We start out with Newton's second law
The first consideration is the one dimensional oscillator. We have one variable x, and the force defined according to:
This leads us to
In other words we are looking for a function whose second derivative is the same function with a minus ...
How about sine or cosine?
The following satisfies this equation. The proof is just differentiating it twice.
It is not quite so obvious that with two real numbers A and δ this is also the complete real solution of the differential equation (where A is the amplitude and δ the phase) but also not so important in this case.
The part that has an immediate application is which gives the frequency. k is the force constant and it increases according to: torsion < bend < stretch (single < double < triple). m is the mass. The highest frequencies are for low mass and high force constant, i.e. X-H stretch. Triple bond stretches are high because of high k, then double bond stretches and so on.
This was the one-dimensional case. Next time I want to show how to reduce the general case to isolated one-dimensional equations like shown here. And that's where the Linear Algebra comes in.