Kapitza pendulum
Pyotr Kapitza showed that a pendulum whose pivot oscillates vertically at high frequency can remain trapped near the inverted configuration — a striking example of parametric stabilization. This page integrates a textbook-style driven equation in which a cosine vertical acceleration of the suspension modulates the restoring term. You can compare slow driving (pendulum falls) with faster, larger driving where the inverted state is easier to maintain. The visualization exaggerates pivot motion for clarity; parameters are chosen for teaching, not for a quantitative match to a particular lab bench.
Who it's for: Undergraduate mechanics and anyone learning parametric resonance, separation of fast/slow motion, or inverted pendulum control intuitions.
Key terms
- Parametric driving
- Inverted pendulum
- Kapitza pendulum
- Vertical pivot oscillation
- Stability
- RK4 integration
How it works
Vertical oscillation of the pivot adds a fast modulated “effective gravity” term. At high enough drive frequency and amplitude, the inverted pendulum can stay near θ ≈ π — a classic parametric stabilization demo.
Frequently asked questions
- Is this the full three-dimensional rigid-body model?
- No. The canvas shows a planar pendulum with a prescribed vertical pivot motion and a single angle θ. Real apparatus has finite pivot stiffness, air drag, and motor limits; those are omitted.
- Why does the rule-of-thumb mention Aω² and g?
- In averaged fast-driving models the effective potential develops a minimum near the inverted angle when the drive is strong enough; Aω² ≳ 2g is a common order-of-magnitude mnemonic for the onset of trapping, not a universal threshold for every initial condition.
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