Certain systems do not completely return to themselves when a subsystem moves through a closed circuit in physical or parameter space. A geometric phase, known classically as Hannay’s angle and quantum mechanically as Berry’s phase, quantifies such anholonomy. We study the classical example of a bead sliding frictionlessly on a slowly rotating hoop. We elucidate how forces in the inertial frame and pseudo-forces in the rotating frame shift the bead. We then computationally generalize the effect to arbitrary—not necessarily adiabatic—motions. We thereby extend the study of this classical geometric phase from theory to experiment via computation, as we realize the dynamics with a simple apparatus of wet ice cylinders sliding on a polished metal plate in 3D printed plastic channels.

A cat falling feet up twists its body to land feet down. It accomplishes this zero-angular-momentum turn with topology not dynamics: the cat’s contortions trace out a closed loop in cat-shape parameter space so that it begins and ends with the same shape, even as it rotates in physical space. Such anholonomy is a key idea in recent classical and quantum mechanics. In this article, we analyze and generalize a famous mechanical anholonomy, a bead sliding on a rotating hoop, transforming it from idealized example to realized experiment.

Cyclically but slowly change a quantum system and its complex wave function returns to its original state multiplied by a complex exponential. This phase factor is the product of a dynamic factor that depends on the time of the adiabatic change and a geometric factor that does not. The 1983 discovery of the geometric phase by Berry,1 anticipated in optics by Pancharatnam,2 provides elegant insight into diverse quantum phenomena, including polarization changes to light traveling in coiled optical fibers and phase changes to charged particles circling magnetic fields.3 

Berry’s colleague Hannay quickly discovered classical analogues.4 These include a Foucault pendulum spun by Earth’s spin in a circle of constant latitude, a flywheel constrained to move tangent to a surface,5–7 and most famously a bead sliding frictionlessly on a horizontal noncircular hoop.8 

A slow cyclic rotation restores the hoop to its original state but unavoidably shifts the moving bead by an angle that depends only on the hoop’s geometry. Rotating a circular hoop-with-bead does not shift the bead at all, but rotating a noncircular hoop indelibly imprints its geometry on the bead’s motion.

This article generalizes the unexpected bead-on-hoop dynamics from adiabatic to arbitrary motions and thereby facilitates accessible laboratory experiments. A new mathematical analysis of the hoop and bead dynamics enables computer simulations to connect theory and experiment and convert a theoretical gem to a practically measurable effect. Section II derives exact equations of motion for a bead sliding on a spinning ellipse. Section III numerically integrates the equations to elucidate the motion and generalize Hannay’s shift from adiabatic to nonadiabatic conditions. Section IV reports an elementary experimental realization of Hannay’s shift. Section V includes suggestions for future work. The  Appendix provides a self-contained derivation of Hannay’s shift, which emphasizes the importance of the Euler pseudo-force.

In the history of geometric phases,9 the elegance of the adiabatic cases may have delayed the study of related non-adiabatic cases, which are asymptotic to them but are accessible only by numerically integrating nonlinear differential equations.10 As a prelude to such integration, this section derives the initial value problem for a bead sliding frictionlessly on a rotating elliptical hoop.

Assume that the ellipse has major and minor radii a>b. In the hoop frame, parameterize the bead position by

rh[ϵ]=acosϵbsinϵ0,
(1)

where ϵ is the eccentric anomaly, as in Fig. 1. Rotate the hoop with respect to the lab by φ[t]. Represent the rotation by the matrix

Rz[φ]=cosφsinφ0sinφcosφ0001.
(2)

In the lab frame, parameterize the bead position by

r=Rz[φ]rh=acosϵcosφbsinϵsinφbsinϵcosφ+acosϵsinφ0.
(3)

Neglecting gravity, the Lagrangian is simply the kinetic energy

L=12mr˙r˙=12ma2sin2ϵ+b2cos2ϵϵ˙2+a2cos2ϵ+b2sin2ϵφ˙2+2abϵ˙φ˙,
(4)

which depends on the bead mass m. The Euler-Lagrange equation

ddtLϵ˙=Lϵ
(5)

simplifies to the nonlinear differential equation

(a2b2)ϵ˙2+φ˙2sinϵcosϵ+a2sin2ϵ+b2cos2ϵϵ¨+abφ¨=0,
(6)

which is independent of the bead mass m. (For circular hoops, a=b implies ϵ¨+φ¨=0 in general and ϵ˙+φ˙=0 from rest, so the bead does not move in the lab frame.)

