Ciro Santilli Lagrangian Mechanics Example: The Compound Atwood Machine by Michel van Biezen (2017)
. Source. Part of lagrangian mechanics lectures by Michel van Biezen (2017).The simplest harmonic oscillator system.
The period does depend on the initial height/angle, except for the very small angle approximation:
youtu.be/Ca7c5B7Js18?t=803 compares Lagrangian mechanics equation vs the direct x/y coordinate equation.
resonance in a mechanical system.