The ellipse a pendulum turns by itself

Hang a heavy bob on a long wire from a pivot that lets it swing in any direction, pull it aside and let it go, and it swings back and forth in a plane. Give it the slightest sideways nudge as it is released and it swings instead in an ellipse. Watch the ellipse for a few minutes and it turns, slowly, in the same sense as the bob is going round it. Nothing is turning the pendulum. The pivot is fixed, the room is still — and even in a calculation from which the Earth’s rotation has been removed entirely, the ellipse turns. It is one of the small consequences of the lie that makes the pendulum simple: a swing’s period does depend on its size, and a pendulum allowed a second dimension shows that dependence as a rotation.

Why a small swing closes #

A pendulum free to swing in any direction is a bob moving on the inside of a sphere, and for a small swing the sphere near its lowest point is flat enough to be treated as a bowl with a parabolic cross-section. The restoring force is then proportional to the displacement and points back towards the centre, whatever the direction of the displacement. The motion east–west and the motion north–south are two independent harmonic oscillations with exactly the same frequency.

Two oscillations of the same frequency at right angles combine into an ellipse, whose shape depends only on their amplitudes and the phase between them: in step they make a straight line, a quarter of a cycle apart they make an ellipse with axes along the two directions, and every other relation makes a tilted ellipse. Because the frequencies are identical, the phase never drifts, and the ellipse closes on itself every period and repeats for ever. It is the same closed orbit that a linear spring gives any central-force motion — one of the two force laws, with the inverse square, for which every bounded orbit returns to its starting point.

That closure is a property of the approximation, and it is fragile in a specific way. It requires the east–west and north–south frequencies to be equal and independent of amplitude. The first is guaranteed by the symmetry of the pivot. The second is the small-swing approximation, and a pendulum’s period grows with its swing — by about one per cent at twenty-three degrees. A real spherical pendulum is a slightly softer bowl than a parabola, and the question is what that does to an ellipse.

A flat swing and a circle do not keep the same time #

The most direct way to see the mechanism is to compare the two extreme ellipses: the degenerate one, a flat swing through the lowest point, and the round one, a circle traced at constant height.

The flat swing slows as its amplitude grows, because the restoring force is $mg\sin\theta$ rather than $mg\theta$ and falls short of proportional at large angles; the exact period is an elliptic integral. The conical pendulum does the opposite. A bob going round a horizontal circle at angle $\theta$ is held on it by the horizontal component of the rod’s tension, and working through the geometry gives a period of $2\pi\sqrt{L\cos\theta/g}$, shorter than the small-swing period, because the bob is higher and the effective length of the pendulum is $L\cos\theta$.

An ellipse is a motion that goes both along its long axis, like a flat swing, and round, like a circle. For a small ellipse both components have the small-swing period and stay in step. For a larger one the in-and-out motion runs slightly slow and the going-round runs slightly fast, and the bob arrives at the far end of its long axis a little after it has already gone round to where that far end used to be. The far end is therefore a little further round each time. That is the precession, and its sign follows from which of the two runs fast: the going-round wins, so the ellipse turns in the sense of the going-round.

The mismatch vanishes in both limits, and the vanishing is why the effect depends on the ellipse’s area. A flat swing has no going-round, so there is nothing to be out of step with and no axis turns. A perfect circle has no in-and-out, so there is no axis to see turn. Only an ellipse with both length and width shows it, and the rate is proportional to both.

Airy’s rate #

George Airy, then the Astronomer Royal, worked out the rate in 1851, when Foucault’s demonstration had turned half of Europe’s physicists into pendulum builders and the problem of stray elliptical motion had become urgent. For an ellipse with semi-axes $a$ and $b$, small compared with the length $L$, the long axis turns at an angular rate

$$\Omega_{\text{Airy}} = \frac{3}{8}\,\omega\,\frac{ab}{L^2},$$

where $\omega = \sqrt{g/L}$ is the pendulum’s angular frequency. Per half swing — from one end of the ellipse to the other — that is $3\pi ab/8L^2$ radians.

The integration and the formula agree closely where the formula should hold, and the figure shows the two properties that matter in practice. The turn is proportional to $b$, so a narrower ellipse turns more slowly in exact proportion, and a perfectly flat swing does not turn at all. And at fixed shape the turn grows with the square of the swing.

