The Hamiltonian

Starting with Noether's theorem for a continuous time symmetry, we'll examine the Hamiltonian. We compare the "state space" description of Lagrangian mechanics to the "phase space" description of Hamiltonian mechanics.
Music "Everything" by Vi Hart
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[@Ron_Shvartsman]

Great video, thanks! Could you clarify the differences between state space and phase space? In many elementary applications, we have q-dot proportional to p, so they would seem to be identical. However, in electromagnetism, the canonical momentum is now Pi + qA, where Pi = m q-dot is the kinetic momentum, and A is the vector potential. What would be the differences in the Lagrangian and Hamiltonian approach in this case?