We can expand any potential
Here’s what’s going on in this formula. First of all, if you’re sitting right at
is a very good approximation to our function near
But we don’t need an infinite number of terms to get a good approximation to our potential near the stable equilibrium point. We just need the first non-zero one—the quadratic term:
That’s the potential energy of a spring, with spring constant
And this is why the simple harmonic oscillator is so prevalent. Systems tend to settle into stable equilibrium, and small disturbances make them oscillate around it. So the first thing we should do with any potential energy function is find its stable equilibrium points, and then ask what happens when we perturb slightly away from them. A particle that’s released there will oscillate around the equilibrium in simple harmonic motion, with natural frequency
To understand the physics farther away from the equilibrium points is usually much harder. The bigger
You’ve seen all this in action before if you’ve studied the simple pendulum, though you may or may not have seen it presented in this language. If a pendulum is inclined at an angle
The stable equilibrium is of course at
The second derivative is meanwhile
Note that it’s positive—that’s how we know we’re at a stable equilibrium, as opposed to the top of the arc, where the slope of the potential also vanishes but the second derivative is negative. That’s an unstable equilibrium because if you take a tiny step away from it the pendulum will swing far away.
Our Taylor expansion around equilibrium is then
describing a simple harmonic oscillator with natural frequency
which is the familiar frequency of a pendulum! Taylor expanding around the equilibrium in this context is often called the small angle approximation, because it amounts to expanding the
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