Principle of Least Action: Path Relaxation (2D)

Trial Path

Gravity g 9.8 m/s²

Relaxation rate 1.0×

Path resolution (nodes) 24

Live State

Action S

—

Gradient steps

Deviation from exact

—

Status

relaxing…

How it works

Hamilton's principle says a real particle travels the path q(t) that makes the action functional stationary (a minimum for this setup):

S[q] = ∫₀ᵀ L(q, q̇) dt,   L = ½m|q̇|² − mgy
δS = 0  ⇒  Euler–Lagrange:  d/dt(∂L/∂q̇) − ∂L/∂q = 0  ⇒  ẍ=0, ÿ=−g

This is the planar (x, y) reduction of the 3D solver's own mechanism — the 3D version's unforced z axis obeys the exact same force-free law as x (ẍ=0, z̈=0) and carries no independent physics, so dropping it loses nothing and lets the whole trial path live natively on one 2D canvas instead of behind a 3D camera. The path is still cut into N straight segments with fixed endpoints, and the discretized action is still minimized directly by gradient descent — no equation of motion is hard-coded, only S itself:

S ≈ Σᵢ [ ½m|pᵢ₊₁−pᵢ|²/dt − mg·dt·(yᵢ+yᵢ₊₁)/2 ]
∂S/∂pᵢ = m(2pᵢ−pᵢ₋₁−pᵢ₊₁)/dt − mg·dt·ŷ   (interior nodes)
pᵢ ← pᵢ − η·∂S/∂pᵢ

Perturb Path — re-randomizes the interior nodes into a jagged trial curve and restarts the relaxation.
Gravity g — sets the target curvature: the exact minimizer is the parabola y(t) = y₀+(y₁−y₀)t/T + ½g·t(T−t).
Relaxation rate — scales the gradient-descent step size η.
Path resolution — number of free interior nodes; more nodes track the true curve more closely but relax more slowly.
Drag a node — grab any interior node (glowing dot) and pull it; it's pinned exactly where you drop it and the rest of the path relaxes around your perturbation until you drag it again.
Pan / zoom — drag empty canvas space to pan, scroll or pinch to zoom around the cursor.

This relaxation method is the same numerical idea used to solve boundary-value Euler–Lagrange problems in finite-element and optics ray-tracing codes: turn "find the stationary path" into "descend the gradient of S".

Trial path (relaxing) Exact minimizer

drag node to pin · drag canvas to pan · scroll to zoom