The stone that turns back against its spin

The egg that keeps turning after it is stopped showed how a spinning object can be stopped with a finger and still turn afterwards, because what was stopped was only the shell. A rattleback is a harder case, since nothing inside it can store anything. It is a single piece of stone, wood or plastic, about the size of a large finger, flat on top and curved underneath like an ellipsoid. Set it on a table and spin it clockwise, and it spins steadily, slowing down gradually. Spin it anticlockwise, and it turns for a moment, starts to shudder at one end, stops dead, rocks and then spins back clockwise. It looks like a magic trick. It is sold as a toy, and archaeologists gave it its older name, the celt, after the stone axe-heads of similar shape that were first observed to behave like this by a naturalist called Walker in 1896.

It is difficult to imagine that this behaviour does not involve something hidden, and it has been tested often. Sawn open, a rattleback has no mechanism. With uniform plastic there is no moving part at all. The effect comes from the bare shape and the way it rolls.

What the sum has to be #

The mechanics of a rolling object are subtle, since a body rolling without slipping cannot be described by a potential energy and a list of coordinates in the usual way. Its constraint connects velocities without connecting positions: how the stone can move next depends on where it is but the route by which it got there leaves it in a different orientation. Least action, except that it is not least and its sequel relied on constraints of a simpler kind, and they do not apply here without care. For present purposes, it is enough to write Newton’s laws, in the body’s own axes, for the rotation about the centre of mass and for the centre itself, with the contact force unknown, and then to eliminate that force using the fact that the point of contact does not slide.

Call $\mathbf{r}$ the vector from the centre of mass to the contact point, in body axes. If the stone does not slip, the centre’s velocity is $-\boldsymbol{\omega}\times\mathbf{r}$. Then differentiate that, substitute for the force in the equation for rotation, and rearrange to obtain

$$\left(\mathsf{I} + m\left(r^2\mathsf{E} - \mathbf{r}\mathbf{r}^{\mathsf T}\right)\right)\dot{\boldsymbol{\omega}} = m\,\mathbf{r}\times(\mathbf{a}_0 - g\,\mathbf{d}) - \boldsymbol{\omega}\times\mathsf{I}\boldsymbol{\omega},$$

with $\mathbf{d}$ the downward direction in body axes, and $\mathbf{a}_0$ collecting the terms that do not involve the angular acceleration. It is a three-by-three solve at each moment, with no approximation for small tilts. The contact point follows from the shape: on an ellipsoid with semi-axes $a$, $b$, $c$, the point whose outward normal is $\mathbf{d}$ has a closed form. A stone with those axes, an inertia tensor, a centre of mass below the centre of curvature, and one small extra parameter is everything needed.

That extra parameter is the angle $\delta$ between the axes of the inertia tensor and the axes of the curved base. The body’s mass distribution is a little rotated about the vertical, relative to the shape of its underside. In a commercial rattleback, this is done with a pair of small weights placed off the long axis, or the top is cut slanted, or the curve itself is ground at an angle. In the stones that started the study, it arose by accident in grinding.

The result is that the behaviour appears directly, with nothing having been put in by hand. The stone, started level, spins for a couple of seconds, and its spin falls to zero. It is doing nothing but rock. Then it picks up speed in the opposite direction, up to about the same size as it started with. There is no friction in this integration, in the sense of any energy loss. The only role of the contact is to prevent slipping. So the reversal cannot be explained by dissipation, as many first explanations claim.

Angular momentum is not the thing that is kept #

The first reaction to such a graph is that angular momentum has gone somewhere, in the spirit of the quantity that survives a change of shape. It has not gone anywhere, in the sense that it was never conserved. A rigid body on a table is acted on by the contact force, which is applied at a point away from the centre of mass and so exerts a torque. Only for a very special symmetric object does the vertical component of that torque average out. So the Earth can take up or give the stone any angular momentum, by an amount far too small for it to notice. The angular momentum of the Earth and the stone together is conserved, as it must be, and in the reversal of a toy the Earth’s rotation changes by a very tiny fraction.

Energy, in contrast, is conserved exactly by the rolling constraint, since the contact force does no work on a point that is not moving. The energy check in this calculation is one of its main tests, and it holds to within two per cent over forty seconds. A rolling constraint does no work for the reason the floor that does no work gave for a pushing floor. So what the stone does must be understood as a passage of energy between motions, spin to rocking and rocking to spin.

