is rotating about a horizontal axis suspended at one end.
There is no apparent relationship between the two. Is there any way to relate these two things without using sums and indices?
I'd like to see Hamiltonian mechanics as a complete, standalone theory, without the crutches of the Newtonian model or Lagrangian formalism. So for example I'd like to see that angular momentum (defined in Hamiltonian theory) has the properties that we expect from it, namely it changes the behavior of a rigid body against external effects. Shortly: how can describe Hamiltonian mechanics phenomena like the bicycle wheel precession?