The swing that turns a step at a time

The spring that swings when it was only bounced hung a weight on a spring tuned so that its bounce was exactly twice as fast as its swing as a pendulum, pulled it straight down and let go. The bounce died away and the weight began to swing; then the swing died and the bounce returned; and the exchange went on, regular while the energy was small and chaotic when it was large. That essay drew the motion in a plane. It ended by noting a thing a plane cannot show, which Peter Lynch and Darryl Holm explained in 2002: in three dimensions the swing does not come back along its old line. Each swing returns in a new direction, and the direction turns by the same angle every time, in a series of abrupt steps.

The steps are worth seeing for themselves, because what turns them is not a force and not a torque. Nothing pushes the swing round. The plane turns because the motion has gone round a closed loop in a space of its own variables and come back changed — the mechanical form of the phase the phase that is only a shape found in polarised light.

A staircase, not a slope #

The first figure is computed from the motion itself. The spring, its weight and gravity are integrated in three dimensions, with no approximation beyond a small time step, and the direction of the swing is read off one swing period at a time as the axis along which the weight moves most. The weight is started bouncing almost straight up and down, displaced four thousandths of its length to one side and given a sideways velocity of a fraction of a millimetre a second in the other direction.

For the first hundred or so units of time — a unit is the time a pendulum of the spring’s length takes to swing through a radian — the energy is in the bounce, and the little swinging there is points nowhere in particular. Then the swing grows, drawing its energy from the bounce, and as it grows it settles on a direction: about twenty degrees from where the sideways nudge pointed. It keeps that direction while it swings hard. As the energy flows back into the bounce the swing fades, and while it is faint its direction slides; when it grows again it has settled on a new direction, 43.6 degrees round from the old. And again. Nine swings, nine steps of the same size, all turning the same way.

This is unlike every other turning of a pendulum’s plane. A Foucault pendulum’s plane turns steadily, at a rate set by the Earth’s rotation; the ellipse a pendulum turns by itself found a slow, steady turning of a pendulum’s elliptical path, set by its amplitude. The swinging spring’s plane does not turn steadily. It stands still while the swing is strong and jumps while it is weak, so the record of its direction is a staircase.

The spring seen from above #

From above, the motion is a star. Each swing is a narrow line through the point beneath the support, the mass running out and back hundreds of times along it before the swing fades; between swings the mass bounces up and down almost on the spot, hardly visible from above. The second figure draws five swings, and their lines fan round the centre like the spokes of a wheel, each turned by the same angle.

The star turns in the sense set by the original nudge. That sideways velocity gave the weight a small angular momentum about the vertical, and that angular momentum is conserved exactly, whatever the spring and gravity do, because both of them act along lines through the vertical axis and can exert no torque about it. The conservation law a symmetry hands over traced that kind of law to a symmetry: the spring and gravity look the same from every direction around the vertical, so the angular momentum about it can never change. The calculation behind the figures checks that it is conserved to the eighth decimal place.

The sense of the turning holds a small surprise. The nudge in the figures gives the weight an angular momentum anticlockwise, seen from above, and the swing’s plane steps round clockwise — against the sense of the twist, not with it. The ellipse of the ellipse a pendulum turns by itself turns the same way its bob goes round; the swinging spring’s plane steps the other way. Reverse the nudge and the star turns the other way, by the same steps.

Why a twist that cannot change makes the plane turn #

How can an angular momentum that never changes produce a turning that happens only at certain times? The answer is in what the angular momentum means at the two stages of the exchange.

While the weight swings hard along a line, a small angular momentum about the vertical is a small sideways component of its motion: the swing is not quite a line but a very thin ellipse. While the energy is in the bounce, the same angular momentum is carried by a swing that is tiny. A tiny swing with a fixed angular momentum must be a fat ellipse, nearly a circle, because a small motion can carry a given angular momentum only by going round rather than back and forth. So in the middle of each exchange the faint swing is not a line at all but a small loop, and a loop has no direction. When the swing grows again, drawing energy from the bounce, the growth picks out a new line from that loop, and the new line is turned from the old.