The picture has a character that a dissipative explanation would never produce. A large pitching motion grows, reaches its peak just as the spin hits zero and then dies away as a spin in the other direction grows. With no friction, the sequence would repeat for ever, passing from one sense to the other. A real stone rolls with some small rolling resistance and comes to rest after a reversal or two, so it never gets to complete this, and it is only the first reversal that people see.

Why one way differs from the other #

Mirror symmetry says the stone can treat the two senses differently only if it is not itself mirror-symmetric. The angle $\delta$ is that lack of symmetry. It makes the stone chiral: reflected in a mirror, it becomes a different stone, as a left hand is different from a right. Toy rattlebacks of this kind are sold with a preferred direction fixed at manufacture, and a mirror-image one can be cast from an inverted mould.

This was the finding most at odds with the popular account. The popular statement is that a rattleback has a good direction and a bad direction. In this conservative model there are only a fast direction and a slow direction, a ratio of about twelve with the parameters used. The other direction will also reverse in the end, after more than half a minute, when a real stone would have run down to nothing. Some rattlebacks of careful design have been made which show reversal in both directions, and it can be shown with a demonstration on a very smooth surface. So the rule in the toy shop belongs to the toy and to the dissipation, not to the mechanics.

The calculation that first made sense of this was by Bondi in 1986, using a small-amplitude expansion. The way to see what is going on is as a coupling. Rolling about the long axis and pitching about the short axis are two oscillations, at very different frequencies. Rolling back and forth is fast and pitching is slower, or the reverse, depending on the shape. A spin about the vertical carries those motions round, and with the axes misaligned, a small roll produces a small torque about the vertical, and a spin in one sense pumps pitching, like a parent pushing a swing at the right moment. So a spin one way feeds energy into pitching at a high rate. Spinning the other way feeds rolling, at a much lower rate.

Tap it and see #

That coupling can be read in the opposite direction. If spin drives rocking, then rocking drives spin. So a stone at rest, tapped lightly, should begin to turn with no spin given to it at all.

Tap a rattleback on one end, and it rocks up and down and starts turning in one direction. Tap it at the side and it rolls, and starts turning the other way. A child with a toy can verify this in a few seconds, and it is often more convincing than the reversal. The figure uses the same equations as before, with nothing changed except the starting velocities. The rolling case shows a rapid fine shudder on top of the trend, and that matches the visible rattle that gives the object its name. Moffatt and Tokieda in 2008 reduced this coupling to a compact model in which the two amplitudes and the spin drive each other through quadratic terms, and the same picture emerges: pitch and roll work the spin in opposite senses with a strength proportional to the misalignment.

The same logic says something about why pitch is more efficient. In the integration the spin from a pitching tap is the steadier of the two. It is a large, slow oscillation that delivers torque with a consistent sign, while roll is a smaller, rapid oscillation with a tiny effect per cycle. Together they explain the asymmetry between the two reversal times. A strong instability exists only for the sense of spin which feeds the mode that delivers spin back most effectively.

One degree is all it takes #

A cheap test of whether the angle is truly the cause is to vary it and nothing else.

The behaviour vanishes at zero angle. The stone with aligned axes has no reversal in either direction, for as long as the calculation runs. It is a very smooth, very ordinary body of revolution for these purposes. At an angle of barely a third of a degree the reversal takes about ten seconds, and larger angles shorten it. It is a simple test that the integration passes, and a useful warning. The mass distribution of a stone a few centimetres long needs to be misaligned from its geometry by only about a degree for the whole effect to be there. It is hard to see that by eye. A toy maker can adjust the angle with two small weights and the difference between the ends and the sides can be undetectable to a person’s fingers.

The toy has also a less obvious requirement in the numbers. The stone must have its centre of mass below the centres of curvature of its base, so that it sits stably on the table, and the two principal curvatures must differ noticeably. In this integration the base has semi-axes of 20 cm by 3 cm in plan and a depth of 2 cm, with the centre of mass 1 cm below the centre of the ellipsoid. All of that goes into the rates, and some shapes have reversal rates too slow to see. Finding good proportions took early manufacturers some experiment.

Three toys that are not the same trick #

It is natural to file the rattleback alongside other objects that do surprising things when they spin. The comparison shows what kind of effect this is. A tossed book or a tennis racket, spun about its middle axis, tumbles: the axis that will not hold showed that this needs only the free rotation of a rigid body with three different moments of inertia, with no contact and no forces. Energy and angular momentum are both conserved. The body’s axis wanders because a steady spin about the middle axis is an unstable solution, and it does so whichever way it is spun. A rattleback’s chirality is absent there.

