Pendulum Contained by Cycloid moves along Cycloidal Path

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Theorem

Let a pendulum with a flexible rod be suspended from a point $P$.

Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs.

CycloidPendulum.png

Then the bob is constrained to move such that its path traces the arc of a cycloid.


Proof

From Evolute of Cycloid is Cycloid, the evolute of a cycloid is another cycloid.

From Curve is Involute of Evolute, the involute of a cycloid is another cycloid as well.

But by the definition of involute, the path defined by the pendulum as described is the involute of the cycloid.

Hence the result.

$\blacksquare$


Historical Note

The discovery that a Pendulum Contained by Cycloid moves along Cycloidal Path was made by Christiaan Huygens during his work on developing a reliable and accurate pendulum clock.

Unfortunately, the technique proved impractical, as the energy losses caused by the mechanics of the system compromise the pendulum's ability to swing reliably for a practical length of time.


Sources

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