 Hannay Angle
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Hannay Angle

In classical mechanics, the Hannay angle is a mechanics analogue of the whirling 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.

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 $I_{\alpha }$ is a constant. After introducing a periodic perturbation $\lambda (t)$ , the action variable $I_{\alpha }$ becomes an adiabatic invariant, and the Hannay angle $\theta _{\alpha }^{H}$ for its corresponding angle variable can be calculated according to the path integral that represents an evolution in which the perturbation $\lambda (t)$ gets back to the original value

$\theta _{\alpha }^{H}=-{\frac {\partial }{\partial I_{\alpha }}}\oint \!{\boldsymbol {p}}\cdot {\frac {\partial {\boldsymbol {q}}}{\partial \lambda }}\mathrm {d} \lambda$ where ${\boldsymbol {p}}$ and ${\boldsymbol {q}}$ are canonical variables of the Hamiltonian.

Example

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 $\omega$ under the effect of Earth's rotation whose angular velocity is ${\vec {\Omega }}=(\Omega _{x},\Omega _{y},\Omega _{z})$ with amplitude denoted as $\Omega =|{\vec {\Omega }}|$ . Here, the $z$ direction points from the center of the Earth to the pendulum. The Lagrangian for the pendulum is

$L={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2})-{\frac {1}{2}}m\omega ^{2}(x^{2}+y^{2})+m\Omega _{z}(x{\dot {y}}-y{\dot {x}})$ The corresponding motion equation is
${\ddot {x}}+\omega ^{2}x=2\Omega _{z}{\dot {y}}$ ${\ddot {y}}+\omega ^{2}y=-2\Omega _{z}{\dot {x}}$ We then introduce an auxiliary variable $\varpi =x+iy$ that is in fact an angle variable. We now have an equation for $\varpi$ :
${\ddot {\varpi }}+\omega ^{2}\varpi =-2i\Omega _{z}{\dot {\varpi }}$ From its characteristic equation
$\lambda ^{2}+\omega ^{2}=-2i\Omega _{z}\lambda$ we obtain its characteristic root (we note that $\Omega <<\omega$ )
$\lambda =-i\Omega _{z}\pm i{\sqrt {\Omega _{z}^{2}+\omega ^{2}}}\approx -i\Omega _{z}\pm i\omega$ The solution is then
$\varpi =e^{-i\Omega _{z}t}(Ae^{i\omega t}+Be^{-i\omega t})$ After the Earth rotates one full rotation that is $T=2\pi /\Omega \approx 24h$ , we have the phase change for $\varpi$ $\Delta \varphi =2\pi {\frac {\omega }{\Omega }}+2\pi {\frac {\Omega _{z}}{\Omega }}$ 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
$\theta ^{H}=2\pi {\frac {\Omega _{z}}{\Omega }}$ 