Summary: If a family of tori in phase space is driven by a time-dependent Hamiltonian flow in such a way as to return after some time to the original family, there generally results a shift in the angle variables. One realisation of this process is in the cyclic adiabatic change of a classical Hamiltonian, and the angle change has previously been shown to separate naturally into a dynamical part and a geometrical part. Here the same geometrical angle change is extracted when the return is achieved non-adiabatically, and the ‘dynamical’ remainder calculated. Two examples are given: the precession of a spin and the rotation of phase-space ellipses.