Classical dynamics with curl forces, and motion driven by time-dependent flux

For position-dependent forces whose curl is non-zero ('curl forces'), there is no associated scalar potential and therefore no obvious Hamiltonian or Lagrangean and, except in special cases, no obvious conserved quantities. Nevertheless, the motion is nondissipative (measure-preserving in position and velocity). In a class of planar motions, some of which are exactly solvable, the curl force is directed azimuthally with a magnitude varying with radius, and the orbits are usually spirals. If the curl is concentrated at the origin (for example, the curl force could be an electric field generated by a changing localized magnetic flux, as in the betatron), a Hamiltonian does exist but violates the rotational symmetry of the force. In this case, reminiscent of the Aharonov-Bohm effect, the spiralling is extraordinarily slow.