The top that nods before it settles

Hold a spinning top at forty degrees from the vertical and let go. What happens next, for about a fifth of a second, is not in most accounts of the subject. The axis does not begin to travel sideways. It falls — visibly, as though the spin had done nothing at all — then swings sideways, rises back to exactly the angle it was released at, stops, and falls again. The path its tip draws is a row of cusps hanging under a circle, and the circle is nowhere in the motion. It is the average of it.

The four panels are the same top, the same field, the same tilt and the same spin. What separates them is one number: the sideways rate the axis was given at the instant of release. Only one value of it produces the smooth circle, and it is not zero. So the picture that is drawn to explain why a torque turns the axis instead of toppling it shows a motion that requires the top to be launched with precisely the right flick — which nobody does, and which no account of the subject mentions.

The word for the extra motion is nutation, from the Latin for nodding, and the name is older than the mechanics: it was given to a small periodic wobble in the direction of the Earth’s axis long before anybody wrote down the equations of a top. That the two are the same phenomenon is not obvious and is the sort of coincidence of naming that turns out not to be a coincidence at all. Both are a fast oscillation of a spinning body’s axis about the slow path a torque is dragging it along, and in both the fast part was noticed first because it is periodic and the slow part was noticed first because it is large.

What the released top is actually doing #

The reason a released top falls is easier to state than the reason it comes back up. At the moment of release the axis has no sideways motion, so the angular momentum about the vertical is whatever the spin about the axis contributes and nothing more. Gravity’s torque is horizontal and at right angles to the axis, so it does not change the vertical component of angular momentum at all. That component is therefore fixed for the whole motion, and it is what forces the return.

As the axis falls, its own spin angular momentum tilts further from the vertical and contributes less to the vertical component. Something has to make up the difference, and the only thing available is sideways motion of the axis itself. So the fall generates the sideways swing, and the sideways swing is what the observer calls precession. The two are not separate phenomena with the second explained by the first. They are one motion in which a conserved quantity is being traded between two ways of carrying it.

The return is then a matter of energy. The top has lost height, so it has gained kinetic energy, and the extra kinetic energy is in the sideways swing. But a sideways swing at that tilt carries more vertical angular momentum than the budget allows once the axis has fallen far enough — so the fall stops, reverses, and the axis climbs back to exactly its starting angle with exactly zero sideways rate, which is where it began. From there the whole thing repeats. The cusps are not a decoration on the precession; they are the shape of a motion in which two conserved quantities are being satisfied at once, which is the same accounting that produces the phase portrait of a pendulum and for the same reason.

There is a subtler point in the panels worth pausing on. The path that loops — the one where the axis briefly travels backwards — comes from launching the top with more sideways rate than the steady value, not less. Overshooting the steady precession makes the axis rise, and rising costs it sideways rate until the rate reverses. Undershooting makes it fall. The steady circle is the knife edge between the two, which is a good reason to be suspicious of any account in which it is the natural motion.

The nod has a frequency, and gravity is not in it #

The obvious guess about the nod is that it is a gravitational oscillation: the top falls, gravity pulls it back, and the frequency is set by the strength of the field, as it is for a pendulum. The guess is wrong, and the figure below is the test.

The measured period follows the free body’s own wobble rate, and the agreement improves as the spin rises. A symmetric body spinning about an axis that is not exactly a principal axis wobbles at I₃ω₃/I₁ with nothing acting on it — no gravity, no pivot, no field. That is the free rotation of a body whose axis will not hold still, and it is the same rate. Gravity, in the top, does not cause the nod. It only sets the tilt the nod is about and the depth it goes to.

This is worth stating plainly because it inverts the usual hierarchy. A top is normally introduced as a gravitational problem with a spin in it. The measurement says the opposite: it is a free-rotation problem with a small gravitational perturbation, and the perturbation is small in exactly the sense that Mgl is small compared with I₃²ω₃²/I₁. At 2,400 revolutions a minute for this rotor that ratio is a few per cent, which is why the nod is nearly the free rate and not exactly it.

The consequence for measurement is direct. The Earth has a free nutation of its own, the wobble of its rotation axis in the body, with a period near fourteen months. That period is not the rigid-body value of about ten months, and the difference is the whole content of the observation: the Earth is not rigid, and its elastic yielding lengthens the free wobble by a third. A quantity that seems to be about the shape of a body turns out to be a probe of how stiff it is, because the frequency is a ratio of moments of inertia and the moments respond to the deformation.