The pull above the line at large swings is the formula’s domain of validity made visible. Airy’s rate is the first term of an expansion in the size of the swing, and at fifty degrees the second term is worth a seventh of the first. The integration does not care; it is the exact motion of a bob on a rigid rod, with the rod’s tension solved at every step from the requirement that the bob stay at distance $L$ from the pivot, and it simply keeps turning faster than the leading term says.

The same turning, much smaller #

At the small swings that careful pendulums use, the effect is small but relentless.

A fifth of a degree per half swing sounds negligible, and a pendulum a metre long makes a half swing every second. The ellipse therefore turns by about twelve degrees a minute — more than seven hundred degrees an hour. The Earth’s rotation turns a pendulum’s plane at a rate that is, at Paris, a little over eleven degrees an hour. A metre-long pendulum released with an ellipse of this shape would show the Earth’s rotation buried under a turning sixty times faster, in the same sense for one direction of going round and the opposite sense for the other, and nothing in the appearance of the swing would say which was which.

What this does to Foucault’s demonstration #

Foucault’s pendulum turns because the Earth turns underneath it: the plane of swing stays fixed relative to the stars, apart from the local vertical, and the floor rotates beneath it at the Earth’s rate multiplied by the sine of the latitude. It is the same rotation of the horizontal plane that makes a body left alone in the ocean trace a closed inertial circle in exactly half a pendulum day. For the demonstration to show that rotation, the pendulum’s own turning has to be small by comparison.

The tolerance falls steeply with length, and the reason is in the formula. Airy’s rate goes as $\omega ab/L^2$, and $\omega$ itself grows as $L^{-1/2}$, so for a swing that is a fixed fraction of the length the rate goes as $L^{-1/2}$ times the ratio $b/a$ — and for a swing of fixed size, as $L^{-5/2}$. Foucault’s choice of a 67-metre wire in the dome of the Panthéon was not showmanship. A long pendulum swinging slowly is exactly the one whose own turning is smallest, and even so a release had to be done by tying the bob aside with a thread and burning the thread, so that no sideways push was given at all.

For a short pendulum no release is careful enough. An ellipse a twentieth of a millimetre wide on a pendulum a metre long is a sideways speed of a fraction of a millimetre per second at release, and it grows during the swing from any asymmetry in the suspension. Every short Foucault pendulum that works therefore does something to kill the ellipse as it forms. The oldest remedy is Charron’s ring, a collar the wire rubs against at the top of each swing, which damps the sideways motion preferentially because the wire touches it most when the ellipse is widest. Modern demonstration pendulums use electromagnetic drives that push the bob only along its line of swing, restoring energy without adding width. Both exist because of Airy’s term.

The effect has another consequence that a demonstration can exploit rather than suffer. Because Airy’s turning is in the sense the bob goes round, and the Earth’s is fixed in sense, a pendulum that develops a slight ellipse in one sense turns a little faster than the Earth alone would turn it, and one whose ellipse goes the other way turns a little slower. The ellipse itself develops partly because of the Earth’s rotation, which gives a swing launched in a plane a slight sideways component. The two effects are coupled, and the careful analysis of long pendulums since the nineteenth century — some of it done by people trying to use pendulums to measure latitude — has had to treat them together.

Two nearly equal frequencies, in other clothes #

The mechanism is general enough that it is worth seeing in the other places it turns up, because each one makes a different part of it obvious.

A pivot that is not quite symmetric. Suppose the suspension is very slightly stiffer north–south than east–west, so that the two small-swing frequencies differ by a part in ten thousand. A swing launched flat along a diagonal is then the sum of two oscillations drifting slowly out of phase, and the drift turns the flat swing into an ellipse, then into a flat swing along the other diagonal, then back. The swing’s shape beats between its two normal modes, exactly as two coupled pendulums hand their energy back and forth, at the difference of the two frequencies. An asymmetric pivot does not turn an ellipse; it manufactures one, and then Airy’s term turns it. The two effects look alike on the floor and have different signatures in time, which is how a careful builder tells them apart.

