Forget watching a system in time — plot its STATE instead. For a pendulum, one axis is position, the other is momentum, and the entire motion becomes a single point tracing a curve. Closed loops mean it repeats forever; spirals mean it’s dying out. The shape of the curve tells you everything, at a glance. It’s the map ScaleSpaceSynth navigates. Slide the energy and watch the orbit grow.
Phase space represents a system’s complete state as a point whose coordinates are the positions and momenta; its evolution traces a TRAJECTORY. For a conservative oscillator (energy E = ½(x²+p²) constant) the trajectory is a CLOSED orbit (an ellipse) — each energy a nested loop; damping spirals inward, driving can form limit cycles or chaos. Liouville’s theorem says the flow preserves phase-space volume for Hamiltonian systems. It is the natural stage for dynamics, from a pendulum to statistical mechanics. A fail-loud self-check throws unless the oscillator’s energy stays constant (a closed orbit). ◆ real dynamical systems, node-verified.
A conservative harmonic oscillator is shown (closed ellipses); real systems add damping, forcing, and higher dimensions. Energy conservation and the closed-orbit shape are exact (symplectic integration).