Lyapunov Descent
(2 · exp(−γ · u) · (cos(ω · u) + γ/ω · sin(ω · u)), ((γ² + ω²) · (2 · exp(−γ · u) · (cos(ω · u) + γ/ω · sin(ω · u)))² +(−2 · exp(−γ · u) · (γ² + ω²)/ω · sin(ω · u))²)/2, −2 · exp(−γ · u) · (γ² + ω²)/ω · sin(ω · u))
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What it draws
A damped oscillator, x″ + 2γx′ + (γ² + ω²)x = 0, let go from x = 2 at rest. The x and z expressions are its exact solution: the position x and the velocity x′, the state, drawn across the floor. The damping γ opens at 0.1 and the frequency ω at 1.
The height is the energy, V = ((γ² + ω²)x² + x′²)/2: the spring's share and the motion's share. So the curve is the state's path lifted onto the bowl of V, and it starts at V = 2.02.
Why it goes downhill
Lyapunov's idea is to prove a rest point stable without solving anything. Find a function that is 0 at the rest point, positive everywhere else, and never increases along the motion. Then the state can never climb away.
Here the rate of change of V along the motion is −2γx′². For γ > 0 it is never positive, so the curve only goes down. Each turn of u keeps e^(−4πγ) ≈ 0.29 of the energy.
The rate is 0 at each end of a swing, where x′ = 0. So the descent is not strict: it levels off twice a turn. Lyapunov's own test then proves only that the state stays near the bottom, not that it gets there.
LaSalle's invariance principle settles it. The only motion that keeps x′ = 0 for good is resting at the bottom, so the state cannot stay on a level and reaches the rest point anyway.
History
Aleksandr Lyapunov set out the method in his 1892 thesis on the stability of motion. Joseph LaSalle extended it in 1960 to rates that are only never positive, the case drawn here.
Try
- Drag γ to 0. The rate is 0 everywhere, and the curve circles on one level ring at height 2: stable, but not asymptotically.
- Drag γ to −0.1. The energy grows about 3.5 times a turn, and the curve climbs out of the bowl: unstable.
- Drag γ to 1. Each turn would keep about 3.5 millionths of the energy, so the state slides to the bottom within its first swing.
- Press Front in the deck. The height falls fastest as the state crosses the middle and levels off at each end of the swing.