Physics
Stationary Action
Nature does not follow forces step by step, or so the other story goes: given where it starts and where it ends, it takes the path whose action cannot be improved on. Bend the path yourself and watch the number get worse, then let it fall back.
Drag the curve in the top panel to bend it anywhere; the action is computed from your actual path, not from a formula
Action of your path—
—
What it moves in
push downpush up
Numbers
Action of your path—
Action of the real path—
Your action minus theirs—
Mean Lagrangian—
Mean KE and PE—
Energy KE + PE swings by—
Worst F = ma error—
What to observe
- Bend the path up or down and the action rises either way. The real path sits where the S curve is flat: its slope is zero, which is what "stationary" means.
- Press Let it relax and the path rolls downhill in S back onto the real one. Each step enforces ma = −dV/dx a little more, because the gradient of the action is exactly that error.
- With nothing to move in the real path is a straight line at constant speed: Newton's first law out of a principle about whole paths.
- With a spring and a long enough trip the S curve turns into a hill: a broad bend now lowers the action. The principle only ever promised that small changes stop mattering, never the smallest S.
Shortcuts:space run/pause · s step · r reset · f fullscreen