Brachistochrone Spectral Ritz Flow (2D)

A genuinely 2D-native companion to the vertex-mesh brachistochrone gradient flow: the trial path is a Fourier sine series with a handful of free coefficients instead of dozens of free points, so gradient descent operates directly in mode space. The canvas shows three linked views at once — the reconstructed curve against the analytic cycloid, a live bar chart of the mode spectrum shrinking toward the cycloid's own Fourier projection, and a running trace of the descent-time functional converging onto its known closed-form floor — turning the abstract idea of a truncated basis expansion into something you can watch settle.