The attracting centre is O = (0, 0), the source is A = (−2.4, 0) m, the particle mass is 1 kg, and μ = GM = 8 m³/s². The acceleration has magnitude μ/r² and points inward. Fixing launch speed while varying direction produces a family of Kepler ellipses with one energy and one semimajor axis. The speed slider stays below escape speed, √(2μ/rₐ) ≈ 2.58 m/s.
For each classical route, W = ∫p · dq is its abbreviated action measured from A. Each local sheet obeys |∇W|² = 2(E − V) and p = ∇W. The displayed curves have S = W − Et = kΔS at a common phase time. They cross trajectories at right angles. Particles cross the moving fronts and need not travel at their speed.
We set V(r) = μ(1/rₐ − 1/r), so V(A) = 0 and E = v₀²/2 is positive. This additive constant leaves every force, orbit, momentum, and W unchanged. It makes the drawn phase levels advance along the rays. With the conventional zero at infinity, the same ellipses instead have negative energy ε = E − μ/rₐ, and levels of W − εt move the other way. Phase-line speed depends on this reference; the particle motion does not.
At fixed negative ε, every ellipse has a = −μ/(2ε). Two ellipses can join A to a generic B, and each can be traversed in either direction. That gives up to four arrival routes before a full circuit. Colors indicate whether each arriving ray has passed its caustic, rather than the identity of its ellipse. Both before-caustic routes can have cyan fronts at the same position with different momenta; likewise for violet. They remain separate action sheets.
The outer envelope is the ellipse |B| + |B − A| = 4a − |A|, with foci O and A. Routes merge there. All surviving trajectories return to A after T = 2π√(a³/μ), another focus where a unique action gradient does not exist. At A no arbitrary arrival arrow is assigned.
This view shows only the first circuit. Further circuits repeat the geometry with added time T and added action 2π√(μa); there are infinitely many such later histories. M marks a point mass: its drawn disk is not a collision surface, so close orbits pass through it and continue. Only approaches within 10⁻⁶ m of the centre terminate, avoiding the singularity at r = 0. Elsewhere the force is exactly inverse square, with no softening or artificial precession.
Writing the position as a complex square z = u² and using dt = |u|² ds turns Kepler motion into u″ = (ε/2)u. We evaluate this oscillator analytically, including its physical time. Along it W = 2εt + 2μs. For each action level, we find the corresponding point on every launch direction and join neighboring points on each sheet. Folding fronts can overlap without being averaged into a single field. This is a classical phase construction: amplitudes, interference, diffraction, and Maslov phase corrections are not simulated.
Launch fan emits one set of 32 equal-speed particles. Launch A → B sends one dot along each visible route, with each dot waiting at B on arrival. Both use the same physical time as the flowing fronts. Play/Pause and Restart act on both sets; changing scenes or refreshing clears the experiment.