The Dynamics scene varies a whole history between fixed events. Here we evaluate the action on the classical history for each endpoint. That endpoint function is Hamilton’s principal function. We keep m = 1 kg, g = 9.81 m/s², A = (0, 0), and positive y downward, so V = −mgy.
The classical history from A to (x, y) in a duration T > 0 has principal action S(x,y,T). Using t as the duration in the formula,
Fixing the full initial velocity leaves just one ray. Instead, fixing its magnitude v₀ and allowing every direction fixes E = mv₀²/2. For each classical route, the abbreviated action W = ∫p · dq gives a separate phase branch S = W − Et. The moving lines are S = kΔS for a continuous source. The phase clock repeats one period ΔS/E; it is not a common flight time for the two rays.
This is the mechanical eikonal equation: the optical index is proportional to the momentum magnitude √[2m(E − V)]. It is not the inverse particle speed used in the brachistochrone travel-time problem. Hamilton–Jacobi reads ∂S/∂t + |∇S|²/(2m) − mgy = 0.
For a point inside the reachable region, two positive flight times solve v₀² = x²/T² + (y/T − gT/2)². Each time gives its own trajectory and its own W = S(x,y,T) + ET. They coalesce at the envelope
Above that parabola, no real classical ray at this energy reaches the point. The two colors keep the branches separate; their overlap is not interference. At the top of the caustic the particle stops momentarily, p = 0, and a front normal is undefined. The zero-duration source branch also has no unique momentum direction.
Fronts and particles generally travel at different speeds. A particle crosses fronts; it does not ride a crest. The faint ray fan shows trajectories, while the colored curves through B show its classical routes. Equal spatial scales make the right angles meaningful. Use the mouse wheel to zoom; changing the physics never changes the camera scale.
Launch A → B animates a dot on each visible route with the same elapsed physical time as the fronts. The short route arrives first and its dot waits at B; the waves continue. Launch fan instead sends one simultaneous set of 32 particles along the displayed rays, all with the chosen launch speed. It emits only once per click. Particles follow exact ballistic paths and cross the existing fixed-energy fronts.
Fan particles are cyan on the short-time action branch and violet on the long-time branch. An upward ray changes branches when it touches the caustic, at v₀² + gvᵧ₀t = 0; the instant of contact is amber. The branch buttons filter both fronts and fan particles. The colors describe the local action sheet, while the highlighted A → B paths retain their route colors. Play/Pause controls all particles and fronts together; Restart returns the current sets to A. Unresolved contour levels may be omitted at distant zoom; a label states the sampling.
This is an exact classical phase construction, not a numerical wave-equation solution. It does not calculate amplitudes, diffraction, interference, or the wave treatment that resolves a caustic. Stationary action is not generally a minimum. The fixed-time uniform-gravity problem is a special case with a positive quadratic excess action.