Illustrated Physics
physics, drawn
Concept

Noethers theorem — where it appears

The theorem that every continuous symmetry of a system's action comes with a conserved quantity. Symmetry under shifts in time gives energy, under shifts in space momentum, and under rotations angular momentum — which is why a medium changing in time conserves momentum and not energy.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

optics · Refraction

Named alongside it

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

SymmetryActionAngular momentumBoundary conditionConserved quantityDispersion relationEnergyEnergy conservationGalilean invarianceLagrangianLaplace runge lenzMomentum

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