# An Inverted Pendulum: Defying Gravity (and Intuition)

**Figure 1.**The pendulum is a point mass on a massless rigid rod.

Wishing to observe the Kapitsa effect directly, I had originally planned to build a device to vibrate the pivot but fortunately realized, before spending time and money, that I have this device at home. The resulting demonstration can be seen in this video.

The inverted pendulum was V.I. Arnold’s favorite demonstration; he used an electric shaver which has a reciprocating arm inside.

**Figure 2.**The leading order approximation of the motion.

Figure 2 shows the pivot oscillating between two points (the amplitude is greatly exaggerated in this figure). We assume that the pivot’s acceleration is very large (thus so is the frequency); this means that *the rod is under great tension or compression for most of the period*. This great force acting upon the bob is aligned with the direction of the rod. As the leading order approximation, we assume that the bob actually moves in the direction of this force. This assumption forces the bob to travel in an arc \(AB\) of a tractrix, i.e. the pursuit curve.^{1}

**Figure 3.**The tractrix, or the pursuit curve: all tangent segments have the same length.

\[\overline {mkv^2}>mg \enspace \sin \theta, \qquad (1) \]

**Figure 4.**The centrifugal force \(mkv^2\) of the non-existent constraint is responsible for stability.

^{2}for small \(\theta\):

\[\overline{u^2}>Lg, \qquad (2)\]

the linearized stability criterion (see [3] for more details). Although this non–rigorous calculation looks suspiciously easy, it does give exactly the same result as the formal derivation due to Kapitsa, as reproduced in Landau and Lifshitz [2] and almost a page long. Incidentally, (2) turns out to be equivalent to the stability condition \( \vert {\rm det}\; F \vert <2\) of the Floquet matrix \(F\), providing a physical interpretation of this condition in terms of the centrifugal force of a non–existent constraint. To conclude, and as a side remark, a purely topological explanation of stability of the inverted pendulum, in a different regime, can be found in [4].

^{1} The tractrix is defined by the property that all the tangent segments connecting it to a straight line have the same length, Figure 3.

^{2} making an additional assumption of small amplitude, allowing us to treat \(k\) as a constant along the short arc of the tractrix.

**References**

[1] Arnold, V.I. (1992). *Ordinary Differential Equations*. Berlin: Springer-Verlag.

[2] Landau, L.D. & Lifshitz, E.M. (1976). *Mechanics*. New York: Pergamon Press.

[3] Levi, M. (1999). Geometry and physics of averaging with applications. *PhysicaD, 132*, 150-164.

[4] Levi, M. (1988). Stability of the inverted pendulum – a topological explanation. *SIAM Review, 30*, 639–644.

[5] Paul, W. (1990). Electromagnetic traps for charged and neutral particles. *Revs. Mod. Phys., 62*, 531-540.

[6] Stephenson, A. (1908). On a new type of dynamical stability. *Manchester Memoirs, 52*, 1-10.