The period that depends on the swing, computed exactly

The small-angle formula says a pendulum’s period does not depend on how far it swings. That is false, and the previous rung on this ladder measured how false it is at small amplitudes and left the exact answer as an unopened box.

This is the box. It contains a function that cannot be written with the symbols of elementary algebra, that was tabulated by hand in the eighteenth century, and that has an evaluation method so fast it is startling.

The equation that has no solution #

The problem is stated in one line and refuses to be solved in any number of them.

A pendulum of length $L$ displaced by an angle $\theta$ obeys

$$\ddot{\theta} = -\frac{g}{L}\sin\theta,$$

and the presence of $\sin\theta$ rather than $\theta$ is the entire difficulty. A differential equation with the unknown inside a sine has no solution in elementary functions — not because nobody has found one, but because it can be proved that none exists.

Energy provides the route round it, and does so without any approximation at all. The total energy is fixed, so at angle $\theta$, having been released from rest at $\theta_0$,

$$\tfrac{1}{2}L^2\dot{\theta}^2 = gL(\cos\theta - \cos\theta_0),$$

which rearranges into an expression for $\dot\theta$ and therefore for the time to travel any small piece of the arc. Adding those times over a quarter swing gives the quarter period as an integral:

$$T = 4\sqrt{\frac{L}{g}} \int_0^{\theta_0} \frac{d\theta}{\sqrt{2(\cos\theta - \cos\theta_0)}}.$$

The integral is the time taken to cross the allowed region, accumulated point by point — and the reason it has no elementary answer is visible in the potential: the particle slows as it approaches the turning point, the integrand diverges there, and the divergence is integrable but not in terms of anything with a name. That is what “no closed form” means here, and it is why the answer is an elliptic integral rather than a formula.

That integral is where the trouble is concentrated. With the half-angle substitution it becomes the complete elliptic integral of the first kind,

$$T = 4\sqrt{\frac{L}{g}}\, K(k), \qquad k = \sin\frac{\theta_0}{2},$$

and $K$ is one of a small family of functions that arose from problems like this one and turned out to be worth naming. Legendre spent forty years on them. They are not solvable in elementary terms and they are perfectly well understood, which is the ordinary condition of most functions in physics.

How the curve was actually computed #

The figure is not a fit and not a lookup table. Every point on it comes from an algorithm that is worth describing, because it is one of the fastest in numerical analysis and it was found by accident.

Take two numbers and replace them, repeatedly, by their arithmetic mean and their geometric mean. The two sequences converge — toward each other, and toward a common limit called the arithmetic–geometric mean. Gauss noticed at fifteen that the convergence is quadratic: the number of correct digits doubles at each step. Starting from 1 and $\sqrt{1-k^2}$, four or five iterations give more precision than double-precision arithmetic can hold, and the limit is related to $K(k)$ by

$$K(k) = \frac{\pi}{2\,\mathrm{agm}(1, \sqrt{1-k^2})}.$$

So each point of the curve above costs about five square roots. The generator evaluates it that way, and during development the result was checked against a series expansion of the same function — two independent routes to the same number, which is the check worth running whenever a figure depends on an algorithm rather than on a formula.

What the numbers say #

The curve is worth reading as a table, because the interesting content is how slowly it leaves one and how suddenly it stops being close to anything.

At 5°, the exact period exceeds the small-angle period by 0.048 per cent. On a clock that is 41 seconds a day — enormous by horological standards, and a constant, which means it is absorbed by cutting the pendulum slightly short.

At 10° it is 0.19 per cent, at 23° just over 1 per cent, and at 45° exactly 4 per cent. A pendulum swinging through 45° each way keeps time 4 per cent slow, which is an hour a day.

At 90° it is 18 per cent. At 150° it is 76 per cent. At 179° the period is nearly four times the small-angle value, and at 180° it is infinite.

The second-order correction $1 + \theta_0^2/16$ tracks the truth to 0.03 per cent at 30° and is 2.2 per cent low by 90°. That is the usual shape of a truncated series: excellent where it was designed to be used, and confidently wrong beyond it — and, crucially, wrong in a way that gives no warning, since the correction formula is a perfectly well-behaved parabola all the way to 180° and simply describes something else there.

