@naturallyInconsistent Indeed the relationship between velocity and momentum is obtained from the Hamiltonian, but it remains true that this is an input to the theory. If you use the Hamiltonian $H = p^2/2 +x^2/2$, your system evolves like a harmonic oscillator and $p= \dot q$. If you use the Hamiltonian $H = (p-A(x))^2/2 + x^2/2$, then that proportionality is lost and your system evolves like a harmonic oscillator in a magnetic field. The nature of the relationship between (canonical) momentum
and velocity cannot be determined from the phase space or configuration space - it constitutes a decision being made by the physicist writing down the Hamiltonian. It is an independent input to a model, so there is no procedure for "getting the correct Hamiltonian" that does not at some point involve choosing which physics you want to model and working out which Hamiltonian to use in that specific instance.
Discussion between J. Murray and natu…
Discussion between J. Murray and natu…