The growth that picks it is the swing that is pumped, not pushed: the bounce varies the length of the pendulum twice in every swing, which is the condition for parametric amplification. A parametric pump amplifies a swing along a particular phase of its own motion, and when the swing is a small loop the phase the bounce favours selects a direction. Each pass through the bounce stage is a loss of direction followed by a choice of one, and the choice is made the same way every time.

How long an exchange takes #

The staircase has a second rhythm besides its steps: the time between them. In the first figure the energy goes from bounce to swing and back nine times in fifteen hundred units, about a hundred and sixty units for each full exchange, while a single swing of the pendulum takes a little over six. The exchange is slow because the coupling that drives it is weak. The bounce changes the pendulum’s length by a few per cent, and a parametric pump of a few per cent needs dozens of swings to move the energy across.

That slowness is what makes the steps clean. The swing’s direction is chosen while it is small, over many of its own periods, and then held while it is large, over many more; the turning and the holding are separated in time because the exchange is slow compared with the motion that carries it. Bounce the spring harder and the coupling strengthens in proportion: at a bounce of 0.30, nearly four times larger, the exchange takes about forty-five units, only seven swings. The separation of times that the steps depend on is then nearly gone, which is the blurring the last figure shows. The same separation is what licenses the averaged equations at all: they describe how the slow amplitudes change over many fast swings, and they are exact in the limit where the exchange is infinitely slower than the swing.

A demonstration on a desk #

The tuning the spring needs is easy to state and easy to arrange. A pendulum of length l swings at $\sqrt{g/l}$ and a weight on a spring bounces at $\sqrt{k/m}$, so the bounce is twice the swing when $k/m = 4g/l$. But the weight hangs at rest with the spring stretched by $mg/k$, and the condition becomes a statement about that stretch: the spring must stretch under the weight by a quarter of its loaded length. A soft coil spring, a weight adjusted until the stretch is right, and a nudge that sets it bouncing with a trace of twist are all it takes, and the first step of the staircase appears within a minute.

Lynch, a meteorologist, came to the spring because of the weather. The atmosphere’s large-scale motions include fast gravity waves and slow rotational flows, and the way energy passes between them through resonant triads of waves has the same mathematical form as the bounce and the swing. The swinging spring is the smallest model of that exchange, simple enough to solve and visible enough to watch, and the stepwise turning of its plane is a property its weather-scale relatives share.

A turn set by a ratio #

The size of each step depends on how much twist there is. The third figure repeats the calculation with the same bounce and different sideways nudges. With a twist of 0.12 × 10⁻⁵ the swing turns nine degrees an exchange; with five times as much, forty-four; with eight times as much, sixty-seven. With no twist at all the motion stays in a plane and the swing returns along its old line every time, which is the planar spring of the earlier essay. The steps grow with the twist and, at small twist, in proportion to it.

But the twist is not the whole of it. Lynch and Holm’s analysis of the averaged motion — the equations that follow the slowly varying amplitudes of the bounce and the swing, rather than the fast motion itself — found that the averaged system has two conserved quantities, the energy of the oscillation and the angular momentum about the vertical, and that the step is set by their ratio.

The fourth figure tests that directly, against the full motion rather than the averaged equations. Two springs are bounced to different heights, the second to 0.6 of the first, so its oscillation carries 0.36 of the energy; their sideways twists are scaled so that each carries the same angular momentum in proportion to its energy. Their twists are very different, and plotted against the twist their steps would lie on two different curves. Plotted against the ratio, the steps fall on one curve, pair by pair, within a degree and a half over the whole range.

