Principle of Least Action: Path Relaxation

Trial Path

Gravity g 9.8 m/s²

Relaxation rate 1.0×

Path resolution (nodes) 24

Live State

Action S

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Gradient steps

Deviation from exact

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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, z̈=0, ÿ=−g

The path is cut into N straight segments with fixed endpoints. The discretized action is 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.
The x and z coordinates carry no potential, so their extremal path is simply the straight line — watch them straighten out exactly like y bows into a parabola.

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

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