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Mathematics that Swings: The Math Behind Golf

 MAA Distinguished Lecture: The Model Golf Game  

All mathematical models are wrong. The late statistician George Box famously said so, and Douglas Arnold (University of Minnesota) reiterated the sentiment at the MAA Carriage House on October 29 in his talk “Mathematics that Swings: The Math Behind Golf.”

For suppose you cleverly folded into a model every detail of, say, a golf swing. 

“Simulating that and watching it on the computer would be like going out and watching Phil Mickelson hit a golf stroke,” Arnold told his Carriage House audience.

“You wouldn’t know how the hell he did it!”

You must be selective in the parameters you include in a model, Arnold explained. You want enough that the model replicates with sufficient accuracy the aspect of the system you’re interested in, but not so many that the model defies analysis.

“So all models are wrong,” he summarized. “But some are useful.” 

The Swing

One useful way to model a golf swing is as a double pendulum. The system’s essential elements are two pivots (the golfer’s shoulder-center and wrist), two straight connecting lines (shoulder-center to wrist, and the club), and a mass (the club head). The variables are torque at the shoulders and at the wrist, and the goal is for the club head to hit the ball at high speed.
The differential equations used to model this simplified system can’t be solved analytically, but they can be studied from a dynamical systems perspective or approximated by computer. And interrogating the equations yields insights: A golfer’s wrist should be a loose fulcrum, Arnold said, with all the force of the swing applied from the shoulders.

Mere observation of a double pendulum (watch one in action) suffices, however, to drive home one point: It takes skill to get a golf swing right. A double pendulum is a complicated dynamical system, sensitive to minute variations in initial conditions. 

“That’s part of the reason golf is so difficult,” Arnold said.

Impact 
Once the swing—what Arnold calls “the only time the golfer has anything to do with the golf game”—is completed, the laws of physics take over.

To study how fast the ball will travel when it leaves the club head, you can think of the ball as a very, very compressed spring. Break out your Newtonian mechanics, your conservation of momentum, and you can write two equations with two unknowns, namely the post-impact velocities of the club and the ball. 

This allows you to reason your way to the fact that, as Arnold put it, “everything depends on the ratio of the mass of the club to the mass of the ball.”

“No matter what you do, no matter how you design your golf clubs,” Arnold explained, “that ball is not going out any faster than twice the speed of the club.”

The model overestimates the ball’s speed, however, since it fails to account for the fraction of the club’s kinetic energy that is lost as heat and friction. 

Flight

Your textbook mathematics and physics don’t really fail you, though, until you try to model the trajectory after impact. The path of Phil Mickelson’s golf ball bears little resemblance to the downward-facing parabola familiar from first year calculus. But why? 

Look to the fine print that prefaces many a homework problem, Arnold suggested. “It says, ‘Ignoring air resistance, find the trajectory…’ So we ignore air resistance; our students ignore air resistance. The golf ball does not ignore air resistance.”

And therein lies the rub.

For the model of the ball’s flight to be useful, it must factor in the force of air on the ball, must account for both drag and lift. And that is not simple.

“This is a long, long story,” Arnold said, “Basically this is fluid dynamics.”

And what a story! In sketching it for listeners, Arnold covered the relative thickness of blood and water, yes, but also blood and corn syrup. (In decreasing order of thickness: corn syrup, blood, water.) This in his discussion of Reynolds number.

He described Gustave Eiffel’s construction—on the grounds of his eponymous tower, no less—of the first wind tunnel.

Arnold expended the bulk of his narrative energy, however, on the masterful resolution by Ludwig Prandtl (1875-1953) of the so-called drag crisis. Prandtl was able to explain why, for instance, as the speed of movement through air increases, the drag does up, up, up, down, up.

The Elusive Quest

Now it turns out that that counterintuitive (but nonetheless beneficial!) decrease in drag is more likely to come into play in your golf game if the ball is a roughened sphere rather than a smooth one. Hence the dimpled surface of the balls in use today. But should the dimples be round or hexagonal or triangular? How should they be distributed? 

“That’s in fact a huge space to explore,” Arnold said. “You’ll never be able to do it by trial and error. So there’s still going on the elusive search for the best dimple pattern.”

So get modeling! —Katharine Merow

Arnold’s talk was part of the Distinguished Lecture Series funded by the National Security Agency.