Mechanics analogue of the geometric phase
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.
Consider a one-dimensional system moving in a cycle, like a pendulum. Now slowly vary a slow parameter formula_1, 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 formula_2. This angle depends on the precise details of the motion, and it is of order formula_3. 
The second order is Hannay's angle, which surprisingly is independent of the precise details of formula_4. It depends on the trajectory of formula_1, but not how fast or slow it traverses the trajectory. It is of order formula_6.
Hannay angle in classical mechanics.
The Hannay angle is defined in the context of action-angle coordinates. In an initially time-invariant system, an action variable formula_7 is a constant. After introducing a periodic perturbation formula_8, the action variable formula_7 becomes an adiabatic invariant, and the Hannay angle formula_10 for its corresponding angle variable can be calculated according to the path integral that represents an evolution in which the perturbation formula_8 gets back to the original value
formula_12
where formula_13 and formula_14 are canonical variables of the Hamiltonian, and formula_15 is the symplectic Hamiltonian 2-form.
Example.
Foucault pendulum.
The Foucault pendulum is an example from classical mechanics that is sometimes also used to illustrate the Berry phase. Below we study the Foucault pendulum using action-angle variables. For simplicity, we will avoid using the Hamilton–Jacobi equation, which is employed in the general protocol.
We consider a plane pendulum with frequency formula_15 under the effect of Earth's rotation whose angular velocity is formula_17 with amplitude denoted as formula_18. Here, the formula_19 direction points from the center of the Earth to the pendulum. The Lagrangian for the pendulum is
formula_20
The corresponding motion equation is
formula_21
formula_22
We then introduce an auxiliary variable formula_23 that is in fact an angle variable. We now have an equation for formula_24: 
formula_25
From its characteristic equation
formula_26
we obtain its characteristic root (we note that formula_27)
formula_28
The solution is then
formula_29
After the Earth rotates one full rotation that is formula_30, we have the phase change for formula_24
formula_32
The first term is due to dynamic effect of the pendulum and is termed as the dynamic phase, while the second term representing a geometric phase that is essentially the Hannay angle
formula_33
Rotation of a rigid body.
A free rigid body tumbling in free space has two conserved quantities: energy and angular momentum vector formula_34. Viewed from within the rigid body's frame, the angular momentum direction is moving about, but its length is preserved. After a certain time formula_35, the angular momentum direction would return to its starting point. 
Viewed in the inertial frame, the body has undergone a rotation (since all elements in "SO(3)" are rotations). A classical result states that during time formula_35, the body has rotated by angle formula_37 
where formula_38 is the solid angle swept by the angular momentum direction as viewed from within the rigid body's frame.
Other examples.
The heavy top. 
The orbit of earth, periodically perturbed by the orbit of Jupiter. 
The rotational transform associated with the magnetic surfaces of a toroidal magnetic field with a nonplanar axis.
References.
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