Drawn on the same nought-to-ninety axis as the period curve, the small-angle error and the period correction can be read against each other — and they are not the same size. The error in $\sin\theta$ is over twenty per cent at sixty degrees while the period is out by less than five, because the period is an average over the swing and the extremes are visited briefly. The correction is smaller than the approximation it comes from.

Comparing the two error curves is instructive and is the reason both are on this page. The substitution error at 45° is about 10 per cent, and the period error at 45° is 4 per cent. The period error is not the substitution error; it is an average of that error over the swing, weighted by how long the pendulum spends where. Approximating a term in an equation and approximating the answer that comes out are different operations with different sizes, and no rule of thumb converts one into the other.

The infinity at a half turn #

The most striking feature of the curve is at its right-hand end, and it is a real prediction rather than a numerical artefact.

As the amplitude approaches 180°, the period grows without limit. A pendulum released from exactly upside down never comes back — not because it takes a very long time, but because the time is infinite.

The boundary between swinging and spinning is where the period diverges, and the two loops drawn inside it are the ninety- and hundred-and-fifty-degree swings quoted above. A pendulum released from just below the vertical takes arbitrarily long to arrive, because it spends arbitrarily long near the top — so the period runs to infinity at half a turn, which no series expansion can show.

The reason is visible in the landscape. At the top of the potential the slope is exactly zero, so the restoring force vanishes, so a pendulum arriving there with exactly no speed left has nothing to move it. It is an unstable equilibrium, and an unstable equilibrium is approached asymptotically: the pendulum gets closer and closer to vertical, more and more slowly, and never arrives.

This is the separatrix, and the divergence of the period is how the boundary between two kinds of motion announces itself in a formula. On one side the pendulum swings and the period is finite. On the other it circulates, and it has a period again — of a different kind, and finite. Between them sits a single trajectory belonging to neither, whose period is infinite, and any quantity that is finite on both sides and infinite between them is a reliable sign that a boundary in behaviour has been crossed.

The same divergence at the same kind of boundary appears throughout physics: in the lens equation as the object reaches the focus, where the image distance runs away; in the response of an undamped oscillator driven exactly at resonance; and in the time a light ray takes to reach a black hole’s horizon. In every case the infinity is the mathematics reporting a change of regime rather than a mistake.

What the exact answer costs #

Having the truth is not obviously better than having the approximation, and the trade is worth stating plainly.

It cannot be reasoned with. $T = 2\pi\sqrt{L/g}$ can be inverted, differentiated, substituted into other expressions and argued about in a sentence. $K(\sin(\theta_0/2))$ can be evaluated. The elementary formula supports thought; the exact one supports computation, and those are different services.

It hides the dependencies it does show. The small-angle formula displays immediately that the period is independent of mass, and proportional to the square root of length. The exact expression has those facts in it too, in exactly the same way — the prefactor is untouched — but they are no longer the first thing visible.

It is exact about a model that is not. This is the sharpest point. The elliptic integral is the exact period of an idealised pendulum: massless rod, point bob, no air, rigid pivot, no rotation of the Earth. Every one of those idealisations is worth tenths of a per cent or more in a real instrument. So at amplitudes below about 10°, where the amplitude correction is under 0.2 per cent, computing $K$ to fifteen digits buys precision that the rest of the model cannot support. The exact solution of an approximate problem is not the exact answer, and knowing which of the two an improvement improves is most of the skill in applying any of this.

Beyond about 30° the situation reverses and the exact form is the only honest one, because the amplitude term is then the dominant error by a wide margin.

Where the model stops #

The exact curve is exact about one specific idealised object, and two of its assumptions fail in ways that matter well before the interesting end of the graph.