FIG. 1.

In the lab frame, elliptic hoop of radii {a,b} rotates or spins through angle φ, while constrained to the hoop a frictionless bead moves or slides though arc length separation s, arc length angle σ, and eccentric anomaly angle ϵ.

FIG. 1.

In the lab frame, elliptic hoop of radii {a,b} rotates or spins through angle φ, while constrained to the hoop a frictionless bead moves or slides though arc length separation s, arc length angle σ, and eccentric anomaly angle ϵ.

Close modal

The exact motion depends on the bead initial conditions and the rotation strategy. Take

ϵ[0]=ϵ0,
(7a)
ϵ˙[0]=ω0,
(7b)

where ϵ0 and ω0 are the initial bead angle and angular velocity, respectively. As a simple example, rotate the hoop once sinusoidally in a time T by

φ[t]=2πsin2π2tT=π1cosπtT
(8)

starting and stopping at rest. Introduce the dimensionless time tt/T and angle ϵ[t/T]ϵ[t] and rewrite the initial value problem as

e2ϵ˙2+π4sin2[πt]sinϵcosϵ+1e2cos2ϵϵ¨+π31e2cos[πt]=0,
(9a)
ϵ[0]=ϵ0,
(9b)
ϵ˙[0]=ω0T,
(9c)

where the dimensionless eccentricity e=1b2/a2, and the dimensionless product ω0T=2πT/T0 is proportional to the ratio of the times for the hoop to spin once and for the bead to slide around a circle. If the initial bead angle ϵ0=0 for concreteness, the dimensionless shape and time parameters e and ω0T completely determine the dynamics.

Numerically integrate the initial value problem [Eq. (9)] to generate animation stills (Fig. 2), in both the hoop and lab frames. Time increases upwards. Directed rectangles represent kinematical and dynamical quantities; the rectangles’ long sides point in the vectors’ directions and their areas are proportional to the vectors’ magnitudes. (Directed rectangles thereby better represent the required large range of magnitudes and directions.)

FIG. 2.

Bead sliding on a hoop in the hoop (left) and lab (right) frames, with constraint force F, Coriolis, centrifugal, and Euler pseudo-forces F, Fc, Fe, and velocity and acceleration v and a. Black curves are bead trajectories. Areas of narrow rectangles are proportional to vector magnitudes. Time increases upwards. Hoop’s high curvature kinks bead’s lab path. Dimensionless parameters e=3/2 and ω0T=0.

FIG. 2.

Bead sliding on a hoop in the hoop (left) and lab (right) frames, with constraint force F, Coriolis, centrifugal, and Euler pseudo-forces F, Fc, Fe, and velocity and acceleration v and a. Black curves are bead trajectories. Areas of narrow rectangles are proportional to vector magnitudes. Time increases upwards. Hoop’s high curvature kinks bead’s lab path. Dimensionless parameters e=3/2 and ω0T=0.

Close modal

In the lab frame, the hoop rotates. Due to lack of friction, the mechanical constraint force F and the bead acceleration a, but not the bead velocity v, are always perpendicular to the hoop. Heuristically, because the hoop moves, as the acceleration updates the velocity from zero, the present velocity is perpendicular to the past hoop but not to the present hoop. So, the bead starts sliding along the hoop in a complex lab trajectory that can include multiple kinks. In the hoop frame, three pseudo-forces accompany the mechanical constraint force: the Coriolis pseudo-force F is perpendicular to the bead velocity v, which is tangent to the hoop; the centrifugal pseudo-force Fc is radially outward from the hoop’s center; and the Euler pseudo-force Fe is perpendicular to the centrifugal pseudo-force (as the hoop rotates and accelerates in its own plane). In the hoop frame, the Euler pseudo-force starts the bead moving.

The spacetime diagrams shown in Fig. 3 provide another representation. Time increases upwards. The hoop’s worldtube is an elliptic cylinder in the hoop frame but is twisted like a screw in the lab frame. The bead’s worldline hugs the hoop’s worldtube in both frames.