That collapse is the signature of a geometric effect. The step depends only on a shape — the ratio of two conserved quantities, which fixes the path the averaged motion traces in its own space of variables — and not on how big the oscillation is. In the averaged picture the state of the spring at any moment is a point on a sphere whose coordinates are combinations of the bounce and swing amplitudes, and each exchange carries the point once round a closed loop on that sphere. The swing’s plane comes back turned by an angle made of two parts, one set by the timing of the exchange and one equal to the area that loop encloses — a geometric phase, the same kind of statement as the polarisation essay’s: a system taken round a closed path in the space of its states returns with a phase it did not have, equal to a solid angle. Here the phase is the direction of a pendulum’s swing, and it can be seen with the naked eye.

Where the steps blur #

The averaged equations hold while the exchange is slow compared with a single swing, which is to say while the motion is small. Bounce the spring harder and the exchange speeds up, the bounce and the swing interfere within a single period, and the averaging fails. The last figure follows a spring bounced nearly four times harder. Its swing’s direction no longer holds still between steps; it slides during the swings as well, the steps lose their fixed size, and successive steps differ by twenty degrees and more.

That is the way into the chaos the spring that swings when it was only bounced found in the planar spring at high energy. The integrable, averaged motion is the small-amplitude limit of a system that is not integrable. As the energy rises the regular structure loosens, first blurring the steps and then dissolving them, until the swing’s direction wanders with no pattern at all, and two springs started a hair apart turn in entirely different directions. The error that doubles on a schedule measured how fast such a separation grows once the motion is chaotic; the steps are what the motion looks like on the regular side of that boundary.

Where the model holds #

The figures treat the spring as massless, perfectly linear and perfectly tuned: its bounce exactly twice its swing. A real spring has mass, which shifts both frequencies; its stiffness varies slightly with stretch; and a tuning a few per cent off the two-to-one ratio slows the exchange and changes the step, until, far enough off, the bounce and the swing barely exchange at all. Real springs are also damped, by internal friction and by the air, and the steps survive only as long as the exchange does — tens of exchanges, typically, in a careful demonstration. The integration itself is exact enough not to matter: it conserves the angular momentum about the vertical to the limit of arithmetic and the energy to a part in a hundred thousand over the longest runs drawn.

What the figures cannot show #

The figures show the swing’s direction and the size of each step. They cannot show the space in which the step is a solid angle: the sphere of averaged states on which Lynch and Holm drew the exchange as a closed loop. Drawing it would mean drawing the averaged equations, which the figures were built to avoid relying on; the collapse of the fourth figure is the evidence that those equations describe the real motion, not a picture of them. The figures also cannot show why the step at a given ratio has the value it has. The figures measure it; the formula that predicts it from the ratio belongs to the averaged theory and involves integrals that have no simpler form.

Still open: what the steps do when the spring is driven #

The free spring’s steps are explained. A swinging spring driven periodically from its support, or damped, or coupled to a second spring — the forms in which the same resonance appears in engineering, in coupled oscillators and in the atmospheric waves that the spring was once used to model — is a different system, and whether its swing still turns in steps, how a drive locks or unlocks the turning, and how the geometric phase of the free system survives damping are questions studied model by model. The analogous effect in other two-to-one resonances, from molecules whose vibrations exchange energy to the coupled waves of fluids and plasmas, is expected on the same grounds and has been found in some, and how robust the steps are to the imperfections of each system is not settled.

The habit worth carrying away is to look for the quantity a motion cannot change, and then for what that quantity must look like when the motion is small. A swinging spring tuned so its bounce is twice its swing passes its energy back and forth, and in three dimensions an angular momentum about the vertical that nothing can change turns the swing a fixed step each time the energy passes through the bounce — 43.6° for a twist of 0.6 × 10⁻⁵ at a bounce of 0.08, the same step for any spring with the same ratio of twist to energy, within 1.3° — because a small swing carrying that angular momentum must be a loop, and a loop forgets its direction. The plane is turned by geometry, not by any torque.