A Gentle Perturbation — An Introduction to Perturbation Methods (1)
To tackle real-world problems, we always resort to all sorts of idealized models. Generally speaking, the closer a model comes to the actual phenomenon, the more complicated it tends to be. If we ignore most of the small disturbances and keep only the dominant terms, we usually end up with a fairly simple model that can be solved exactly. Starting from such an exactly solvable approximate model, and gradually feeding the effects of the small terms back in so that our answer becomes more and more accurate — that is precisely the idea behind the perturbation method, also known as "perturbation theory." This approach originated from solving the N-body problem in celestial mechanics, and has since developed into a fairly systematic discipline, finding applications in many fields such as quantum mechanics and electron theory.
It is not hard to notice that there are plenty of examples of this sort in real problems: when we want to work out some phenomenon, we first consider the most prominent effect, and only afterwards consider the finer details. For instance, to compute the Earth's orbit, we first treat it as a pure two-body system together with the Sun, and then add in the various small effects one by one — the influence of the Moon, the influence of the major planets, and even the effects arising from the Earth's irregular shape. Of course, it is not only these kinds of complicated "big problems" that call for this treatment; some of the "small problems" we run into in everyday work may also benefit from the perturbation method. This post is an attempt to introduce the perturbation method to readers.
The main steps of the perturbation method are: first ignore the small influence (set the small parameter to 0) and find the exact solution; then express the desired solution as a power series in the small parameter. This method can be applied to solve algebraic equations, differential equations, and many other kinds of problems. Let us start with a simple algebraic equation to illustrate the idea:
I. Solve the equation: $\varepsilon x^3+x^2=p^2$more
This is a simple example, and it does in fact have an exact solution. However, since the general formula for the roots of a cubic equation is mostly of theoretical interest, from a practical standpoint we need to look for a more effective method. If $\varepsilon $ is a small quantity, then we can first ignore this term and solve $x^2=p^2$, obtaining $x=\pm p$; without loss of generality, let us take $x=p$ as our starting point. Suppose the exact solution is
$$x=p+a_1 \varepsilon +a_2 \varepsilon^2 +a_3 \varepsilon^3+...$$
Substituting and expanding:
$$\begin{eqnarray*} \varepsilon (p+a_1 \varepsilon +a_2 \varepsilon^2 +...)^3+(p+a_1 \varepsilon +a_2 \varepsilon^2 +...)^2=p^2 \\ \varepsilon [p^3+3a_1 p^2 \varepsilon +3(a_1^2 p+a_2 p^2)\varepsilon^2 +...]+[p^2+2a_1 p \varepsilon +(a_1^2+2a_2 p) \varepsilon^2 +...]=p^2 \end{eqnarray*} $$
If we only keep terms accurate up to $\varepsilon^2$, then we have
$$\begin{eqnarray*} (p^3 \varepsilon +3a_1 p^2 \varepsilon^2)+[p^2+2a_1 p \varepsilon +(a_1^2+2a_2 p) \varepsilon^2 ]=p^2 \\ (p^3+2a_1 p) \varepsilon +(3a_1 p^2 +a_1^2+2a_2 p)\varepsilon^2 =0\end{eqnarray*} $$
We simply require the coefficient of each term to vanish, which gives
$$\begin{eqnarray*} p^3+2a_1 p=0 \\ 3a_1 p^2 +a_1^2+2a_2 p=0 \end{eqnarray*} $$
Solving these:
$$\begin{eqnarray*} a_1=-\frac{p^2}{2} \\ a_2=\frac{5p^3}{8} \end{eqnarray*} $$
To sum up, the approximate solution of $\varepsilon x^3+x^2=p^2$ is $x=p-\frac{p^2}{2} \varepsilon+\frac{5p^3}{8}\varepsilon^2$. It's clear that for this to work well, we still need a relatively small value of p. Although this series need not converge, it nevertheless serves as a rather good approximation for numerical purposes. For example, for $0.1 x^3+x^2=1$, it gives $x=0.95625$, whereas the exact solution is 0.9554....
**The perturbation method is really just an idea, one that can be applied across many areas of mathematical physics. In fact, in quantum mechanics almost all problems are too complicated to be solved exactly, and perturbation theory together with the variational method are the two most fundamental and effective approximation methods in quantum mechanics — which shows just how significant the perturbation method is. In the next post, we will try to explore the application of the perturbation method to solving differential equations.
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.