Concept

Integrability

The property of a dynamical system that enough conserved quantities exist to confine each trajectory to a low-dimensional set. An integrable system's orbits are orderly and describable in closed form; the typical system has no such quantities.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

analysis · The integral

A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear.

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

dynamics · Billiards

A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem.

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

dynamics · Billiards

Named alongside it

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

Billiards Caustic Periodic orbit Reflection Area Conic Convergence Counterexample Dirichlet Dirichlet's function Equidistribution Focus

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