The swing that is pumped, not pushed

A child on a swing does not push against anything. There is nothing to push against: the seat, the chains and the rider are one object, and internal forces cannot change the system’s momentum. What the rider does is stand up at the bottom and sit down at the ends, which alters the length of the pendulum rather than applying a force to it — and alters it twice in every full period. Internal forces cannot move a system’s centre of mass, so whatever is happening is not a push.

An equation with nothing on the right #

A driven oscillator obeys

$$\ddot x + 2\beta\dot x + \omega_0^2 x = \frac{F}{m}\cos(\Omega t),$$

and its response is the familiar peak: amplitude $F/k$ times a resonance factor, largest near $\Omega = \omega_0$, with a width set by the damping. A pumped one obeys

$$\ddot x + 2\beta\dot x + \omega_0^2\left[1 + \varepsilon\cos(\Omega t)\right] x = 0.$$

The difference is where the modulation sits. In the first, the driving is added; in the second, it multiplies the displacement. The consequence is immediate and is the whole subject: $x = 0$ is an exact solution of the second equation at every $\varepsilon$ and every $\Omega$. A swing that is perfectly still stays perfectly still no matter how energetically the rider crouches and rises. Pumping is not a way of putting energy into an oscillator; it is a way of making the state of rest unstable, so that whatever disturbance is present grows.

The driven case is worth holding up beside it. A force produces a response at every frequency and every amplitude — there is no threshold anywhere on a resonance curve, and the response is proportional to the force. Everything about the pumped case is a departure from that picture: a threshold instead of proportionality, a wedge instead of a peak, exponential growth instead of a steady state, and a frequency at twice rather than at one times.

The second equation is Mathieu’s, written down in 1868 for the vibrations of an elliptical drumhead and turning up since in far more places than that.

The distinction between the two kinds of excitation has a name worth using because it separates two things that are otherwise easy to confuse. Forcing adds a term; parametric excitation modulates a coefficient. A tuning fork struck with a hammer is forced. A guitar string whose tension is wobbled at twice its own frequency is pumped. The first responds to any hammer; the second responds to nothing at all until the wobble is deep enough, and then responds to it explosively. Both are called resonance in ordinary speech and they share almost nothing beyond the word.

Why twice #

Take the modulation to be the rider rising and falling. Standing up raises the centre of mass, which shortens the effective pendulum and raises $\omega_0$; sitting down lengthens it and lowers $\omega_0$. The rider stands at the bottom of each pass and sits at each end — and there are two bottoms per full period, so the stiffness goes up twice and down twice per swing. The modulation frequency is $2\omega_0$.

The energetics are worth doing, because they explain both the factor of two and the exponential. The tension at the bottom of a swing of angular amplitude $\theta_0$ is $mg(3 - 2\cos\theta_0)$, which exceeds $mg$; at the ends it is $mg\cos\theta_0$, which is less. Raising the centre of mass by $h$ at the bottom costs $mg(3 - 2\cos\theta_0)h$ and lowering it at the end returns $mg\cos\theta_0 h$. The surplus per half-period is

$$\Delta E \approx 3mgh\,(1 - \cos\theta_0) \approx \tfrac{3}{2}mgh\,\theta_0^2,$$

and the energy of the swing is $\tfrac12 mgL\theta_0^2$. So the fractional gain per half-period is $3h/L$ — a constant, independent of amplitude. A fixed fraction added per stroke is a geometric series, which is an exponential; and that is why a swing that is already going gains far more per pump than one that has just started, and why the first few pumps feel useless. It is the same compounding that makes an exponential in a barrier height the difference between instantly and never: a proportional rule and an additive one behave differently in kind, not in degree.

Where the surplus comes from is the tension. Along the chains it has to supply both the radial component of the weight and the centripetal requirement, so it is largest at the lowest point and smallest at the extremes — at 34° from the vertical the difference is a factor of about two. The rider does work against that tension when standing up, and does more of it at the bottom than is recovered at the top, because that is where the tension is greater. The asymmetry exists only because the swing is already moving, which is why a stationary swing cannot be pumped at all.

