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.

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