Thread by @j_bertolotti

#PhysicsFactlet (308)
There are not many problems in Physics that can be solved exactly, so we tend to rely on perturbation theory a lot. One of the problems with perturbation theory is that infinities have the bad habit of popping up everywhere when you use it.
(A thread 1/ )

If you know anything about Physics you are probably thinking about quantum field theory and all the nasty infinities that we need to "renormalize". But quantum field theory is difficult, so let's look at a MUCH simpler problem: the anharmonic oscillator.
2/

Disclaimer: I can't know how much you (the reader) know about this. For some of you this thread will be full of obvious stuff. For others there will be so many missing steps to be hard to follow. I will do my best, but I apologize with everyone in advance.
3/

The (simple) pendulum equation has the form θ''+ω₀² sin θ=0 (where θ is the angle with respect to the vertical, θ'' its second derivative with respect of time, and ω₀² is a constant).
Despite its apparent simplicity, this equation is hard to solve.
4/

What is often done is to use the "small angle approximation", i.e. assume that we are looking at small oscillations around the position of equilibrium, i.e. we can approximate sin θ with θ.
The resulting equation (θ''+a θ=0) is an harmonic oscillator, which is easy to solve.
5/

What if our oscillations are not that small? Then the solution of the harmonic oscillator won't be very accurate, and we want to find a better one.
Perturbation theory allows us to find better and better corrections to our simple solution, leading to a better approximation.
6/

The second term in a Taylor expansion around θ=0 of sinθ is proportional to θ³ so [skipping a few steps of perturbation theory] we find the differential equation satisfied by the first-order correction to the simple harmonic oscillator.
7/

This equation can be solved (in my mind the simplest way is by performing a Laplace transform), and we can find our correction.
(Notice that we only care about the inhomogeneous solution, as the homogeneous one is the same harmonic oscillator as above.)
8/

The first term of the solution is easy: it is just a correction to the amplitude of the of the harmonic oscillator.
The second term is weirder, but still ok.: an oscillation at 3 times the frequency of the harmonic oscillator.
9/

As our system is not harmonic, new frequencies will appear when we include higher and higher perturbative terms. Might surprise you the first time you see it, but it is perfectly Physical and easy to measure in the lab.
10/

What is weird is the third term: it is an oscillation at frequency ω₀, but its amplitude grows linearly with time.
This is clearly wrong, as it violates conservation of energy, and leads to a divergence.
11/

It is tempting to declare it an unphysical solution and discard it, but it is instructive to look at it a bit more careful, as it is a simple example of the kind of infinities you get in Physics any time you use perturbation theory to try to find a solution to your problem.
12/

Is this infinity a hole in our theory of simple pendula? Should we throw Newtonian mechanics away because it leads to divergences?
No, no need to panic. This infinity is just an artefact of our perturbative expansion.
13/

The sum of all perturbative corrections give us the solution to our original problem, but what can happen is that terms that appear at one perturbative order will cancel with terms that appear at another perturbative order. Nothing too scary. Newtonian mechanics is safe!
14/

But, short of calculating all the infinite perturbative corrections, what can we do to get a solution that makes sense?
One thing we can do is to change slightly how we perform the perturbation.
15/

The infinity is an artefact coming from us implicitly imposing that all oscillations must happen at integer multiple of ω₀. This is very restrictive and leads to all sort of divergences.
What if we allow θ₁ to oscillate at a slightly different frequency ω=ω₀+ω₁?
16/

Including this extra degree of freedom, we find an equation for θ₁ very similar to the previous one, but with an extra term, proportional to ω₁.
17/

The trick is that we can choose ω₁ to be whatever we want. After all, we are deciding what the oscillation frequency of θ₁ should be in our new expansion. In particular, we can choose ω₁ such that the terms proportional to sin(ω₀ t + ϕ) cancel.
18/

If we solve this equation we find a new solution for θ₁, which doesn't have any term diverging to infinity.
Where did the infinity go? Nowhere, it wasn't truly there to begin with. It was just a calculation artefact due to us trying to fit a round peg in a square hole.
19/

This is known as the "Poincaré-Lindstedt method". The method used in quantum field theory (renormalization group) is more complicated, but follows the same general logic: allow some wiggle room in your expansion, and use it to cancel the problematic terms.
20/

The end.
Hope not too many typos found their way to the thread, and not too many mistakes found their ways to the equations.
Thank you for reading 🙂

@massimosandal I wrote this thread with you and your bafflement for how Physicists manipulate Mathematics in mind 🙂

Black line - Sinusoidal solution to the harmonic oscillator.
Gray line - Numerical solution of the anharmonic oscillator.
Red (dashed) line - A sinusoidal wave with its frequency adjusted to fit the anharmonic solution.

(bonus panel 1/2)

If you try to describe the gray line with the frequency of the black line, you are going to be in troubles. The red line is not the same as the gray one, but it is VERY similar to it, so the new frequency is a better way to describe the anharmonic oscillation.

(bonus panel 2/2)