Illustrated Physics
physics, drawn
Concept

Phase portrait — where it appears

A plot of velocity against displacement in which a whole history is one curve, and a conserved energy is a closed level set. Trajectories on it never cross, so the qualitative fate of every initial condition can be read off the picture without solving anything.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The pendulum's phase portrait. Angle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix.

The pendulum, and the small lie that makes it simple

A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.

mechanics · Pendulum A harmonic well. Potential energy against position, with a horizontal line at the total energy. The motion is confined to where the line lies above the curve, and the turning points are the intersections — computed by solving for them, not marked by hand.

The hill that gives it back, and the forces that do not

Potential energy turns a question about motion into a picture of a landscape. It works for gravity and springs, it fails for friction, and the difference between those two cases is the whole of what makes energy useful.

mechanics · Energy Every path the body can take, at one angular momentum. The angular momentum vector, drawn in the body's own frame on the sphere its length confines it to, for a body whose principal moments are 3.068e-3, 6.817e-3 and 9.750e-3 kg m². Each closed curve is one motion, traced by integrating Euler's equations rather than by solving for the intersection of the sphere with the energy ellipsoid, so a curve closes only if the physics closes it. The low-energy curves circle the greatest-moment axis and the high-energy ones circle the least; both sets are small loops that stay near their axis, which is what stability looks like. Between them is the one curve that is not a loop at all — four arcs, drawn heavier, meeting at the intermediate axis and leaving it again. A body spun about that axis is balanced on the crossing point of paths that go somewhere else, which is the whole of why it does not stay.

The axis that will not hold

A book spun about its long edge keeps spinning about it. Spun about the axis through its covers, it keeps spinning about that. Spun about the third axis, it flips end over end, again and again, with nothing touching it. Three numbers decide, and what matters is only their order.

mechanics · Rotation One equation, and the number that decides what it does. The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2.

The three ways of coming to rest

An oscillator with friction in it can swing and fade, arrive and stop, or crawl back so slowly it looks stuck. One equation and one number decide which. The number is not what most people would guess, and neither is the value that returns fastest.

mechanics · Pendulum Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve.

The error that doubles on a schedule

Two double pendulums released a hundred-millionth of a degree apart follow one curve for ten seconds and then have nothing to do with each other. The separation grows exponentially the whole time, including while the picture shows a single trace — which turns unpredictability into a rate, and makes the length of a forecast the logarithm of the precision rather than anything proportional to it.

mechanics · Chaos

Named alongside it

The objects these essays reach for when they reach for this one.

SeparatrixExponential sensitivityPotential energySimple harmonic motionAngular momentumChaosCharacteristic equationConservation lawsConservative forceCritical dampingDampingDeterminism

All concepts