From Infinite Subdivision to Rotation Matrices: How I Derived the Equation of Simple Harmonic Motion by Hand
High-school physics tells us that the displacement in simple harmonic motion (SHM) is a sinusoidal function. But textbooks never explain why — they assume a solution of the form $x(t)=\sin(\cdots)$, substitute it into the equation, verify that it works, and call it a day. That is a working backwards from the unknown to the known approach, and it has always left me uneasy. So I decided to do the opposite: starting from Hooke’s law as the sole axiom, I would analyse the evolution of velocity step by step and see whether I could derive the unknown from the known — letting that sine (or cosine) emerge on its own. This post records my two attempts: the first using brute-force infinite subdivision, the second using matrices and geometry. From complicated to clean, I ended up colliding head-on with Euler’s formula. ...