Coffee in Edinburgh and Galilean Extremism

Yesterday, over coffee with some friends in Edinburgh, we ended up in one of those conversations that start with society and drift off anywhere. True to my style, I ended up talking about complex systems. I told them that one of the best ways to analyse them is through what I call Galilean extremism: pushing a system to an impossible case in order to understand the possible.

Galileo did this brilliantly. He observed that a stone and a feather do not fall in the same way, but asked: what would happen in a perfect vacuum, without air? That impossible scenario revealed the essential: all bodies fall with the same acceleration. The impossible allowed him to make sense of the possible.

In nonlinear physics, we do something similar: we go to extremes (such as infinite friction in a pendulum) to discover the hidden rules that organise behaviour.

Try it yourself: Drop a light piece of paper and a coin at the same time. Then put the paper on top of the coin and drop them together — they fall almost at the same speed. That’s Galileo’s trick in your hands.

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What “nonlinear” means

In linear physics, if I know two solutions, their sum is also a solution. It’s like a tidy Excel sheet: ordered, predictable. But the real world doesn’t behave that way.

Bend a sheet of paper slightly: it springs back. Bend it too much: it creases. The linear relationship breaks. The same with a motorway: if one car enters, traffic adjusts smoothly; if a hundred enter at once, a jam appears.

That jump to behaviours that are not proportional or additive is what we call nonlinearity.

The pendulum as a portable laboratory

The pendulum is a classic, powerful example. Mathematically, it is described as:

\[ \ddot{\theta} + \mu \dot{\theta} + \frac{g}{L}\sin\theta = 0 \]

where \(\theta\) is the angle, \(L\) the string length, \(g\) gravity, and \(\mu\) air friction.

The crucial bit is the \(\sin\theta\). That’s the seed of nonlinearity. If we replace it with \(\theta\), we get the linearised version, valid for small angles:

\[ \ddot{\theta} + \mu \dot{\theta} + \frac{g}{L}\theta = 0 \]

And what do we mean by “small angles”? Think of a swing. If you give it just the tiniest push, the angle it makes with the vertical is minimal. In that range, \(\sin\theta \approx \theta\). Try it on your calculator: type in \(\sin(5^\circ)\) and compare it with \(5^\circ\) converted into radians (\(\approx 0.087\)). They’re almost identical.

This approximation is so good that, believe it or not, many real-world structures —buildings, roads, bridges— are designed using it. When an engineer models the vibration of a beam or the sway of a tower, they often swap \(\sin\theta\) for \(\theta\). That “white lie” makes an otherwise unmanageable problem solvable. And it works… until the extremes arrive: earthquakes, hurricanes, unexpected loads. That’s when nonlinearity barges in and simple physics no longer suffices.

Phase space: the Google Maps of a system

To describe the pendulum’s state, \(\theta\) alone isn’t enough. We also need the angular velocity \(\dot{\theta}\). Together they form the pair \((\theta,\dot{\theta})\), which lives in phase space.

It’s like looking at traffic in a city: it’s not enough to know where the cars are, you also need to know where they’re heading and how fast. Phase space is that extended map showing all possible trajectories.

• With \(\mu=0\) (no friction), the trajectories are closed orbits: eternal oscillations.
• With friction, those orbits spiral inward, like cars funnelled into a roundabout that eventually come to rest.

Try it yourself: Sketch a graph with \(\theta\) on the x-axis and \(\dot{\theta}\) on the y-axis. Mark a few points from your pendulum experiment and connect them. You’ll see the spiral emerge.

Eigenvalues and eigenvectors: the DJs of motion

Near equilibrium (\(\theta=0\)), the system reduces to the linearised equation:

\[ \ddot{\chi} + \mu \dot{\chi} + \frac{g}{L}\chi = 0 \]

Its solution depends on the eigenvalues:

\[ \lambda_{1,2} = \frac{-\mu \pm \sqrt{\mu^2 - 4\frac{g}{L}}}{2} \]

These numbers tell us whether the system vibrates, dies out quickly, or dies out slowly.

• If \(\mu\) is small, \(\lambda\) are complex and damped oscillations appear.
• If \(\mu = 2\sqrt{g/L}\), we have a critical case: the system fades without oscillating.
• If \(\mu\) is very large, two time scales emerge: a fast one (\(\lambda \approx -\mu\)) and a slow one (\(\lambda \approx -\tfrac{g}{\mu L}\)).

Each eigenvalue comes with an eigenvector, indicating the direction in phase space. It’s like DJs: one plays fast tracks that end abruptly, the other plays slow rhythms that linger and dominate.

Try it yourself: Work out the eigenvalues for a pendulum with \(L=1\) m and \(\mu=0.1, 2, 10\). You’ll see how the nature of the motion changes with friction.

The slow-time trick

When friction is large, the fast dynamics vanish so quickly they barely register. To focus on what remains, we change the clock:

\[ t = \mu \tau \]

With this scale, the equation becomes:

\[ \frac{d\theta}{d\tau} = -\frac{g}{L}\sin\theta \]

What looked like a two-dimensional problem collapses into one.

Intuitively, it’s like the sea. If you want to understand climate change and sea-level rise, you don’t need to calculate every wave crashing on the shore or every turbulent eddy under the water. You can ignore the fast processes (waves of seconds, turbulence of milliseconds) and keep the slow ones: tides lasting hours and sea-level rise over decades. The trick is the same: separate scales, filter out the fast, and focus on the slow choreography.

The centre manifold: the slow lane

In phase space, all trajectories end up doing the same thing: first they plunge quickly towards a “fast” direction, and once that motion dies out, they drift slowly along a curve. That curve is the centre manifold.

Close to equilibrium it looks like a straight line, the slow eigenvector. But because the system is nonlinear (the equation contains \(\sin\theta\)), that straight line bends into a curve.

Think of a park full of cyclists taking different routes. At first, each follows their own path, but soon they all converge onto the same main trail. That trail is the centre manifold: the slow lane where long-term dynamics concentrate.

There’s a beautiful scene in The Theory of Everything —the Stephen Hawking film— where Roger Penrose is teaching a class outdoors (something very British, and not unusual at Edinburgh). At one point he says: “This is new to us, and it’s called topology. Not equations, just geometry!” That’s exactly the spirit here.

What matters is no longer the full, messy differential equation but the shape of the surface where the dynamics settle. The centre manifold is precisely that: a geometric object that captures the effective dynamics. In nonlinear physics, this way of thinking pops up everywhere — instead of focusing on raw equations, we look at the topology of the space they create.

It’s a shift in mindset: from equations to geometry, from algebra to shape. And it’s this shift that makes nonlinear physics so powerful, because once you’ve found the slow lane, the whole story of the system unfolds there.

Conclusion

Nonlinear physics forces us to think like Galileo: exaggerate the impossible to uncover the essential. In the overdamped pendulum, the limit of infinite friction reveals that everything flattens onto a single dimension, a slow lane where the real dynamics live. It’s the same principle that allows us to study sea-level rise without tracking each wave, or to understand a traffic jam by looking at the slow tide of cars rather than each individual honk.

The lesson is clear: behind complexity, there are always hierarchies of times and scales. The fast fades, the slow governs.

And this is only the beginning. In the next part, we’ll see how these ideas lead us to even more surprising territory: chaos, bifurcations, and attractors that can turn an innocent system into something unpredictable.