Bouncing Shapes Concave Surface

Build a picture out of thousands of little balls, hold it in the air, and let go. Gravity does not care what the picture was: every ball falls into the bowl and bounces. On each bounce the velocity reflects about the local surface tangent, the normal part flipped and scaled by the restitution $e$, so with $e = 1$ the balls bounce forever and energy is conserved, while $e \lt 1$ bleeds energy away on every hit until the shape pools into a still layer at the bottom. The bowl profile sets the motion: a parabola gives near simple-harmonic swings, a V-bowl constant-acceleration sweeps, a steep quartic strongly amplitude-dependent ones. The diagnostic tracks the total energy, flat when nothing is lost and a descending staircase when it is.

Figure 1. Balls forming a shape, dropped into a selectable concave bowl. Top: the shape shatters on the surface and bounces, each impact reflecting velocity about the local tangent with restitution e. Bottom: total energy over time, conserved at e = 1 and stepping down at e < 1. Method: gravity plus tangent-plane restitution.

bowlparabola

shapestar

restitution e0.88

curvature a0.55

balls320

WHAT TO TRY

Pick a shape and drop it: it holds for an instant, then shatters into the bowl and pools at the bottom.
Set restitution e = 1: nothing is lost, the balls bounce forever and the energy line stays flat.
Lower e: each bounce loses a fraction of the normal energy, the energy line steps down, and the shape settles faster.
Change the bowl: a parabola swings like a spring, a V-bowl sweeps at constant acceleration, a steep quartic traps the balls near the bottom.