Overview: Motivating the harmonic oscillator

Why bother with the harmonic oscillator?

The modest little problem of a mass hanging on a spring is covered in almost every introductory physics class. This might seem strange. We almost never encounter a mass hanging on a spring in everyday life and certainly not in biology. Why should we care?

The reason we do this problem in such great detail — considering multiple similar systems, adding damping, adding driving forces, doing all the math — is because, while the mass on a spring system is rarely seen, oscillations are everywhere. They occur in biological organisms in the detection of sound in the ear, the beating of the heart, and they are the critical phenomenon in the functioning of the brain. Coupled oscillators (especially non-linear ones) are the basis of important models of the functioning of neural nets in the brain. Oscillations are even observed in a variety of chemical reactions.

Our modest little model is the toy model that provides insight into the mathematics that describe all oscillations. Perhaps even more important, the math of a single oscillator is extended to infinitely many oscillators to allow us to develop the mathematics of waves, including sound, light, and the propagation of nerve signals. The powerful tools of spectral and Fourier analysis, used in many biological applications, are based on the math of the simple harmonic oscillator like the mass on a spring. 

The physics that leads to oscillation

There are three core situations that lead to a system oscillating.

1. The system has a stable point where all the forces on the system are balanced (net force = 0).
2. If the system deviates away from that stable point (for whatever reason) it experiences a force that tends to push it back to where it started.

That force tends to accelerate the system back towards its stable point. If the motion of the system is governed by anything like Newton's second law, the force makes it accelerate towards the stable point. As it approaches the stable point, the net force weakens since it vanishes at the stable point. But if the object is moving, it will continue to move through the stable point, not stopping, since there is no force there to stop it. This gives us our third core concept necessary for oscillation:

3. The system overshoots the stable point. The force will then build up to slow it down, but not before it has gone the other half of the oscillation. The situation will then repeat.

We'll go through this in detail in specific cases so it might make more sense then.

If there is a stable point, then the system's potential energy should look like a well with a minimum point. The (classical) system will sit at the minimum point when it is stable. (A quantum system never can sit exactly at rest at a minimum point, which is why we draw our molecular bound states not exactly at the bottom of the potential well.) If the restoring force can be treated as linear — or equivalently, if the potential energy can be treated as a parabola — then the motion is called harmonic. It's called "harmonic" because the solution of Newton's second law (a second order differential equation that determines the motion of the object) are sines and cosines of time with a particular frequency — just like the result produced by a pure musical tone heard at a particular point in space. The simplest example of this is a mass on a spring, so we will treat that example in great and gory detail.

The math generalizes!

But harmonic oscillation has value far beyond the case of a mass on a spring. For almost any stable situation, the energy for small enough deviations around the stable point can be approximated by a quadratic (parabola) and this is equivalent to a mass on a spring. What we learn in mathematical modeling is

If the equations for two different systems are the same, then each one serves as a good analogy for the other.

Since the equations that describe the mass on a spring appear in so many other cases, it is a useful analogy, and a valuable conceptual tool to have in your toolkit.

Joe Redish 3/11/12, 5/27/19