When Exact Is the Enemy: The Art of Useful Approximations

The Textbook Lie Nobody Mentions

Look, here’s what they don’t tell you in physics class. Every textbook problem you solve—the harmonic oscillator, the hydrogen atom, the infinite square well—they’re all carefully chosen because they have exact solutions. The professor draws the equation on the board, works through the algebra, and out pops a beautiful closed-form answer. You’re left thinking: This is physics. I solve for the answer.

But here’s the dirty little secret: working physicists almost never do this.

In the real world, you face Einstein’s field equations—sixteen coupled nonlinear differential equations where solving for the geometry requires knowing the matter distribution, but determining how matter moves requires knowing the geometry first. It’s a chicken-and-egg nightmare. Exact analytical solutions exist only for highly symmetric cases—a perfectly spherical, static, electrically neutral mass. That’s the Schwarzschild solution, found in 1916, and it remains one of the rare gems we can write down completely.

Everything else? Supercomputer simulations running for months. Approximations layered on approximations.

Why Physicists Love What Textbooks Avoid

The three-body problem has no general closed-form solution. Turbulence remains unsolved. Protein folding defeated analytical approaches for decades. These aren’t failures of physics—they’re physics.

So what do we actually do? We approximate. And here’s where the real craft comes in.

There’s this lovely theorem that says neural networks can approximate any continuous function to arbitrary precision. Sounds powerful, right? But the theorem tells you nothing about how to find that solution or how many neurons you’d need. A hundred thousand neurons might fail where a hundred thirty succeed. Existence isn’t discovery.

Same story everywhere. The Laplace limit tells us that Lagrange’s infinite series for Kepler’s equation only converges for eccentricities below 0.6627—below that threshold, more terms mean better accuracy; above it, adding terms makes things worse. Halley’s Comet, at eccentricity 0.97, laughs at your elegant series.

The trick isn’t calculating more. It’s knowing when your method breaks and switching to something else—Taylor features near a point, Fourier sines across a range, or just straight numerical computation when the math gets ugly.

The Real Skill They Should Teach

Here’s what textbooks should really hammer home: useful beats exact every time.

The Schwarzschild solution works beautifully for spherical stars, not because real stars are perfectly spherical—they’re not—but because they’re close enough. sin(θ) ≈ θ for small angles isn’t a cop-out; it’s the move that makes the pendulum solvable and reveals the underlying physics. That √(l/g) period formula? Pure approximation magic.

The art of physics isn’t finding the exact answer. It’s finding the right approximation—knowing what to keep, what to drop, what level of precision actually matters. An exact solution buried in uninterpretable mathematical machinery tells you less than a rough estimate that reveals physical intuition.

This isn’t a bug in how physics works. It’s the feature. The real skill is judgment, not calculation. And that—not the quadratic formula—is what makes a physicist.