FIG. 3.

Spacetime diagrams of a bead sliding on a hoop that rotates once in its plane, in the hoop (left) and lab (right) frames. Large white tubes represent hoops, thin cyan tubes represent beads, thin yellow tube is bead in the absence of hoop spinning, and black arc is the generalized Hannay’s shift. Time increases upwards. Same parameters as Fig. 2.

FIG. 3.

Spacetime diagrams of a bead sliding on a hoop that rotates once in its plane, in the hoop (left) and lab (right) frames. Large white tubes represent hoops, thin cyan tubes represent beads, thin yellow tube is bead in the absence of hoop spinning, and black arc is the generalized Hannay’s shift. Time increases upwards. Same parameters as Fig. 2.

Close modal

Figure 4 plots the numerically computed shift δs versus the hoop eccentricity e and the dimensionless product ω0T (of initial bead angular velocity ω0 and hoop rotation period T). Slow slide or fast spin ω0T0 is on the left. The traditional Hannay limit [Eq. (A19)] of fast slide or slow spin ω0T is on the right. The trivial circular hoop e0 is on the bottom. The eccentric and dynamically rich elliptical hoop e1 is on the top. The circular limit with its trivial shift connects smoothly to the classic asymptotic adiabatic shift.

FIG. 4.

Ratio of bead shift to asymptotic shift δs/δs versus eccentricity e and dimensionless product ω0T (top) for the sinusoidal rotation [Eq. (8)] with e=0.8 slice (bottom). Ellipses illustrate eccentricities (left) and red-white-blue palette codes shifts (right). In the {ω0T,e} plane, slow slide or fast spin ω0T0 is leftward, fast slide or slow spin ω0T (the traditional Hannay limit) is rightward, circular hoop e0 is downward, and eccentric elliptical hoop e1 is upward.

FIG. 4.

Ratio of bead shift to asymptotic shift δs/δs versus eccentricity e and dimensionless product ω0T (top) for the sinusoidal rotation [Eq. (8)] with e=0.8 slice (bottom). Ellipses illustrate eccentricities (left) and red-white-blue palette codes shifts (right). In the {ω0T,e} plane, slow slide or fast spin ω0T0 is leftward, fast slide or slow spin ω0T (the traditional Hannay limit) is rightward, circular hoop e0 is downward, and eccentric elliptical hoop e1 is upward.

Close modal

For a typical e=0.8 slice, the shift oscillates for slow beads and quick rotations (small ω0 and T), but plateaus for fast beads and slow rotations (large ω0 and T), as predicted by the adiabatic limit [Eq. (A19)]. The details of the nonadiabatic oscillations depend on both the shape of the hoop and the rotation strategy.

Other hoops such as stadia, which consist of two semi-circles of diameter 2r connected by two parallel lines of length 2, produce similar results.

An obstacle to experimentally realize the Hannay hoop-and-bead anholonomy is the need for no or low friction. This section describes a solution: replace the bead with wet ice and the hoop with a plastic channel and a metal plate, as in Fig. 5.

FIG. 5.

Apparatus parts. Purple 3D printed plastic channels fixed to an aluminum-topped turntable with white clay (left). Wet ice cylinder slider topped with purple insulating annulus and steel ball (top right). Insulators and ice molds just after batch printing (bottom right).

FIG. 5.

Apparatus parts. Purple 3D printed plastic channels fixed to an aluminum-topped turntable with white clay (left). Wet ice cylinder slider topped with purple insulating annulus and steel ball (top right). Insulators and ice molds just after batch printing (bottom right).

Close modal

The slider is a cylinder of wet ice under an insulating plastic annulus under a steel ball. The ball adds inertia to lessen the unwanted accelerations due to residual frictional forces. The plastic annulus or collar thermally isolates the ice from the ball preventing the ball from rapidly melting through the ice at room temperature. The inner channel is a plastic ellipse, but the outer channel is a plastic non-ellipse, as two concentric ellipses do not maintain a constant separation. A dedicated constant-width spacer helps arrange the pieces, which are conveniently and efficiently fixed by modeling clay to a 24×24×1/4 5052-grade aluminum plate polished for smoothness and mounted on a turntable for easy rotation. The high-impact polystyrene plastic pieces are designed in Mathematica and 3D printed using fused deposition modeling, a kind of additive manufacturing.11 

Preparing good ice cylinders with diameters up to 4 cm requires special care. 3D printed ice molds are cubes with central cylinders removed. Circular cross sections are essential to avoid jamming in regions of high curvature. Small flexible sheets of silicone superglued and taped to the molds’ bottoms allow easy extraction of the ice cylinders. Placing the molds on metal sheets in a freezer encourages the bottoms to freeze flat. Freezing the water in layers enables the annuli to horizontally freeze to the ice cylinders.