The threshold, and what puts it there #

An undamped oscillator has instability wedges that come to a point on the frequency axis: any modulation at exactly $2\omega_0$, however feeble, will eventually grow. Damping lifts each wedge off the axis and gives it a flat bottom, because the growth has to outrun the loss.

The threshold for the first wedge, worked out to lowest order, is $\varepsilon > 4\beta/\omega_0 = 4\zeta$ — four times the damping ratio. At $\zeta = 0.02$ that is 8%, which is what the boundary in the map comes to when it is located numerically. For a rider on a swing, $\varepsilon$ is roughly $2h/L$ with $h$ the rise of the centre of mass, so a rise of a few per cent of the chain length clears the threshold comfortably, which is why children discover this without instruction.

Why one period of integration is enough #

The map is built by integrating a single period of the modulation and then asking a question about the result, which looks like far too little work for a statement about behaviour over all time. It is exactly the right amount, and the reason is a theorem about equations whose coefficients repeat.

The equation is linear, so what it does to a state over one modulation period is a matrix: start from the two independent states — unit displacement with no velocity, and unit velocity with no displacement — integrate each for one period, and the two answers are the columns. Every subsequent period applies the same matrix again, because the coefficients have returned to what they were. So the behaviour over $n$ periods is that matrix raised to the $n$, and whether anything grows is decided entirely by whether either of its eigenvalues lies outside the unit circle. Nothing about the long run requires integrating for a long time.

There is a constraint on those eigenvalues that makes the picture tidier than it might be. The determinant of the matrix is fixed in advance — it is $e^{-2\beta T}$, an identity about linear second-order equations that holds whatever the modulation does — so the two eigenvalues multiply to that number. With no damping the product is exactly one, which leaves two possibilities and no others: both eigenvalues on the unit circle, in which case the motion is bounded for ever, or one real eigenvalue larger than one and its reciprocal, in which case something grows exponentially. There is no third case and no marginal decay.

That is why an undamped parametric system’s wedges reach right down to the axis. Stability is not a competition between growth and loss when there is no loss; it is a question about where a pair of eigenvalues sits, and an arbitrarily small modulation is enough to push them off the circle at the right frequency. Damping makes the product less than one, which lets both eigenvalues sit inside the circle with room to spare, and that room is the threshold.

Where the factor of four comes from #

The threshold $\varepsilon > 4\zeta$ is quoted above and is worth deriving, because it is two sentences and it explains why the number is not two or one.

At exact tuning the modulation acts on the oscillation twice per cycle of the swing, and only the component of the modulation in step with the motion contributes; the rest averages away over a cycle. Doing that average gives a growth rate of $\varepsilon\omega_0/4$ for the amplitude — the quarter being what survives from resolving a cosine modulation against a sinusoidal motion and keeping the resonant term. The damping removes amplitude at $\beta = \zeta\omega_0$. Growth beats loss when $\varepsilon\omega_0/4 > \zeta\omega_0$, and the frequency cancels.

So the four is an averaging factor and nothing more, and the shape of the statement is what matters: a threshold exists because both processes are proportional to the amplitude, so their competition is decided by a ratio of rates rather than by how large the disturbance happens to be. A swing that has just been nudged and a swing already going strongly are on the same side of the threshold.

The higher wedges, and where they actually sit #

There is a wedge at $\Omega = 2\omega_0/n$ for every integer $n$: the rider could in principle pump once per full swing, or twice every three swings, and get the same result. The thresholds rise steeply — 8.0%, then 40.0%, then 62.5% at this damping — so only the first is available to anything with a modest modulation depth.

The wedges also lean. Their apexes sit at $2\omega_0/n$ only in the limit of vanishing modulation; at a depth of 0.9 the third has moved from 0.667 to about 0.63, because a large modulation changes the mean stiffness as well as varying it. Assuming otherwise is not an academic error: a search for the third wedge that starts at 0.667 and walks outward finds nothing at all, because at that frequency and that depth the system is stable again.

