Classical adiabatic angles and quantal adiabatic phase

A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma _n associated with an eigenstate with quantum numbers n=(n_l); the classical property is a shift Delta theta _l(I) in the lth angle variable for motion round a phase-space torus with actions I=(I_l); the connection is Delta theta _l=- delta gamma / delta n_l. Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q², qp and p²; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.