homework and exercises - Explanation: Simple Harmonic Motion

I am a Math Grad student with a little bit of interest in physics. Recently I looked into the Wikipedia page for Simple Harmonic Motion.

Guess, I am too bad at physics to understand it. Considering me as a layman how would one explain:

• What a Simple Harmonic motion is? And why does this motion have such an equation $$x(t)= A \cos(\omega{t} + \varphi)$$

associated with it? Can anyone give examples of where S.H.M. is tangible in Nature?

Answer

This is all about potential; it is common that a particle movement is described by a following ODE:
$m\ddot{\vec{x}}=-\nabla V(\vec{x})$,

where $V$ is some function; usually one is interested in minima of $V$ (they correspond to some stable equilibrium states). Now, however complex $V$ generally is, its minima locally looks pretty much like some quadratic forms, and so the common assumption that $V(x)=Ax^2$... this makes the last equation simplify to:

$\ddot{x}=-\omega^2x$,

with solution in harmonic oscillations.

The common analogy of this is a ball in a paraboloid dish resembling potential shape; it oscillates near the bottom.