The same equation somewhere else #

Mathieu’s equation is not about swings, and the wedge structure appears wherever a parameter is modulated near twice a natural frequency.

Faraday waves. A dish of water vibrated vertically at frequency $\Omega$ develops standing waves on its surface at $\Omega/2$, and only above a threshold amplitude. Faraday reported it in 1831; the surface waves are the unstable mode and the vertical shaking modulates the effective gravity, which is the stiffness of the restoring force.

The Paul trap. An ion is held in an oscillating quadrupole field, which is a modulated stiffness with the wrong sign half the time. The ion is trapped in the stable regions of the same map — the complement of the wedges — and the operating point of every quadrupole mass filter is a point in a Mathieu stability chart, which is what sorting charged particles by mass looks like when the field alternates instead of holding still.

Optical parametric amplification. A crystal pumped at $2\omega$ amplifies light at $\omega$, above a threshold set by the cavity loss, and the gain is exponential in the crystal length — a device whose whole operation is a wedge on this map, and whose output is coherent for the reason a narrow line is. The phase-sensitivity is the same one the swing shows: a signal in the wrong phase relative to the pump is attenuated rather than amplified.

In an ordinary oscillator energy moves back and forth between two stores, and that is the process the modulation interferes with. A parametric pump acts on the stiffness, so it changes the potential store without touching the kinetic one — and doing that twice a cycle, in step, is how energy gets in without any force ever being applied along the direction of motion. The pump is perpendicular to the motion and still does work, because the work is done against a tension that depends on the motion.

What stops it #

The linear equation predicts a swing that goes over the bar, and no swing does. The thing that stops it is not the damping — the damping was beaten at the threshold and stays beaten — but the fact that a pendulum’s period depends on its amplitude.

What stops the growth is that the period is not a constant. A pendulum’s period rises with amplitude — 0.2 per cent at 10°, 18 per cent at 90° — so a rider pumping at a fixed rhythm is modulating at a fixed $\Omega$ while $\omega_0$ falls, and the ratio $\Omega/\omega_0$ climbs. That ratio is the horizontal axis of the stability map, so the system walks sideways out of the tongue it was growing in. The amplitude stops rising not because the pumping stopped but because the swing drifted out of resonance with it.

The swing therefore walks sideways out of its own wedge. It starts at $\Omega/\omega_0 = 2$, in the middle of the widest part, and as the amplitude grows $\omega_0$ falls and the ratio rises past the right-hand boundary. At that point the growth stops, and the amplitude the swing settles at is the one whose period puts it exactly on the edge. Nothing balances; the system simply leaves the region where growth was possible.

That makes the final amplitude a property of the nonlinearity rather than of the pumping, which is the opposite of the driven case, where the steady amplitude is set by the balance between drive and damping and grows without limit as the damping is reduced.

The harmonic approximation runs out at a definite amplitude, and for a pendulum and a molecular bond it runs out in the same way: both are parabolic at the bottom, neither stays parabolic, and the per-cent departure arrives at a displacement that can be computed rather than guessed. Everything in this essay above that amplitude is a linear account of a system that has stopped being linear — and the departure is what closes the account rather than a nuisance in it.

The same equation standing on its head #

The most striking use of this map reads it the other way round — not the wedges, but everything between them.

Take a rigid pendulum and hold it upside down. The inverted position is an equilibrium and it is an unstable one: displace it and it falls. Now shake the pivot up and down, fast, through a small amplitude. Above a definite shaking speed the inverted position becomes stable: nudge the pendulum and it returns to vertical, upside down, and stays there.

That is the same equation with the sign of the stiffness reversed, and the condition is arrived at in the same way. Averaging over the fast shaking leaves an effective potential with a new term in it, proportional to the square of the pivot’s peak velocity, and that term has a minimum where gravity has a maximum. The inverted position is stable when $(a\Omega)^2 > 2gL$ — the pivot’s peak speed exceeding a threshold set by the pendulum’s own length. For a pendulum twenty centimetres long shaken through a millimetre, that is a few tens of hertz, which is a small motor.

