To calculate practical problems, we always adopt various ideal models. Generally speaking, the closer a model is to actual phenomena, the more complex it tends to be. By ignoring most minor disturbances and retaining only the primary terms, we can usually obtain a fairly simple model that can be solved exactly. Using such an exactly solvable approximate model as a foundation and gradually adding the effects of minor terms to make our answer increasingly accurate is the core idea of the perturbation method, also known as "perturbation theory." This method originated from solving the N-body problem in celestial mechanics and has now developed into a quite systematic discipline, applied to many fields such as quantum mechanics and electronic theory.
In fact, it is not difficult to find many such examples in practical problems: when we want to calculate a certain phenomenon, we first consider the most prominent factors and then consider the details. For example, to calculate the Earth’s orbit, we first treat it as a pure two-body system consisting of the Sun and the Earth, and then add various minor effects, such as the influence of the Moon, the influence of major planets, and even the effects produced by the Earth’s irregular shape. Of course, it is not just these types of complex "large problems"; some "small problems" we encounter in daily life may also allow the perturbation method to come into play. This article attempts to introduce the perturbation method to the readers.
The main steps of the perturbation method are to first ignore the minor influences (set the small parameter to 0) to find an exact solution, and then express the required solution as a power series in terms of the small parameter. This method can be used to solve algebraic equations, differential equations, and problems in various other fields. Below, we first illustrate this with a simple algebraic equation:
I. Solving the equation:
\varepsilon x^3 + x^2 = p^2
This is a simple example that possesses an exact solution. However, since the general root-finding formula for cubic equations is primarily useful for theoretical analysis, we must find a more effective method from a practical perspective. If \varepsilon is a small quantity, we can first ignore this term and solve x^2 = p^2, obtaining x = \pm p. Without loss of generality, we take x = p as our basis. Assume the exact solution is: x = p + a_1 \varepsilon + a_2 \varepsilon^2 + a_3 \varepsilon^3 + \dots
Substitute this into the equation and expand: \begin{aligned} \varepsilon (p + a_1 \varepsilon + a_2 \varepsilon^2 + \dots)^3 + (p + a_1 \varepsilon + a_2 \varepsilon^2 + \dots)^2 &= p^2 \\ \varepsilon [p^3 + 3a_1 p^2 \varepsilon + 3(a_1^2 p + a_2 p^2)\varepsilon^2 + \dots] + [p^2 + 2a_1 p \varepsilon + (a_1^2 + 2a_2 p) \varepsilon^2 + \dots] &= p^2 \end{aligned}
If we only consider accuracy up to \varepsilon^2, we have: \begin{aligned} (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{aligned}
As long as the coefficient of each term is zero, we have: \begin{aligned} p^3 + 2a_1 p &= 0 \\ 3a_1 p^2 + a_1^2 + 2a_2 p &= 0 \end{aligned}
Solving these gives: \begin{aligned} a_1 &= -\frac{p^2}{2} \\ a_2 &= \frac{5p^3}{8} \end{aligned}
In summary, the approximate solution to \varepsilon x^3 + x^2 = p^2 is x = p - \frac{p^2}{2} \varepsilon + \frac{5p^3}{8}\varepsilon^2. It can be seen that for this to be effective, a relatively small p value is also helpful. Although this series does not necessarily converge, it is a very good approximation for numerical calculations regardless. For example, for 0.1 x^3 + x^2 = 1, it gives x = 0.95625, while the exact solution is 0.9554\dots.
The perturbation method is just an idea; it can be applied to various fields of mathematical physics. In fact, in quantum mechanics, almost all problems are so complex that they cannot be solved exactly. Perturbation methods and variational methods are the two most fundamental and effective approximation methods in quantum mechanics, which highlights the significance of perturbation theory. In the following articles, we intend to explore the application of perturbation methods in solving differential equations.
When reposting, please include the original address: https://kexue.fm/archives/1878
For more detailed information regarding reposting, please refer to: Scientific Space FAQ