A sphere rolling on a straight horizontal line corresponds to a spin 1/2 in a constant magnetic field in the direction.
The lollipop, or a sphere rolling counterclockwise on a circle of radius , corresponds to a spin 1/2 precessing in a magnetic field that rotates in the plane.
Equivalence of (a) rolling on a Cornu spiral and (b) the Landau–Zener problem of the spin flip probability on a time dependent field.
Sphere rolling along a curve of zero torsion (meaning that the velocity of the center of the sphere is parallel to the tangent of the curve at the contact point).
When a sphere of radius rolls on the parallel of a second sphere of radius , the angular velocity describes a cone. In the mapping from rolling spheres to spins, the magnetic field also describes a cone. This case of a sphere rolling on a parallel is isomorphic to a spin 1/2 precessing in a time dependent magnetic field and describing a cone at constant rate.
The instantaneous position of two spheres of radius rolling on a sphere of radius . One sphere rolls outside and the other inside the sphere of radius . The plane of the paper corresponds to the instantaneous plane containing the centers of the spheres and the angular frequencies and for inner and outer rollings.
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