Verlet integration
A way to step the equations of motion forward by storing the previous and current position instead of a velocity.
Velocity is held implicitly in the difference between two positions, so the next one comes from x_new = 2·x − x_prev + a·Δt². The scheme is stable, time-reversible and conserves energy remarkably well over long runs. Loup Verlet popularised it in 1967 for molecular dynamics, though Newton and Delambre had done the same arithmetic far earlier.
Games discovered it through Thomas Jakobsen's 2001 GDC talk on the physics in Hitman: Codename 47. Its great strength is that constraints — "these two points must stay a fixed distance apart" — can be enforced by simply moving the positions and letting velocity follow along on its own. That makes Verlet the natural choice for cloth, rope and ragdolls, and a forerunner of position-based dynamics.