this is not your normal text-book pendulum equation. In the simple form, it is assumed that the angle is small. With this approximation, it could be written as: Textbooks like it this way because ...

A basic pendulum is a mass at the end of ... I can get the following expression: There is a simple solution to this differential equation by assuming a small amplitude of oscillation (and thus ...

Absent any exterior forces, a pendulum would swing back and forth in a single plane forever—there would be no gradual angular shift. But the Earth is rotating, so the story isn’t that simple.