A video camera fixed in either the hoop or lab frame records the manually induced motion of the slider and hoop. Section III formalism permits any starting and ending conditions, thereby greatly facilitating comparison of simulation and experiment. In the rotating hoop frame, the camera also records a reference laser spot fixed in the lab frame. Capstone software tracks the motion. A Mathematica notebook, available online as the supplementary material, imports the 2D position data, interpolates the hoop motion to initialize a computer model, and numerically integrates the model to compare theory and experiment.

Multiple trials indicate good agreement between experiment and theory for a broad range of starting conditions, although excessively slow or fast starts, for example, can cause the slider to stall or topple. As an example, the spacetime plots (Fig. 6) compare the model and the data. Residual channel friction slightly slows the real slider compared to the ideal slider, so Fig. 6 comparison includes a simple friction term τf=mκsgnϵ˙ added to the right side of the Euler-Lagrange equations [Eq. (5) ] with κ=0.012 N m/kg to best fit the data. Printing the channel in metal rather than plastic should improve performance, as the metal’s larger thermal conductivity will encourage melting and create a more slippery interface.

FIG. 6.

Bead sliding on spinning hoop in the hoop (left) and lab (right) frames. Red spheres are experiment, cyan tubes are simulation, yellow tubes numerically extrapolates bead sliding in the absence of hoop spinning, and black arcs are the generalized Hannay’s shift. Time increases upwards (top) and outwards (bottom). Ellipse radii {a,b}={14cm,11cm}, start angle ϵ0=2.5 rad, start angular velocity ω0=1.4 rad/s, total time T=2.8 s, and residual specific friction coefficient κ=0.012 N m/kg.

FIG. 6.

Bead sliding on spinning hoop in the hoop (left) and lab (right) frames. Red spheres are experiment, cyan tubes are simulation, yellow tubes numerically extrapolates bead sliding in the absence of hoop spinning, and black arcs are the generalized Hannay’s shift. Time increases upwards (top) and outwards (bottom). Ellipse radii {a,b}={14cm,11cm}, start angle ϵ0=2.5 rad, start angular velocity ω0=1.4 rad/s, total time T=2.8 s, and residual specific friction coefficient κ=0.012 N m/kg.

Close modal

A bead sliding on a noncircular hoop “remembers” a rotation of the hoop with a shift of its motion. In the hoop’s frame, the Euler pseudo-force, lesser known relative of the Coriolis and centrifugal pseudoforces, causes the shift. Compared with previous work, generalizing the shift from adiabatic to arbitrary conditions using mathematics and computation greatly enlarges the opportunities to physically observe the shift. A computer simulation reasonably describes an actual experiment using a simple apparatus, where wet ice sliding on a polished metal base approximates the slider and 3D printed plastic channels approximate the hoop. Future simulations might allow angular momentum transfers between the bead and a freely rotating hoop. Future experiments might 3D print the channels in metal rather than plastic to further reduce friction.

The hoop-and-bead anholonomy is an intriguing and memorable phenomenon, a jewel of classical mechanics that connects to a workhorse of quantum mechanics, Hannay’s shift mirroring Berry’s phase. Geometric phases in physics are a significant recent development. Classical mechanics is not exhausted; classical mechanics can still surprise and delight.

See supplementary material for a Mathematica notebook that generates the simulation from the experimental data and compares them.

This work was funded by the National Science Foundation (Grant No. DMR-1560093) and The College of Wooster. We thank Timothy Siegenthaler for creating the aluminum-topped turntable.

