We consider a classical physical system in a stable
static equilibrium. As previously mentioned this implies that
and that the potential energy is at a local minimum.
Such a system has a restoring force which drives it back
towards equilibrium if we try to move it away from the
stable equilibrium condition. Left alone following such a perturbation
the system oscillates about the stable equilibrium. Because stable
equilibria are very common in our universe so are oscillatory
phenomena.
The position vector in an oscillation is periodic in time
and we can write it as
where A is the amplitude of the oscillation and
is a periodic function of time with the
fundamental periodicity, T
Correspondingly we define the fundamental frequency
which is the number of periods of oscillation per unit time (second).
Especially common are so-called harmonic oscillations. These are oscillations which wherein 
is simply a trigonometric
function:
To understand why trigonometric functions appear in this context
we study the kinematics of harmonic oscillations. First we note that
the fundamental period and frequency are related as follows to
These equations are the same as those we derived
for circular motion and are our first indication that there is
a close relationship between harmonic oscillations
and circular motion. We follow the usual prescription in kinematics
of deriving velocity and acceleration by differentiating
and we get:
The last expression which reminds us of that for centripetal
acceleration in circular motion holds the key to
explaining why harmonic oscillations are so important. From Newton's second law we get
Thus harmonic oscillations result from the simplest
possible restoring force: linear in the displacement away
from the stable equilibrium position. Any system with a linear restoring force
supports small angle harmonic oscillations. Moreover the constant of proportionality, k,
determines the natural frequency for the harmonic oscillation:
Larger restoring forces and smaller masses
produce larger natural frequencies.
Another way we can view Eq. 27 is as a differential
equation:
Solutions to this type of linear differential equation is a trigonometric function of the form shown in Eq. 21.