1. For energy conserving motion with one degree of freedom, we can get the equation
of motion just by requiring energy conservation. For example, if
E = 1/2 m (dx/dt)^2 + V(x), then dE/dt = [m (d^2 x/dt^2) + dV/dx] dx/dt.
As long as dx/dt is not zero, this imples Newton's law  ma = - dV/dx.

2. Using the above, we can relate all harmonic oscillators with one degree of freedom.
If the coordinate labelling that degree of freedom is called Q (for example,
the x-coordinate of a particle, the height of a column of water, the angle of a
physical pendulum, etc.), and the kinetic and potential energies take the form
K = 1/2 A (dQ/dt)^2 and V(Q) = 1/2 B Q^2, then the motion is harmonic,
with angular frequency the square root of B/A.

3. Small motions about an equilibrium configuration are approximately harmonic.
The reason is Taylor's theorem: For any potential V(x), the force vanishes at any x_e where
V'(x_e)=0. That is called an equilibrium point. Making a Taylor expansion of V(x)
about x_e we have V(x) = V(x_e) + 1/2 V''(x_e) (x - x_e)^2 + 1/3! V'''(x_e) (x - x_e)^2 + ...
The constant piece produces no force, and the next nonzero piece has the harmonic oscillator
form, with spring constant k = V''(x_e). For motions with small enough amplitude, the
rest of the terms are negligible, so we have an approximate harmonic oscillator.

4. Multiplication by the imaginary number i is equivalent to counterclockwise rotation
though  Pi/2 radians in the complex plane. More generally, multiplication by e^is
is equivalent to counterclockwise rotation though  s radians.