The right-hand half of the curve requires a rigid rod. A pendulum on a string cannot reach those amplitudes at all, and the reason is worth deriving because it is a one-line application of what circular motion demands. At angle $\theta$ the bob needs an inward force $mv^2/L$; the available inward force is the tension minus the radial component of the weight, so

$$T = mg(3\cos\theta - 2\cos\theta_0).$$

At the release point $v$ is zero and this reduces to $T = mg\cos\theta_0$, which is negative for any $\theta_0$ beyond 90°. A string cannot push. So a string pendulum held out past the horizontal and released does not swing — it falls, the string goes slack, and the bob is a projectile until the string snaps taut again with a jerk. Everything on the curve past 90° describes a bob on a rigid rod and nothing else.

The rod is massless and the bob is a point. A real rod has a moment of inertia of its own, which changes the prefactor and, for a compound pendulum, replaces $L$ by $I/md$. That correction can be tens of per cent, dwarfing the amplitude term at ordinary swings.

That curve is worse news than the amplitude correction, because $k$ is a property of how the mass is distributed and $h$ is measured from a centre of mass nobody can see. A metre bar pivoted 0.4 m from its centre swings as a simple pendulum 0.608 m long — not 0.4, not 0.5, and not any distance a rule can be laid against.

The curve’s own shape is the way out, and it is the reason a pendulum was the standard instrument for measuring $g$ for a century. Because $(k^2 + h^2)/h$ takes every value above its minimum twice, each period belongs to a pair of pivots whose distances from the centre multiply to $k^2$ — and for that pair the equivalent length is exactly the distance between them.

So the quantity that could not be measured is replaced by one that can, without ever learning either of the things it was made of. Kater’s reversible pendulum is that identity built in brass: two knife edges, a weight slid along the bar until the two periods agree, and $g$ from the separation and the period. The defect in the model became the instrument.

Nothing is dissipating. The whole derivation used energy conservation, so the amplitude is assumed fixed. A real pendulum’s amplitude decays, and since the period depends on amplitude, the period drifts as it decays. The exact curve therefore describes a motion that no real pendulum performs: it gives the period of this amplitude, while a real one is always partway between two of them.

And the pivot is at rest on a non-rotating Earth. The plane of the swing is fixed relative to the stars rather than to the room, and a support with any compliance shares energy with its mount.

The tables that came before the algorithm #

The integral in this essay is older than the pendulum problem, and it got its name from somewhere else entirely.

Finding the arc length of an ellipse — a question forced on astronomy the moment Kepler put the planets on ellipses — produces an integral of exactly this form. It resisted every attempt at an elementary solution through the seventeenth century, and by the eighteenth it was clear that a new class of functions was needed. Legendre spent the better part of forty years reducing every integral of that shape to three standard forms and tabulating them, publishing the tables in the 1820s. The name elliptic records the ellipse problem and has nothing to do with the pendulum, the orbit, or anything elliptical in the applications.

Those tables were how the curve on this page would have been produced for the next century and a half. An engineer wanting the period of a large-amplitude pendulum computed $k = \sin(\theta_0/2)$, looked up $K(k)$, and multiplied. The tabulation was the technology, and it was expensive enough that the second-order correction remained the working formula for anyone who did not need better.

What changed is not the mathematics but the cost. Gauss’s arithmetic–geometric mean, sitting unpublished in a notebook from 1799, turns the same evaluation into five square roots — so a figure like the one at the top of this page, which needs the function at several hundred amplitudes, is now cheaper to compute than to look up. The result is a small shift in what a figure can be: a curve of an inelementary function is now something a build step evaluates rather than something an author copies, and copying is where errors live.

Huygens, and the shape that has no amplitude dependence #

The obvious question — whether some other constraint has a genuinely amplitude-independent period — was asked immediately and answered brilliantly in 1673.

Huygens showed that a bob constrained to a cycloid rather than a circle has a period exactly independent of amplitude. The curve is the path traced by a point on a rolling wheel, and it is the tautochrone: a bead released from any height on it reaches the bottom in the same time, exactly, with no approximation anywhere.

The circular constraint gives a restoring force of $mg\sin\theta$ rather than something proportional to displacement, and that is the whole trouble. Huygens’ cycloid fixes it by changing what “displacement” means: measured along a cycloidal arc, the restoring force is proportional to it, so the period is exactly independent of amplitude. The shape has no lie in it, and the practical difficulties it introduced were worse than the one it removed.

