Pendulum Contained by Cycloid moves along Cycloidal Path
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.
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
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid