In applied mathematics and physics, we often encounter equations that depend on a small parameter, usually denoted by .
When is small, the system is almost something we already understand — and perturbation theory is the method that lets us take advantage of that.

At its core:

If a system depends on a small parameter, we can approximate its solution as a power series in that parameter.

This allows us to extract useful information without solving the full complicated problem directly.

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The Setup

Suppose is small, and we want to solve

We assume the solution can be expanded as

Here:

• is the solution to the simpler unperturbed problem.
• capture corrections due to the perturbation.

We substitute this expansion back into the equation and match coefficients of powers of .
This yields a sequence of equations that can be solved order by order.

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A Simple Example

Consider

where is small.
If , we have

So we expect solutions near when is small — the hallmark of a regular perturbation.

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Method 1: Solve First, Then Expand

Use the quadratic formula:

Expand the square root:

Substitute back:

So:

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Method 2: Expand First, Then Solve

Assume directly that

Plug this into the equation, expand, and equate coefficients of powers of .
The result matches the one obtained before — but we never needed an exact formula.
This is the power of perturbation theory.

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Regular vs. Singular Perturbations

Perturbation theory works smoothly as long as the unperturbed problem behaves well — this is called a regular perturbation.

But if the small parameter multiplies the highest derivative, things get trickier.

For example:

If , the equation becomes linear and loses one root — meaning the small parameter fundamentally changes the structure.
That’s a singular perturbation problem.
Such cases require special techniques like matched asymptotic expansions or boundary layer theory.

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Why This Matters

Perturbation theory appears everywhere in applied math and physics:

Field	Small Parameter	What It Explains
Planetary motion	Mass ratio or relativistic term	Perihelion precession
Quantum mechanics	Weak external fields	Stark & Zeeman effects
Fluid dynamics	Reynolds number, viscosity	Boundary layers
Engineering vibrations	Damping coefficient	Frequency shifts

It lets us extract approximate truths from impossible exact equations.

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The Takeaway

Perturbation theory is more than an approximation tool — it’s a philosophy of analysis.

It says:

• Find what happens when .
• Assume the solution changes smoothly as increases.
• Expand.
• Match coefficients.
• Understand the corrections.

In short, it’s how mathematicians and physicists make sense of almost solvable systems — from atoms to planets.

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Follow-up: From Pendulums to Planets

If you enjoyed this introduction, check out the follow-up post:

From Pendulums to Planets: How Perturbation Theory Explains Perihelion Precession

In that post, we take these same ideas and apply them to planetary motion, showing how tiny relativistic corrections in the equations of gravity explain the slow rotation of Mercury’s orbit — one of the most beautiful triumphs of both mathematics and physics.

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Further reading:

• A. Holmes, Introduction to the Foundations of Applied Mathematics (Springer, 2019)
• C. Bender & S. Orszag, Advanced Mathematical Methods for Scientists and Engineers
• L. Landau & E. Lifshitz, Mechanics, Ch. 5 — Small oscillations and perturbations