Two things are worth taking from it. The first is that a fast modulation can create a restoring force where there was none, by exactly the mechanism that destroyed one in the swing — the sign of what it does depends on which side of the natural frequency the modulation sits, and the swing’s is at twice while the inverted pendulum’s is at many times. The second is that the stability chart is one object read for two purposes: a swing rider wants to be inside a wedge, an ion in a quadrupole trap and an inverted pendulum want to be outside every one of them, and it is the same chart.

Where the model stops #

The equation is linear, and growth cannot be unlimited. Nothing in the Mathieu equation stops the amplitude at any value, so it predicts a swing going over the bar. What stops a real one is nonlinearity: the period lengthens as the amplitude grows, so the swing drifts out of step with a rider pumping at a fixed rate, and the growth saturates when the detuning takes the system out of the wedge. The amplitude a parametric system settles at is set by whatever nonlinearity arrives first, not by a balance of drive against damping.

The modulation is sinusoidal. A rider’s rise and fall is nearer a square wave, which contains harmonics at $3\Omega, 5\Omega, \dots$ and therefore overlaps more than one wedge. The thresholds change by factors of order one and the structure does not.

The stiffness is the modulated quantity. Modulating the damping instead gives a different equation with different stability regions, and modulating the mass gives another. The wedge-at-twice structure belongs to the term that multiplies $x$.

The oscillator has one degree of freedom. A real swing can also twist and can move out of the plane, and pumping excites those too — which is what a rider who is not concentrating discovers. With two or more coupled modes the stability chart becomes a chart in more dimensions and the wedges can overlap, so that a modulation stabilising one mode destabilises another. The single-mode picture is right for a swing on two chains and wrong for a swing on one rope.

What the pictures cannot show #

The map is a statement about stability and says nothing about how fast. Deep inside the first wedge the growth rate is large and the amplitude doubles in a couple of periods; just inside the boundary it is arbitrarily slow, and the difference between “unstable” and “stable” near the edge is not observable in any finite experiment. The boundary is sharp in the arithmetic and gradual in practice.

Neither does the map show phase. The growing solution has a definite phase relative to the modulation — it grows when it is in step and decays when it is a quarter cycle out — and a figure plotting a region of a plane has thrown that away. It is the reason a rider who mistimes the crouch slows the swing down rather than merely failing to speed it up.

The phase note above has a consequence that is worth more than a caveat. A parametric amplifier amplifies a signal in step with the pump and attenuates one a quarter cycle out of step, by the same factor. That is not a defect to be engineered around: it means the amplifier has a preferred axis, and a quantity that is small along one axis and large along the other has been produced from one that was symmetric. Where the input is the noise a field carries as a matter of principle, the output is a state with less uncertainty in one quadrature than the vacuum has and more in the other — which is squeezing, and it is the mechanism behind the interferometers that measure lengths below what unmodified noise would allow. All of it is the phase sensitivity a mistimed crouch demonstrates on a swing.

And the growth figure plots a displacement, so its envelope is the visible thing while the energy is what grows exponentially at twice the rate. Nothing in the picture distinguishes those two rates.

Where the ladder goes next #

The rung below asked what a system does when it is pushed at its own frequency. This one asks what it does when it is changed at twice its own frequency, and the answers differ in every respect that can be checked: a threshold instead of proportionality, a wedge instead of a peak, exponential growth instead of a steady amplitude.

The rungs above go two ways. One is toward the nonlinear saturation that a real swing reaches, and toward the amplitude-dependent period that causes it — a question about where the harmonic approximation stops. The other is toward stability charts in general: the same map, read as a region of allowed parameters rather than forbidden ones, is what makes a Paul trap a trap, and the habit of asking whether an equilibrium is stable rather than what the response is turns out to be the more useful question far more often than the resonance curve suggests.