What’s a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel?
Poe’s story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function of time L(t).
Using the small angle approximation sin(θ) ≈ θ, the equation of motion is
L θ″ + L′ θ′ + g θ = 0.
If we assume the length is increasing at a constant rate, then with a change of variables described in [1], this differential equation becomes Bessel’s equation and its solutions are Bessel functions.
The authors of [1] state that their primary interest is whether a pendulum can have the motion that Poe describes. But as they indicate in their opening sentence,
The study of the motion of a steadily descending pendulum has practical applications (e.g., to the oscillations of a load being lowered by a crane).
A more recent application would be describing the motion of a package being delivered by a Zipline drone.
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[1] Robert L. Borrelli, Courtney S. Coleman, and Dana D. Hobson. Poe’s Pendulum. Mathematics Magazine, Vol. 58, No. 2 (Mar., 1985), pp. 78–83
My favourite thing about this problem is that you can analyze it successfully using quantum mechanics despite it being classical. If the lowering is slow enough then you can think of the pendulum as a quantum harmonic oscillator that doesn’t jump energy levels. The quantum number associated to the energy level is the thing you can treat as constant and other properties then follow.