Pull a mass on a spring, let go, and it oscillates forever — in the ideal. One line, m x″ = −k x, and out falls a frequency that ignores amplitude, an energy that never leaks, and three named ways the motion dies when you add friction. This is the linear heartbeat hiding inside almost every physical model: the terms go in, the integrator runs, the proven law comes out. The blue team builds and defends it; the red team tries to break it.
source Robert Hooke, Lectures de Potentia Restitutiva, or Of Spring (1678) — the law ut tensio, sic vis ("as the extension, so the force") — archive.org/details/bim_early-english-books-1641-1700…hooke…1678. Rendered, not quoted. AMBER facsimile, no canonical DOI.
Hooke's law says the restoring force is linear: F = −k x. Newton turns that into m x″ = −k x − c x′ — a mass, a spring, an optional dashpot. Four things fall straight out:
1 angular frequency ω = √(k/m), so period T = 2π√(m/k) — independent of amplitude. 2 undamped energy E = ½k x² + ½m v² is constant. 3 the closed form x(t) = A cos(ωt + φ) solves it exactly. 4 with damping, c vs 2√(km) sorts the decay.
Live values for the panel's current spring:
Take the-pendulum: θ″ = −(g/L) sinθ. For small θ, sinθ ≈ θ, and it becomes this spring — the same m x″ = −k x with k/m = g/L.
That is the whole trick of linear physics: near any stable equilibrium, the potential looks like ½k x², so the motion looks like a spring. It is the normal mode of every vibration — the string, the crystal lattice, the LC circuit, the molecule. Each sphere reduces, at small amplitude, to this one.
The blue team's live check: re-run the physics — undamped energy bounded, the three regimes sorted, the tamper caught. If red flips the spring sign, this badge is where it shows.
Three numbers fix the whole motion, plus where you start it:
| symbol | meaning | units |
|---|---|---|
| m | mass | kg |
| k | spring stiffness | N/m |
| c | damping (dashpot) | N·s/m |
| x₀, v₀ | start position, velocity | m, m/s |
Everything else — frequency, energy, regime, the entire trajectory — is computed, never entered. Feed these into the engine below.
Symplectic integrator (semi-implicit Euler, det J = 1) — energy stays bounded, never invented.
Left: the mass on its spring, live. Right: the phase portrait (x, v) — a closed ellipse when undamped, a spiral to the origin when damped. Move any slider; the motion recomputes from m x″ = −k x − c x′.
What the machine proves, checked live on load: simple harmonic motion at ω = √(k/m) with period T = 2π√(m/k), amplitude-independent; the closed form A cos(ωt+φ) satisfies the equation exactly; undamped energy is conserved (bounded to <1% over 20 periods); and c vs 2√(km) sorts every damped motion into under-, critically-, and over-damped, with the discriminant c²−4km changing sign at the boundary.
The blue witness (left) confirms it live; the red team (right) flips the spring sign to try to make it lie.
Real damping is rarely a clean c x′ either — dry (Coulomb) friction is constant in magnitude, air drag goes as v². And no macroscopic oscillator is truly undamped; "energy conserved" is the idealization the next line of the model has to pay back. The spring-mass is not the physics; it is the first term of every physics.
"A heavier mass swings slower, so it also swings with a shorter reach." Cut. Frequency drops as √(m), but amplitude is set by release, not by mass — they are independent. The engine holds A fixed while ω changes.
"A bigger push gives a longer period." Cut. SHM is isochronous: T = 2π√(m/k) has no amplitude in it. (This is exactly what fails once the spring goes nonlinear — see the adversary.)
"Critical damping is the fastest return, so it overshoots least by oscillating a little." Kept, corrected. Critical (c = 2√(km)) returns fastest with no oscillation at all; the discriminant is exactly zero — the boundary case, computed here.
The red team's move: flip the spring's sign to +k x — a force that pushes away from equilibrium instead of back. The motion stops oscillating and runs away exponentially; energy explodes. The blue witness (window 7) is watching.
Flip the sign and the SHM / bounded-energy check fails: the mass diverges, the phase portrait flies open, and the witness recomputes, disagrees with the conserved law, and turns red. Nothing is faked; the attack is real and it is caught.