Illustrated Physics
physics, drawn
Concept

Conserved quantity — where it appears

A number computed from a system's state that does not change as the state evolves. Noether's theorem attaches one to every continuous symmetry of the action — energy to time translation, momentum to space translation, angular momentum to rotation — and the correspondence runs both ways.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

The same start, four force laws. Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

The orbit that does not come back to itself

A bounded orbit under any central force oscillates between a smallest and a largest radius for ever. That it should also return to the same point is a further demand, and only two force laws in existence meet it — the inverse square, and the linear spring.

astrophysics · Orbit stability Three quantities that do not move while everything else does. On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

The conservation law a symmetry hands over

Energy, momentum and angular momentum are usually presented as three separate empirical facts that happen to hold. They are one fact three times: every continuous symmetry of a system's action supplies a quantity that does not change, the correspondence is exact, and it runs both ways.

mechanics · Least action Two ways of destroying a pulse, and their cancellation. The same starting pulse, carried forward three times by three equations that differ only in which terms are present, all shown at t = 0.097. Dotted: where it began. With the dispersive term alone the pulse spreads and sheds an oscillating tail, because its Fourier components travel at different speeds and drift out of step — the profile departs from what it was by 27 per cent of its own height. With the nonlinear term alone the tall part overtakes the shallow part and the front leans forward: the steepest gradient is 9.3 times what it started as and is on its way to vertical. With both, neither happens — the profile has moved to the right and is otherwise identical to what it was, to 2.7e-12 of its own height. The balanced run is a pseudo-spectral integration and the other two are exact, and the integration conserves the two quantities the equation conserves — mass to 2.2e-16 and the squared integral to 4.7e-15 — which is the check that the answer belongs to the equation rather than to the integrator. What the picture cannot show is why the cancellation is stable: a pulse of the wrong height for its width does not persist in the wrong shape, it sheds the excess as a dispersive tail and settles on the shape that works, which is why these objects turn up in canals and optical fibres rather than only in equations.

The pulse two failures keep alive

Dispersion spreads a pulse until it is nothing. Nonlinearity steepens it until it breaks. Each on its own destroys a disturbance, and there is exactly one height for each width at which the two cancel completely — leaving a shape that travels for ever and survives being run into by another one.

waves · Wave packets Two principles, two classes of path, two things left free. On the left, five curves from the same launch point to the same target: the true trajectory of a particle of energy 0.7 in a uniform field, and four deformations of it that share both ends. On the right, four quantities computed along that family and plotted as departures from their values on the true path. Maupertuis' abbreviated action ∫p·ds, computed at fixed energy, is stationary — flat at the centre. Hamilton's action ∫L dt, computed at fixed duration, is stationary too. The other two are not: the time a fixed-energy path takes changes at first order in the deformation, and so does the energy a fixed-duration path carries. That is the whole difference between the two principles. Each holds one of those quantities fixed and lets the other vary, and neither can hold both.

The principle that fixes the energy instead of the clock

There are two principles of least action, they compare different sets of paths, and they are not the same statement. One holds the duration fixed and lets the energy vary; the other holds the energy fixed and lets the duration vary — and written that way, mechanics turns into optics with a refractive index.

mechanics · Least action A hump that is not a soliton comes apart into solitons. A single smooth hump of height 6, shaped as the square of a hyperbolic secant, released into the Korteweg–de Vries equation and followed by a pseudo-spectral integration, drawn at times 0.00, 0.15, 0.35, 0.60, each snapshot raised above the last. The hump is too tall for its width to be a soliton, and it separates: by the last time there are 2 crests, of heights 8.00 and 2.00, running apart at different speeds, with a small ripple left behind. Read as a potential well, the same hump holds 2 bound states, at κ = 2.000 and 1.000, and a soliton of height 2κ² belongs to each: 8.00 and 2.00. The integration conserved the hump's area to 3.1·10⁻¹⁵.

The solitons a hump already contains

A soliton is one height for one width. Release a hump of any other shape and it does not keep that shape or simply spread — it comes apart into a fixed number of solitons of fixed heights, running off in order of size, with a ripple left behind. The number and the heights can be read off before anything moves, by treating the hump upside down as a well and counting the levels it holds.

waves · Wave packets A rosette: the orbit of a body whose inertia is its energy. An orbit of eccentricity 0.45 under inertia from energy, Newton's pull, integrated for 7 revolutions from the closest approach of a body with k/Lc = 0.35. The dashed curve is the Newtonian ellipse from the same starting position and momentum, which closes on itself. The integrated orbit does not: its closest approach moves forward by 24.3° each revolution, against the exact 24.3° of 2π(1/Γ − 1) with Γ = √(1 − k²/L²c²). The dots are the successive closest approaches.

The orbit special relativity cannot close

An inverse-square orbit closes on itself because of a conserved vector that nothing else has. Give the orbiting body an inertia that grows with its speed, as special relativity requires, and the vector turns — the orbit becomes a rosette, advancing by a sixth of Mercury's famous 43 arcseconds a century. A different theory that respects special relativity just as well turns the rosette backwards by the same amount. The 43 is not special relativity plus a correction; it is a measurement of what gravity pulls on.

astrophysics · Orbit stability The swing's plane, turning a step at a time. The direction in which a swinging spring swings, seen from above, against time, for a spring whose bounce is twice as fast as its swing, started bouncing almost vertically with a trace of sideways motion. Points are coloured by how much of the energy is in the swing (darker: more). While the swing holds the energy the direction is fixed; while the bounce holds it, the direction changes, and the swing returns in a new direction each time — 9 steps in 1500 units of time, each 43.6° on average (43.6, 43.6, 43.6, 43.6…), always in the same sense. In a plane the swing could only come back along its old line. In space it comes back turned, and the turn is the same every time.

The swing that turns a step at a time

A weight on a spring whose bounce is twice as fast as its swing trades its energy back and forth: the bounce becomes a swing, and the swing a bounce again. Let it move in space rather than a plane, with the slightest sideways twist, and each swing comes back in a new direction, turned by the same angle every time. The turn happens all at once, while the weight is only bouncing and the swing is too small to see, and its size is set by a single ratio, of the twist to the energy carrying it. The steps are a geometric phase made mechanical, and they blur into chaos when the spring is bounced too hard.

mechanics · Chaos

Named alongside it

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

Angular momentumPrecessionActionCentral forceDispersionEffective potentialLagrangianSolitonApsidal angleAveragingBertrand theoremBound state

All concepts