Concept

Conservation law

A quantity, such as energy or angular momentum, that keeps the same value all along every motion of a system. Each one confines an orbit to a level set, and enough of them make the motion regular rather than chaotic.

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

Two blocks and a wall count to 31. Space-time diagram of a heavy block of mass 100 driving a block of mass 1 against a wall: 31 elastic collisions in all.

Two blocks and a wall count π

Slide a heavy block towards a light one resting by a wall, and count every collision until they part for good. A heavy block 100 times the mass gives 31; ten thousand times, 314; a million, 3,141. The digits are π because the two blocks are secretly one billiard ball in a wedge, and a ball in a wedge bounces as many times as the wedge's angle fits into half a turn.

dynamics · Billiards

Islands of order in a chaotic sea. Poincaré section of the Hénon–Heiles system at energy 1/8: orbits started on a line of the section, crossings coloured by a separation test, 16 regular and 4 chaotic.

A conserved quantity that is not there

In 1964 Michel Hénon and Carl Heiles asked whether a star in a galaxy obeys a third conservation law besides energy and angular momentum. They cut its orbit by a plane and looked. At low energy every orbit drew a clean curve, as if the law held; at higher energy the curves dissolved into dust. Integrating their model again shows the share of chaotic orbits climbing from nothing at energy 1/12 to almost all at 1/6.

dynamics · Sensitive dependence

Named alongside it

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

Billiard Chaos Elastic collision Hamiltonian system Kam theorem Lyapunov exponent Pi Poincare section Reflection Unfolding

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