Hannay angle

In classical mechanics, the Hannay angle is a mechanics analogue of the geometric phase (or Berry phase). It was named after John Hannay of the University of Bristol, UK. Hannay first described the angle in 1985, extending the ideas of the recently formalized Berry phase to classical mechanics.^[1]

Consider a one-dimensional system moving in a cycle, like a pendulum. Now slowly vary a slow parameter ${\displaystyle \lambda }$, like pulling and pushing on the string of a pendulum. We can picture the motion of the system as having a fast oscillation and a slow oscillation. The fast oscillation is the motion of the pendulum, and the slow oscillation is the motion of our pulling on its string. If we picture the system in phase space, its motion sweeps out a torus.

The adiabatic theorem in classical mechanics states that the action variable, which corresponds to the phase space area enclosed by the system's orbit, remains approximately constant. Thus, after one slow oscillation period, the fast oscillation is back to the same cycle, but its phase on the cycle has changed during the time. The phase change has two leading orders.

The first order is the "dynamical angle", which is simply ${\displaystyle \int _{0}^{T}\omega (\lambda ){\dot {\lambda }}dt}$. This angle depends on the precise details of the motion, and it is of order ${\displaystyle O(T)}$.

The second order is Hannay's angle, which surprisingly is independent of the precise details of ${\displaystyle {\dot {\lambda }}}$. It depends on the trajectory of ${\displaystyle \lambda }$, but not how fast or slow it traverses the trajectory. It is of order ${\displaystyle O(1)}$.^[2]