THE PRINCIPLE OF LEAST ACTION ·

Of all conceivable paths between two fixed events, nature takes the one whose action S = ∫ L dt is stationary (δS = 0). Fix the endpoints, wiggle the path, and the action rises — the physical trajectory sits at the bottom of that valley, and Newton's laws fall out of it.
source Maupertuis, «Accord de différentes lois de la nature» (1744); Euler, Methodus inveniendi… Additamentum II (1744); Hamilton, “On a General Method in Dynamics”, Phil. Trans. R. Soc. (1834). Rendered, not quoted.
Blue Team · builds & defends
3The Model
A path is a function x(t) on [t₀,t₁] with the endpoints pinned. Its action is the time-integral of the Lagrangian L = T − V:
S[x] = ∫ ( ½ m ẋ² − V(x) ) dt
We discretize into N segments (natural units m=1), so S = Σ ½(Δx/Δt)²Δt − Σ V̄Δt. The engine solves for the path where the gradient of S vanishes at every interior node — that is the extremal. Two systems: the free particle (V=0) and the harmonic oscillator (V=½kx², k=1, over t∈[0,1] < ½ period, so the extremum is a genuine minimum).
5The Lineage
δS = 0 is not a new law — it is the law, restated as geometry. Setting the variation to zero yields the Euler–Lagrange equation d/dt(∂L/∂ẋ) − ∂L/∂x = 0, which for L = T−V is exactly mẍ = −V′(x) — Newton's F = ma. This sphere is the variational root feeding the-lagrangian: the panel proves numerically that the stationary path satisfies E–L node by node (aᵢ = −V′(xᵢ) to 1e−9).
Maupertuis → Euler → Hamilton: one principle, all of mechanics.
7The Witness
Live re-check of the claim “the declared path is the physical one.” It recomputes the gradient of S on the currently-declared path and compares its action to the true extremal. If a neighbour has lower action, the claim is false and this badge flips red.
…
The Machine
4Data In in ↓
endpoints x₀,x₁1.0 → 0.3
interval t0 → 1
segments N40
step Δt0.025
potential V½kx², k=1
free-particle V0
Endpoints fixed; the path between them is the unknown.
▼ SOLVE δS=0 ▼
0The Panel LIT
S(extremal)—
∇S norm—
best wiggle S—
E–L residual—
Accent curve = extremal (physical) path · faint curves = endpoint-fixed wiggles, all with higher action.
▼ PROVEN ▼
8Data Out out ↓
The stationary path minimizes S among all endpoint-fixed neighbours and satisfies Euler–Lagrange exactly.
…
Red Team · attacks & breaks
1The Adversary WALL
“Least” overstates it. The path is stationary, not always minimal. Past a conjugate point — for the oscillator, beyond half a period — the classical path becomes a saddle of S, and cheaper neighbours exist. The panel deliberately stays under half a period so the extremum is a true minimum.
It is a boundary-value problem, not an initial-value one: fixing two endpoints can admit zero, one, or many extremals. It does not mean a particle “sniffs out” the future.
Discretization is not free — coarse N inflates S by O(Δt²). Comparisons here use a fixed N and identical grids, so the wiggle margins are real, not artifacts of the mesh.
2The Graveyard
Nature always minimizes action.
Nature makes it stationary (δS=0); minima, maxima, and saddles all qualify.
The particle tries every path and picks the cheapest.
The stationarity condition is local and equivalent to Newton's law — no teleology, no global search.
Least action is a separate axiom on top of F=ma.
It is equivalent to F=ma via Euler–Lagrange, not additional to it.
Any wiggled path with the same endpoints is “close enough” to physical.
Only the extremal has ∇S=0; every wiggle raises S — that is exactly what the tamper proves.
6The Tamper
Disclosed planted void. Press tamper to declare a wiggled, non-extremal path as “the physical one.” Its action is no longer stationary — the true extremal, a nearby path, has strictly lower S — so the Witness (7) catches it live.