An orbit. A planet’s ellipse closes because the inverse square has a conserved vector pointing at the perihelion, and the two-dimensional harmonic oscillator — the small-swing pendulum — has an analogous hidden conserved quantity, a tensor rather than a vector, that fixes the orientation of its ellipse. Any perturbation that spoils the exact form of the force spoils the conservation, and the ellipse turns. For a planet the spoiling term comes from the other planets or from relativity, and the turning is a rosette with a step per orbit; for the pendulum it is the $\sin\theta$ that differs from $\theta$, and the step is Airy’s. The pendulum is the version that fits on a table and can be watched in an afternoon.

A more careful clock. The spherical motion also contaminates measurements that use a pendulum for something other than showing the Earth turn. A reversible pendulum used to measure gravity without measuring a length relies on the period of a flat swing; any ellipticity adds the going-round’s slightly different period to the measurement. And the cycloidal cheeks that make a flat swing’s period independent of its size only work in one plane. A pendulum isochronous in a plane is not isochronous in every direction, so no suspension removes the mismatch between the flat and the circular periods, and Airy’s turning survives any cure for the amplitude dependence of a flat swing.

The common thread is a pair of frequencies that the ideal model makes exactly equal and the real system makes nearly equal. Whenever that happens, the motion that should repeat instead drifts slowly through a family of shapes or orientations, at a rate set by the small difference, and the drift is often far easier to measure than the difference itself. Airy’s turning is a measurement of the anharmonicity of the pendulum, made by watching an ellipse rotate rather than by timing a swing to a part in a thousand.

Where the idealisation is still an idealisation #

The rod is rigid and massless and the pivot is perfect. A real wire stretches slightly under the changing tension, and its stretch is larger at the bottom of the swing, which adds its own small change to the period. A real pivot is never perfectly symmetric, and a pivot that is slightly stiffer in one direction than the other gives the north–south and east–west motions different frequencies, which turns a flat swing into an ellipse on its own. Asymmetry of the suspension is usually a larger source of stray ellipticity than the release.

There is no air. Air drag damps the swing and damps the width of the ellipse at a slightly different rate, which changes $b/a$ over a long run and with it the Airy rate. A demonstration pendulum running for a day loses much of its amplitude, so the rate is not constant through the day.

The Earth does not turn in these figures. Every integrated path here has a fixed pivot in a non-rotating frame, which is what isolates Airy’s effect. Adding the Earth’s rotation adds a slow uniform turning of the whole pattern, and in the rotating frame a Coriolis force acts on the swinging bob. For small swings the two turnings add; for larger ones they interact.

The swing is followed for a dozen periods. Over that time the integration’s own error is far below the turning being measured. Over a day of swinging, the real limits are the pendulum’s, not the arithmetic’s.

The height a plan view flattens #

The figures show the bob’s path projected onto the floor, which is where the turning is visible, and they hide the third dimension that causes it. The bob moves on a sphere, so it rises as it moves outward, and its height at the far ends of a large ellipse is part of why the in-and-out motion is slow there. A plan view flattens that away, and the conical-pendulum argument above has to supply what the drawing cannot.

Nor does the plan view carry time. A dot for each half swing marks where the far end of the ellipse has got to, but not how long it took, and at large swings the period has itself grown by several per cent; the turning per half swing and the turning per second are different numbers, and only the first is drawn.

Still open: how to keep a short pendulum’s swing flat without disturbing what it measures #

The design of Foucault pendulums is an old problem that is not finished. Every method that suppresses ellipticity acts on the pendulum, and anything that acts on the pendulum can also act on the direction of its plane of swing, which is the quantity being displayed. A Charron ring that is not perfectly centred introduces its own preferred direction; an electromagnetic drive whose coils are slightly asymmetric does the same; and a pendulum that has been made to swing flat by force is no longer a free pendulum whose turning shows only the Earth. How accurately a short pendulum with active control can measure the rate of turning, rather than merely display its sign, depends on the details of the control in ways that have been analysed for particular designs but not in general. Pendulums built to measure the rate precisely have remained long ones.

The next question the spherical pendulum raises is what happens when the thing being swung is not a point. A rigid body swinging on a pivot has rotation of its own about its axis, and a spinning bob on a string couples its spin to its swing in the manner of a gyroscope — which is where the pendulum turns into the top. The habit worth carrying from here is to look for two nearly equal frequencies in any motion that should repeat. An orbit closes only when its two frequencies are exactly equal, and any dependence on amplitude, however small, turns a closed ellipse into a slowly turning one — which is Airy’s pendulum, Mercury’s perihelion and every precessing orbit in between.