Hannay angle explained

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.^[1]

Consider a one-dimensional system moving in a cycle, like a pendulum. Now slowly vary a slow parameter

λ , 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

\int ^{T} _{0} \omega( λ ) • λ dt . This angle depends on the precise details of the motion, and it is of order O(T) .

The second order is Hannay's angle, which surprisingly is independent of the precise details of

• λ . It depends on the trajectory of λ , but not how fast or slow it traverses the trajectory. It is of order O( 1 ) .^[2]

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 λ (t) , the action variable I _{\alpha} becomes an adiabatic invariant, and the Hannay angle \theta ^{H} _{\alpha} for its corresponding angle variable can be calculated according to the path integral that represents an evolution in which the perturbation λ (t) gets back to the original value^[3] $$\backslash theta^H\_\backslash alpha\; =\; -\backslash frac\backslash oint\backslash !\backslash boldsymbol\; \backslash cdot\; \backslash frac\backslash mathrm\backslash lambda\; =\; -\backslash partial\_\; \backslash iint\; \backslash omega$$where \boldsymbol{p} and \boldsymbol{q} are canonical variables of the Hamiltonian, and \omega 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.^[4]

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

Rotation of a rigid body

A free rigid body tumbling in free space has two conserved quantities: energy and angular momentum vector

E, \vec L . Viewed from within the rigid body's frame, the angular momentum direction is moving about, but its length is preserved. After a certain time T , 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

T , the body has rotated by angle $$2ET/\backslash |\backslash vec\; L\backslash |\; -\; \backslash Omega$$

where

\Omega is the solid angle swept by the angular momentum direction as viewed from within the rigid body's frame.^[5]

Other examples

The heavy top.^[6] The orbit of earth, periodically perturbed by the orbit of Jupiter.^[7] The rotational transform associated with the magnetic surfaces of a toroidal magnetic field with a nonplanar axis.^[8]

References

• Book: Reduction, Symmetry, and Phases in Mechanics . Jerrold E.. Marsden. Jerrold E. Marsden . Richard. Montgomery . Tudor S. . Ratiu. Tudor Ratiu . 69 . 0-8218-2498-8 . AMS Bookstore . 1990 .
• Book: C. Pisani . Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials . 3-540-61645-4 . 1994 . Springer . Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society . 282 .
• Book: Physics of Ferroelectrics: a Modern Perspective . Karin M Rabe . Karin M. Rabe . Jean-Marc Triscone . Charles H Ahn . 2 . Springer . 2007 . 978-3-540-34590-9.

External links

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

Notes and References

1. Hannay . J H . 1985-02-01 . Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian . Journal of Physics A: Mathematical and General . 18 . 2 . 221–230 . 10.1088/0305-4470/18/2/011 . 0305-4470.
2. Robbins . J M . 2016-10-28 . The Hannay angle, thirty years on . Journal of Physics A: Mathematical and Theoretical . 49 . 43 . 431002 . 10.1088/1751-8113/49/43/431002 . 1751-8113. 1983/2992186e-5dde-4a3f-a2a9-67377afcadf9 . free .
3. Hannay Angle: Yet Another Symmetry-Protected Topological Order Parameter in Classical Mechanics. Toshikaze Kariyado . Yasuhiro Hatsugai. J. Phys. Soc. Jpn.. 85. 2016. 4 . 043001 . 10.7566/JPSJ.85.043001. 1508.06946. 2016JPSJ...85d3001K . 119297582 .
4. Khein . Alexander . Nelson . D. F. . 1993-02-01 . Hannay angle study of the Foucault pendulum in action-angle variables . American Journal of Physics . en . 61 . 2 . 170–174 . 10.1119/1.17332 . 0002-9505.
5. Montgomery . Richard . 1991-05-01 . How much does the rigid body rotate? A Berry’s phase from the 18th century . American Journal of Physics . en . 59 . 5 . 394–398 . 10.1119/1.16514 . 0002-9505.
6. Park . Changsoo . 2023-05-01 . Heavy symmetric tops and the Hannay angle . American Journal of Physics . en . 91 . 5 . 357 . 10.1119/5.0101149 . 0002-9505.
7. Berry . M V . Morgan . M A . 1996-05-01 . Geometric angle for rotated rotators, and the Hannay angle of the world . Nonlinearity . 9 . 3 . 787–799 . 10.1088/0951-7715/9/3/009 . 0951-7715.
8. Geometric phase, rotational transforms, and adiabatic invariants in toroidal magnetic fields. Bhattacharjee, A.; Schreiber, G. M.; Taylor, J. B. . Phys. Fluids B. 4. 9 . 1992. 2737–2739. 10.1063/1.860145 .