Why nobody sees it #

If the nod is in every released top, and it is, the question is why the textbook picture survived. The answer is in the amplitude rather than the frequency.

The depth of the nod goes as the inverse square of the spin while its frequency goes as the first power. A top at a few hundred revolutions a minute nods through a visible angle, slowly, and anyone watching sees it. The same top at six thousand nods through a fraction of a degree, twenty-five times a second, which is below what an eye resolves in both space and time. It has not stopped happening. It has stopped being observable, and the observation that replaced it — a smooth circular precession — is an average over a motion nobody can see.

That is a specific and common kind of mistake, and it is worth naming. A model is not wrong when its predictions match: the slow drift of a fast top really is the slow root of the quadratic, to a fraction of a per cent. It is wrong about what is happening, and the difference matters as soon as anything asks a question the average does not answer — what the top does in its first tenth of a second, what happens if it is knocked, why a gyroscope’s output has a fast transient in it that the specification calls noise. The same distinction runs through the approximation that lets a pendulum have one period and through every place where a picture that predicts well describes badly.

There is also a real dissipative story, and it is the one that actually removes the cusps from a demonstration. Friction at the pivot damps the fastest motion in the system first, and the nod is the fastest motion by a factor of thirty or so. Within a second or two of release the nod is gone and the slow precession is left, which makes the steady solution the attractor rather than the natural motion. Dissipation selects it. Without friction, an ideal top released from rest nods for ever, and every account that draws the circle has quietly assumed a loss it does not mention. Compare the three ways an oscillator can come to rest, where the damping is the whole subject rather than an unstated ingredient.

The top that stands up #

The most familiar top of all is one that does none of this. Spun hard and set upright, it stands with its axis vertical, apparently motionless, until it slows down and suddenly begins to wobble and falls over. The name for that state is a sleeping top, and it is the limiting case of everything above: the tilt is zero, the torque is zero, and there is nothing to precess.

Whether the vertical position is stable is decided by the discriminant of the quadratic that gives the two precession rates. Real roots exist only when I₃²ω₃² is at least 4I₁Mgl cos θ, and at the upright that is a plain statement about the spin: above a critical rate the top is stable, below it the vertical position is not a minimum of anything and the smallest disturbance grows. For the rotor drawn here the critical rate is a few hundred revolutions a minute; for a child’s top it is a few hundred too, which is why they are spun by hand rather than by machine.

What makes this worth drawing is the shape of the failure. Friction removes spin steadily, so the state point crawls downward across the boundary at a rate that has nothing dramatic in it. The behaviour on the two sides is completely different. Above the line the disturbance from a knock oscillates and dies away; below it the same disturbance grows exponentially. So a sleeping top does not wobble a little more each second as it slows. It stands, and stands, and then wakes — and the waking is fast, because it is an exponential growth that started from whatever tiny disturbance was already there. That is the signature of a system crossing a stability boundary rather than degrading, and it is the same signature as a column that carries its load until it buckles and as the growth that beats a wave when a cloud collapses.

The instrument that has to live with the nod #

Everything above is a curiosity in a toy and a specification in an instrument. A rate gyroscope measures how fast its case is turning by holding a spinning rotor and reading the torque needed to keep the rotor’s axis where it belongs, and the reading is only as good as the assumption that the axis is where the electronics think it is. A knock puts the rotor into exactly the motion drawn in the first figure: a nod at tens of hertz, superposed on whatever the case is really doing.

Two things follow. The nod is at a frequency far above the signal band of most instruments, so it can be filtered — but only if the filter knows it is there, and only if the excursion is small enough that the pick-off stays in its linear range. And the average of a nod is not quite the undisturbed direction once the readout is nonlinear, so a rotor that is quietly nodding registers a small steady error that looks exactly like a real rotation rate. Instrument engineers call it coning error, and the geometry behind it is that a vector sweeping a small cone does not average to the cone’s axis when the thing reading it is not linear in the excursion.

This is a specific instance of a general obligation. A motion too fast to see is not a motion too fast to matter, because averaging is a linear operation and instruments are not, and the first thing a designer needs to know is the frequency and the amplitude of everything that has been averaged away. The figures above give both, as functions of the one number a designer controls, which is the spin.