This appendix provides an elementary derivation of Hannay’s shift in the adiabatic limit of fast beads and slow rotations.1,12 For a hoop vector r rotating in the lab frame

drdt=drdth+ω×r,
(A1)

so the velocity

v=vh+ω×r
(A2)

and the acceleration

a=dvdt=dvhdt+ω×drdt+dωdt×r=ah+ω×vh+ω×vh+ω×r+ω˙×r=ah+2ω×vh+ω×ω×r+ω˙×r=ah+a+ac+ae
(A3)

includes the Coriolis, centrifugal, and Euler accelerations.

If r locates the bead on the hoop, then the velocity

vh=drdt|h=drdsdsdt=τ^s˙,
(A4)

where s is the bead arc-length separation from a fiducial point, and τ^ is the unit tangent vector. The bead’s tangential acceleration

s¨=as=τ^ah=τ^aτ^aτ^acτ^ae.
(A5)

The lab term

τ^a=0
(A6)

vanishes assuming no friction and a horizontal hoop. The Coriolis term

τ^a=τ^2ω×vh=2τ^ω×τ^s˙=0
(A7)

vanishes mathematically because the box product collapses and physically because the Coriolis pseudo-force is always perpendicular to the hoop. If the hoop rotates perpendicular to its plane, then ωr and the centrifugal term

τ^ac=τ^ω×ω×r=drdsωrωωωr=drdsrω2=drdsrω2,
(A8)

where

2rdrds=ddsrr=ddsr2=2rdrds
(A9)

but

d|r|ds=drdsdrds.
(A10)

Since the box product is invariant under circular shift, the Euler term

τ^ae=τ^ω˙×r=ω˙r×τ^=ω˙r×drds.
(A11)

Hence, Eq. (A5) becomes

s¨=τ^acτ^ae=drdsrω2ω˙r×drds.
(A12)

Integrate the arc-length-separation acceleration s¨ once to get

s˙[t]=s˙[0]+0tdt1s¨[t1]
(A13)

and integrate twice to get

s[t]=s[0]+s˙[0]t+0tdt10t1dt2s¨[t2]=s[0]+s˙[0]t+0tdt1dt1dt10t1dt2s¨[t2]=s[0]+s˙[0]t+t10t1dt2s¨[t2]t1=0t1=t0tdt1t1s¨[t1]=s[0]+s˙[0]t+0tdt1tt1s¨[t1]
(A14)

with an additional integration by parts to generalize the familiar s=s0+v0t+a0t2/2 to variable acceleration.

In the adiabatic limit, the mass slides around the hoop many times for every hoop rotation, so replace quantities that depend on the arc-length separation with their hoop averages

f=1L0Ldsf[s]
(A15)

to get

s[T]s[0]+s˙[0]T+0Tdt1Tt11L0Ldss¨[t1],
(A16)

where L is the hoop length and T is the rotation time. In the inner integral, Eq. (A12) centrifugal term implies

1L0Ldsdrdsrω2=ω2Lr0r0drr=ω22Ld(r2)=0
(A17)

and Eq. (A12) Euler term implies

1L0Ldsω˙r×drds=ω˙1Ldr×r=ω˙2LdA=ω˙2AL,
(A18)

where A is the directed hoop area. Thus, in the adiabatic limit, the Euler pseudo-force produces the entire anholonomy. For one rotation, integrate the Euler term by parts to find Hannay’s shift

δs=s[T]s[0]s˙[0]T2L0Tdt1(Tt1)ω˙[t1]A=2L(Tt1)ω[t1]At1=0t1=T0Tdt1ω[t1]A=2LTω[0]dφA=4πAL,
(A19)

assuming the hoop starts at rest with ω[0]=0 and rotates perpendicular to its plane with ωA. Hannay’s shift depends only on geometry and is proportional to the ratio of the hoop’s area to its circumference. Hannay’s angle is the dimensionless ratio δs/LA/L2.

For an ellipse of major radius a and minor radius b=a1e2, area A=πab, perimeter length L=4aE[e2], and Hannay’s shift δs=π2b/E[e2], where E[] is the complete elliptic integral of the second kind.

For a circle of radius R, area A=πR2, perimeter length L=2πR, and Hannay’s shift δs=2πR=L. Thus, as the hoop spins, one circumference forwards in the lab frame, the bead slides one circumference backwards in the hoop frame, undisturbing its lab frame slide.

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Supplementary Material