It is remarkable that in the physics of electromagnetic fields it was not until
the 1950's that measurable significance was given to the concept of potentials.
Students before 1950 were taught that the E and B fields were not uniquely
dependent upon the potentials, and therefor the potentials were just a useful
mathematical construct without "measureable" content.
Then came the BohmAharanov idea.
However, much before BA, the Joukowski theory of flight, and the lift around an airfoil, demonstrated that it was the potentials that caused measurable Lift, not the vorticity in a flow. Better said, Lift is due to the circulation integral being nonzero (the equivalent to the BohmAharanov idea) even though the integration path was in a region of ZERO vorticity (zero magnetic field).
In aerodynamics, vorticity is the primary cause of drag, not lift. Stokes theorem fails for the Joukowski theory. The velocity field is represented by an Action 1form which is closed, but not exact. Outside the topological obstruction defined as the wing, the vorticity of the flow is identically zero (for holomorphic potentials). The topological concept is that the closed integration path around the contour of the wing is a cycle, not a boundary.
In practice, most of the drag of a wing comes tip vortices that do not exist if
the wing is of infinite span. That is why high performance glider wings have
large aspect ratios.
If one measures the coefficent of Lift for a given airfoil as a function of the
angle of attack, a Lift curve is produced that increases as a function of the
angle of attack, and then falls dramatically as the wing reaches a stall point.
What is not widely appreciated is that if airfoil shape is reversed, with the
trailing edge in front, the SAME lift curve as a function of angle of attack is
produced!!! Lift is not dissipative, it is reversible.
