Explanation: Simple Harmonic Motion

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.

Simple harmonic motion (SMH) describes the behavior of systems characterized by a equilibrium point and a restoring "force" (in some generalized sense) proportional to the displacement from the equilibrium.

Example system

A simple mechanical system with this behavior is a mass on a spring (which we will consider in one dimension for ease). There is some point where the mass is stable: it can be left alone with no external forces on it and will not accelerate. That's the equilibrium, call it $x_0$. If you move the mass from that point, the spring exerts a restoring force $F = -k(x-x_0)$. Here $k$ is a property of the spring called the "spring constant"; a stiff spring hs a high spring constant and a weak spring has a low value for $k$. Lets consider the special case where $x_0 = 0$ (with no loss of generality, but just to keep the number symbols to a minimum).

So the acceleration of the mass at any point in time is

$$a = \frac{d^2x}{dt^2} = \frac{F}{m} = -\frac{kx}{m} $$

which is a second order linear differential equation, the solutions to which are of the form you exhibit above (where $A$ and $\phi$) take the part of the two constants of integration needed to make the solution agree with an arbitrary set of boundary conditions.

Let's prove it by explicit substitution: $$x(t)=A \cos(\omega t + \varphi )$$

$$v(t) = \frac{d}{dt} x(t) = A \sin(\omega t + \varphi ) \omega $$

$$a(t) = \frac{d}{dt} v(t) = \frac{d^2}{dt^2} x(t) = -A \cos(\omega t + \varphi ) \omega^2 $$ so substituting back in we get $$ -A \cos(\omega t + \varphi ) \omega^2 = \frac{F}{m} = -\frac{k x}{m} = -\frac{k A}{m} \cos(\omega t + \varphi ) $$ which implies that $$ \omega^2 = \frac{k}{m} $$ which relates the spring constant and the mass of the object to the frequency of oscillation.

General discussion

The thing that is important about SHM is that these restoring "forces" linearly proportional to "displacement" (which we allow that there may be generalized meanings for both "force" and "displacement") are very common in the universe. So a great many phenomena may be described this way (and even more may be if we limit ourselves to small perturbations).

I guess this thread shows everyone has their own tastes when it comes to this topic!

First of all, it's called harmonic motion because sine and cosine are the elementary harmonic functions. Recall that in general a harmonic function is a solution of Laplace's equation (which shows up everywhere in physics), and in we initially study $sin(x+vt)=sin(kx+\omega t)$ since these are the building blocks of solutions of the wave equation. In QM, the principle of superposition (from ODE's) takes on an entirely different, physical meaning.

It's also obvious that the equation of harmonic motion is the projection of $e^{i\omega t}$ onto the real axis, which is a standard trick to derive this equation.

Now an important example of harmonic oscillators in nature are the atoms in a solid. This is basic 1905 Einstein, in 1905 Einstein showed that atoms in a solid can be treated as quantum harmonic oscillators, and that this explains the behavior of specific heat at low temperatures.

Another important example is that the EM field can be considered as three space where each point is a quantum harmonic oscillator. In fact, the EM field is quantized by considering the Fourier components of the field to be creation and inhalation operators for photons of a given frequency $\omega$, which are exactly alike to the raising and lowering operators of the QM harmonic oscillator.

The final standard remark is that almost all small oscillations in nature can be approximated by $F=-kx$, by a Taylor expansion of the force $F(x)$ to first order in $x$.