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.

Kapitza pendulum

Pivot shakes vertically: fast driving can stabilize the inverted equilibrium — parametric pumping in θ̈ + (g/L) sin θ ≈ (Aω²/L) cos(ωt) sin θ.

About this model

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.

Continue with similar topics in this category — or all 85 in Classical Mechanics.

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