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.

, 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.

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Hamiltonian mechanics

The Hannay angle is defined in the context of action-angle coordinates becomes an adiabatic invariant gets back to the original value are canonical variables of the Hamiltonian 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.

under the effect of Earth's rotation

Rotation of a rigid body

, 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)

is the solid angle swept by the angular momentum direction as viewed from within the rigid body's frame.

Other examples

In systems with more than one degree of freedom, Hannay angle may no longer have an interpretation of merely a geometric contribution to the angle variable. An elementary example is the geometric angle in adiabatically driven anisotropic oscillator .

If a frictionless flywheel is fixed to the surface of a rotating disk, and the flywheel is not parallel to the surface, then after a single rotation, the flywheel would rotate. This is essentially a tabletop version of Foucault's pendulum. More generally, if the flywheel is transported over a curved surface such that its axis is always perpendicular to the local surface, then after performing a full loop on the surface, the flywheel would be rotated by the integral of Gaussian curvature enclosed by the curve.

Stated in another way, consider a heavy symmetric top is the Hannay angle.

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

External links

• Professor John H. Hannay: Research Highlights. Department of Physics, University of Bristol.