A tippe top inverts itself when spun, standing on its stem. The axis a leak of energy chooses discussed how a slow loss of energy favours spin about a particular axis, and the tippe top depends on sliding friction at its contact in an essential way. With no sliding there is no inversion at all. Its centre of mass rises in the process, and the energy comes from spin. In a tippe top, then, dissipation does the work, while in a rattleback it does not, and this difference has often been missed in explanations. A spinning hard-boiled egg that stands up on its long axis is similar to the tippe top: it needs friction, as Moffatt and Shimomura showed. The egg in the egg that keeps turning after it is stopped has a liquid inside that holds angular momentum, and a boiled egg does not.

So there are at least three kinds of trick: a free-body instability, a friction-driven inversion and a chiral, rolling-constraint energy exchange. Only the third is a statement about a constraint, which can be neither integrated away nor written as a potential. The push that comes out sideways depends on a similar structural feature. A gyroscope’s response to a torque is at right angles to it, so simple intuitions about pushing fail. Here, the constraint at the contact point is what supplies sideways forces. Those forces do no work, but they direct energy between motions, with a preference that depends on the sense of the motion.

Could such a thing be scaled up? A planet is not a rattleback. But an elongated satellite rolling on nothing cannot do it either, as the contact is required. The object’s scale is limited by how smooth a surface can be made, and a person could in principle make a rattleback the size of a table. A curious use has been suggested in the opposite direction: small chiral bodies, shaken on a vibrating plate, tend to rotate in one sense, in a kind of microscopic rattleback effect. Those have been used to demonstrate a directed motion on a shaking table without any net force. In that form, a thing with no moving parts converts undirected shaking into directed turning, with the same underlying requirement that the body has no mirror symmetry.

What the integration leaves out #

The most important omission is any dissipation. Real stones have rolling friction and a little sliding, and in fact there is a long-standing and now resolved dispute about whether friction is essential. Some early explanations attributed reversal entirely to friction, and those were shown to be wrong by showing, as here, that a frictionless rolling stone reverses. Friction nevertheless shapes what is seen. It ensures that a real stone’s rocking dies out and the stone settles to the preferred spin, and with a large enough spin, sliding may enter and damp it faster. Simulations with realistic rolling and sliding friction, in the work of Garcia and Hubbard and later authors, match high-speed films quite well, once such terms are included.

The results depend on the values of the inertias and the semi-axes, and those here are a representative choice rather than a measurement of one stone. So the reversal times of 2.8 s and 33.8 s and the ratio of twelve are not general numbers. Many rattlebacks reverse faster, particularly thin plastic ones; a good one takes less than a second. What is general is the structure: no reversal without misalignment, opposite senses from pitch and roll, a fast direction and a slow one, and exchange through rocking.

Energy conservation is checked to within two per cent, and that is the size of error accepted here, which in a long enough run will show up as a small drift in amplitude. The surface of the stone is taken as a perfect ellipsoid, the table as a perfect plane, and neither gives way. A soft table would add a dissipation and change its rates, as the grip that needs a little slipping showed that compliance can change matters at a rolling contact.

Still open: whether a rattleback can be chaotic #

With three coupled modes and quadratic couplings, the system does not have to remain regular. Borisov, Mamaev and their colleagues studied long integrations of the conservative rattleback and found that with suitable parameters the reversals become irregular, and the state wanders in a pattern which they have compared to a strange attractor. That is unusual in a system with no ordinary dissipation, and it relates to a peculiar property of non-holonomic systems: they conserve energy but do not generally conserve phase-space volume, so they can behave as if dissipative without losing anything. Such strange attractors have been found numerically for this and related bodies. Whether those could be seen in a real object before friction takes over is not established. Real stones do not rock for long enough. So the toy may have behaviour of a type that its friction hides, and the only place where it has been observed is a computer.

A rolling stone whose inertia axes are misaligned from the curvature of its base by about a degree couples pitch and roll to spin in opposite senses, so it reverses from one spin direction in 2.8 s and from the other in 33.8 s, with energy conserved throughout and no reversal at all if the angle is zero. What looks like a broken conservation law is a torque from a table that nobody noticed. The habit worth keeping is to ask what is actually conserved when something appears to contradict a law: here energy is, angular momentum about the centre never was, and the surprise came from assuming the wrong one.