Lemma
math, backwards
journey · 4 days · physics

Where Change Vanishes

A pendulum hung from above swings, in principle, forever. A raindrop falls to a constant speed and never accelerates again. A car spring jiggles, then settles. Three pages with three different physics, but one question underneath each: where does net force vanish, and how does the system relate to that point?

Equilibrium in Lemma is a shape — a three-step procedure (two opposing forces → balance equation → fixed point) that appears, under different names, across three physics applications. This path opens with the derivative — the tool that names “no change” — and walks the three pages, one each for the three ways a system can relate to its equilibrium: orbit it, approach it, settle into it.

the path · 0/4 · 0%
  1. 1
    module·day 1·→ next
    /modules/derivatives
    Open with the tool that *names* equilibrium. A derivative is the rate of change; setting a derivative equal to zero is what equilibrium *means*. Read Arc 5 — *force balance is exactly where the derivative vanishes here* — and note the pattern before you meet it three times.
    open →
  2. 2
    application·day 2
    /physics/pendulum-clock
    First instance — *orbit*. The pendulum has an equilibrium at θ = 0, but it never settles there in the ideal model — it *orbits* the equilibrium, swinging through it twice per period. The whole 17th-century clock technology lives in this *periodic-around-equilibrium* regime. The equilibrium is the *axis* of the motion, not its end.
    open →
  3. 3
    application·day 3
    /physics/terminal-velocity
    Second instance — *approach*. The raindrop's equilibrium $v_t = g/k$ is never quite reached — gravity pulling down and drag pushing up approach equality asymptotically, never algebraically. The trajectory *bends toward* the equilibrium without touching it. Calculus is the only honest description of *almost there forever*.
    open →
  4. 4
    application·day 4
    /physics/damped-oscillator
    Third instance — *settle*. With both a restoring force and a damping force, the system *reaches* equilibrium — but the shape of the approach depends on the damping ratio. Underdamped: oscillates while shrinking. Critically damped: returns straight, fastest. Overdamped: crawls back slowly. Same equilibrium, three different ways to get there. *Pendulum + terminal-velocity, generalised — and now the equilibrium can actually be reached.*
    open →