Two rates, and the spin below which there are none #

The quadratic deserves to be looked at directly, because the second root is a real motion that almost never gets drawn.

Setting the tilt’s acceleration to zero gives

$$I_1\cos\theta\,\dot\phi^2 - I_3\omega_3\dot\phi + Mgl = 0,$$

which is a quadratic in the precession rate, not an equation for it. Its two roots are two genuinely different steady motions. The slow one has leading term $Mgl/I_3\omega_3$ and is what the standard argument produces, because that argument implicitly assumes the precession is slow enough for its own centrifugal effects to be neglected — which is to say, it assumes the answer it gets. The fast root is what happens if the top is launched with a large sideways rate: the axis then holds its tilt while sweeping round quickly, held up not by the spin’s interaction with gravity but by the centrifugal effect of its own sweeping.

The fast root is hard to demonstrate and easy to compute, and it is worth knowing about for one reason: its existence is why the discriminant condition exists. A quadratic with no real roots is a tilt at which no steady motion is available, and that is what “the top falls over” means mathematically. So the sleeping-top criterion, the two precession rates and the nod are three faces of the same quadratic, and any account that reports only the slow root has thrown away the condition under which the slow root exists.

What the pictures cannot show #

Every figure here draws a single quantity — a tilt, a rate, a boundary — and none of them draws the top. The angular momentum vector, in particular, is not along the axis: it has a component along the axis from the spin and a small component perpendicular to it from the axis’s own motion, and it is the sum that the torque turns. The perpendicular part is exactly what the nod is made of, and a picture that draws angular momentum as an arrow along the shaft has already assumed the nod away. That is a fair approximation for a fast top and it is the assumption under examination.

The traced paths are also drawn as the tip of a unit vector on the surface of a sphere, projected flat. Nothing about them says how much energy is in the nod compared with the precession, and the answer is surprising: at the numbers used here, almost all of the top’s kinetic energy is in the spin, a little is in the nod, and the precession carries less than the nod does. A motion can dominate the appearance of a system while carrying almost none of its energy, which is the same lesson as the drift that is not where the energy is in a very different setting.

And the figures compute an ideal Lagrange top: rigid, axially symmetric, pivoted at a mathematical point, in a uniform field, with no friction anywhere. Each of those does real work. A top whose two transverse moments differ does not have a single nutation frequency; a pivot with a finite contact area exerts a torque that makes the top climb upright, which is why a well-made top rises rather than merely staying where it was put; and the uniform-field assumption fails for anything larger than a room, where the gradient across the body is what a tidal torque means.

Where the model stops #

Rigidity is assumed and is doing more work than it looks. The moments of inertia are constants here, and for a steel rotor at a few thousand revolutions a minute that is excellent. It is not excellent for the Earth, whose free wobble comes out at fourteen months instead of ten precisely because the body yields, and it is not excellent for any rotor near its burst speed.

The pivot is a point that exerts no torque. A real pivot has a contact patch, and the friction in it does two things the equations here do not: it damps the nod within a second or two, and it exerts a small torque that raises the axis. The second is why a top spun on a table rises to the vertical rather than staying at the angle it was set down at, and it is not a gravitational effect at all.

And the closed forms are fast-top expansions. The nutation frequency I₃ω₃/I₁ and the amplitude 2I₁Mgl sin θ/(I₃ω₃)² are leading terms in Mgl/(I₃²ω₃²/I₁), which is a few per cent here and is not small for a slow top. The measured curves in the figures are integrations of the full equations and do not use them; the dashed lines are the approximations, and the gap between the two is the size of the correction.

Where the ladder goes next #

The rotation ladder began with torque as a different quantity from force and where the mass sits deciding the race, went through the axis that will not hold under free rotation and the quantity that survives a change of shape, and reached the sideways answer to a downward push. This rung asks what that answer leaves out. The rungs after it: the gyrocompass, where the same equations plus a constraint pick out true north; the rolling body, where the contact condition removes a degree of freedom rather than adding a force; and the rigid body in three dimensions with all three moments different, where the conserved quantities no longer separate and the motion is not periodic at all.

The habit worth carrying away is about averages. An average over an unobservable motion can be right about every number and wrong about what is happening, and the way to find out which is to ask what the system does in the first instant rather than in the steady state. A top asked that question answers with cusps.