When we study perturbation theory in applied math, it often starts with something humble —
a pendulum swinging gently back and forth.

But the same mathematical ideas explain why Mercury’s orbit slowly rotates around the Sun.
This post connects the two: from the nonlinear pendulum to planetary precession.

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The Pendulum as a Perturbation Problem

Holmes’ Introduction to the Foundations of Applied Mathematics uses the classic example:

with small initial deflection , .

For small angles, expand .

Assume a regular perturbation expansion:

Plugging this into the equation gives the leading-order solution

which is the familiar small-angle harmonic motion.

But the next term corrects the frequency, giving a slightly slower oscillation.
This nonlinear correction causes a gradual phase drift — each cycle takes a little longer.

That same idea, scaled up to planetary mechanics, predicts the rotation of orbits in space.

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The Planetary Analogy

A planet orbiting the Sun follows nearly Keplerian motion:

The pure inverse-square law gives closed ellipses.
But in reality, other effects — relativity, other planets, the Sun’s shape — slightly perturb the potential:

where is small.

Writing the equation in terms of and expanding gives

which looks just like the nonlinear oscillator for the pendulum.
And just as before, that nonlinear term shifts the frequency — the orbit doesn’t close perfectly.

Each revolution, the closest point (the perihelion) moves forward by a small angle .
This cumulative shift is what we call perihelion precession.

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Mercury and the Mystery of 43 Arcseconds

In the 1800s, astronomers used Newtonian perturbation theory to calculate the effects of other planets on Mercury’s orbit.
Most of the precession was accounted for — except 43 arcseconds per century.
No known Newtonian cause could explain that leftover.

Perturbation theory had revealed the phenomenon — a small, systematic shift —
but not the reason behind it.

Then came Einstein.

In general relativity, the effective potential includes a tiny correction term proportional to :

That acts like the pendulum’s cubic term —
it modifies the frequency slightly, leading to precession.

Carrying through the perturbation calculation yields:

which equals exactly 43 arcseconds per century for Mercury —
a perfect match to observation.

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Historical Insight: How Perturbation Theory Bridged Centuries

Step	Who / When	What Happened
1700s–1800s	Lagrange, Laplace, Le Verrier	Developed Newtonian perturbation theory — predicted most orbital precessions.
1840s	Le Verrier	Calculated Mercury’s precession — found 43″/century discrepancy.
1905–1915	Einstein	Derived General Relativity; found an additional correction to Newtonian gravity.
1915	Einstein (published result)	Plugged that correction into the perturbative orbit equation — exactly matched Mercury’s observed precession.

So the story goes like this:

Perturbation theory first told us something was off.
Einstein’s relativity explained why, and gave the missing term that made the math and nature align.

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The Big Idea

Perturbation theory doesn’t just approximate solutions — it reveals structure.
It shows how small nonlinearities accumulate over time to shape the universe’s large-scale behavior.

From a pendulum’s slow drift to Mercury’s century-long dance,
the same mathematics reminds us that every deviation tells a story about the laws beneath.

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Further reading:

• A. Holmes, Introduction to the Foundations of Applied Mathematics (Springer, 2019) — Section 2.7
• A. Einstein, Explanation of the Perihelion Motion of Mercury from General Relativity (1915)
• L. D. Landau & E. Lifshitz, Mechanics — nonlinear oscillations and small perturbations