Huygens built clocks with curved metal cheeks at the suspension, so that the string wrapped against them and the bob followed a cycloid. The result is one of the most instructive failures in the history of instruments: those clocks kept worse time than the ones without. The cheeks added friction, and made the effective suspension length depend on amplitude in a new way, and the errors introduced exceeded the error removed.

The lesson generalises past clocks. A correction is worth applying only if the machinery it requires introduces less error than it removes, and that comparison has to be made with numbers rather than with enthusiasm. The whole later history of precision pendulums went the other way — keep the circle, keep the amplitude tiny and constant, and let the $\theta_0^2/16$ term be a fixed offset absorbed by the length. Every good escapement is an amplitude regulator for that reason, and the drift it exists to prevent is the fifteen seconds a day the previous rung costed out.

The clock that took the other route #

Huygens tried to remove the amplitude dependence by changing the constraint. The instrument that finally won took the opposite approach: keep the circle, and make the amplitude so constant that the correction is a fixed number.

The difficulty is that a pendulum driving a clock is being interfered with constantly — the escapement has to take energy out of it to move the hands and put energy back to keep it going, and both operations disturb the amplitude. The solution, arrived at in 1921, was to stop asking one pendulum to do both jobs.

A free-pendulum clock has two. The master swings in an evacuated cylinder, connected to nothing, doing no work at all: once every half-minute a small arm falls onto it, delivering a precisely repeatable impulse, and is then lifted away by the second pendulum. The slave pendulum runs the clockwork, counts the beats and keeps itself in step with the master by an electrical signal. All the dirty work is done by the pendulum whose accuracy does not matter.

The result kept time to about a second a year, which is a part in $3\times10^7$ — good enough that when several were compared through the 1920s the residual disagreement turned out not to be in the clocks. The Earth’s rotation is irregular, and a pendulum in a vacuum tank was the first instrument to say so.

That is what the arithmetic on this page is worth when the amplitude is nailed down: the $\theta_0^2/16$ term becomes a constant absorbed into the length, and what is left to fight is everything else.

The same equation, drawn in space #

The pendulum’s equation has a twin in a subject with no motion in it at all, and the correspondence is exact rather than suggestive.

Take a thin elastic strip and push its two ends together. Its shape is fixed by the condition that the bending moment at each point is proportional to the curvature there, and writing that out gives

$$\frac{d^2\theta}{ds^2} = -\frac{F}{EI}\sin\theta,$$

with $\theta$ the angle the strip makes with the line of the force and $s$ the distance along the strip. That is this essay’s equation with arc length in place of time.

So every motion of a pendulum corresponds to a shape of a bent rod, and the correspondence is one-to-one. A small-amplitude swing is a gently bowed strip; a large-amplitude swing is a strip bent so far that it doubles back; a pendulum going right over the top is a strip curled into a full loop. And the separatrix — the trajectory that takes forever — is the single shape that straightens out asymptotically at both ends, the classical elastica loop that a flexible ruler takes when its ends are pushed together and crossed.

Kirchhoff pointed this out in 1859 and it is still called the kinetic analogy. Its practical value is that the tabulated elliptic integrals of the pendulum problem solve the buckling problem too — one function, looked up once, describing a swinging weight and the shape of a bent spring, with time in one and distance in the other.

The ladder from here #

Later rungs on this anchor: damping, and the three regimes either side of critical. The driven pendulum and resonance, where the amplitude dependence of the period makes the response curve lean. The double pendulum, where two of these joined together become unpredictable in a technical sense. Foucault’s pendulum, which uses the plane of the swing to detect the rotation of the ground. Kater’s reversible pendulum and the measurement of $g$ to five figures. The physical pendulum and the moment of inertia. The escapement, treated as a control system rather than as a gear train. And the elliptic functions themselves, which invert this integral and describe the motion in time rather than merely its period.

The amplitude dependence also has a general lesson worth carrying: a system whose period depends on its amplitude is nonlinear, and nonlinearity is what makes the superposition of two solutions stop being a solution. Everything in this ladder past this rung is a consequence of that one word — including the sharp overtones of a real piano string, which are the same anharmonicity in a system with boundaries.