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Math of golf analyzed in lecture

The College of Arts and Sciences’ Department of Mathematics sponsored “Mathematics That Swings: The Math Behind Golf,” the first of their annual Distinguished Lecture Series, May 11.

Ron Perline, associate professor in the mathematics department, was in charge of setting up the Distinguished Lecture Series this year.

“It’s a little bit different this year than it was last year. In the past what we’ve had is some fairly prominent and respected mathematician come and give a technical lecture which is supposed to be understandable for a fair amount of faculty,” Perline said.

This year the series was comprised of two lectures: “Mathematics That Swings: The Math Behind Golf” for general math students, and “Finite Element Exterior Calculus: Where Numerical PDE Meets Topology” for graduate students and professors.

“We thought it would be important to have them also communicate something to a more general technical audience. By that I mean we had people here who were freshmen calculus students and I think that they could follow a fair amount of it,” Perline said.

Douglas N. Arnold, former president of the University of Minnesota, was the guest speaker. Arnold was also the McKnight Presidential Professor of Mathematics at the University of Minnesota and the president of the Society of Industrial Applied Mathematics.

“I am using an example of golf to show what you can do in mathematics,” Arnold said.
Arnold began researching the connection between golf and mathematics because he was hired by Exxon Mobil to produce a public service announcement of Phil Mickelson hitting a golf ball with different equations floating around on the screen.

“I was hired as a consultant to provide the equations and make sure they were relevant,” Arnold said. “I did a lot of research on why math is relevant to golf and it turned out to be an interesting story,” he added.

A year later, in 2010, the topic of Math Awareness Month was “Math and Sports.” This caused Arnold to revisit the connections between math and golf.

“I have to admit that I think I understand some of the principles of what you want to do to the ball, but I am not somebody that has the athletic skill to do it,” Arnold said.

The lecture covered how math is related to the golfer’s swing, the impact of the golf club and the golf ball and the aerodynamics of the ball itself.

Arnold began by comparing the golfer’s swing to a double pendulum. The golfer’s club and arms act as the rods of a double pendulum. When the arms and club line up in a certain way the golfer exerts the most force on the ball. He showed diagrams of many different mathematical models of a golf swing to supplement his speech.

“What’s the point of a mathematical model? Well, as George Box said, ‘All mathematical models are wrong, but some are useful.’ The idea is that you’re not trying to get a mathematical model that captures every detail of the exact physical system, you’re trying to get one simple enough that you can do some analysis, do some computation, and somehow draw insight from it and get a better idea from it,” Arnold explained.

Arnold next went on to explain the math behind the impact of the club on the ball. The ball is in contact with the club for two-thousandths of a second, and that’s when the energy is transferred from the club to the ball. The ball acts as a spring and is squished on impact and elongated directly after.

He compared the force exerted on the golf ball by the club with a Newton’s Cradle. The difference is that the balls in Newton’s Cradle are the same mass. A golf club is not the same mass as a golf ball.

With the support of mathematical models, Arnold concluded, “No matter how big a club you use and no matter how hard you hit the ball, it’s not going to go any faster than twice the speed you can accelerate the club to.”

After reminding the audience of Box’s theory, Arnold explained that in a real-life impact, energy is lost and the ball will actually go slower than the mathematical models predict.

Arnold then began to explain the aerodynamics associated with a golf ball and its trajectory path.

According to Arnold, a mathematical model of a trajectory path looks like an upside-down parabola. A golf ball’s path does not follow the model path due to many factors, including the two main components of air resistance: lift and drag.

Lift has to do with the rotation of the ball in the air causing it to rise up. Drag deals with the surface of the ball and the friction and pressure applied to the ball.

Arnold focused more on the drag of the golf ball. He explained that the dimples on a golf ball are there to deter the air resistance that push on the ball. When a smooth ball moves through the air, a coating of air resistance called a boundary layer, builds up and drags the ball. The dimples in the golf ball disrupt the resistance, causing the ball to sail through the air.

“Now this is a very complicated and mathematical problem, an optimization problem. If you think about the space and all the possible patterns you could put on a golf ball; the number of dimples, the diameter of the dimples, the depth of the dimples, the shape of the dimples, whether they’re all the same or you have a distribution of them,” Arnold said.

“The U.S. Patent Office has literally thousands of different patents for dimple patterns,” he added.

Arnold then went on to recognize the development of technology and its ability to show simulations of golf balls going through the air with air flowing over the dimples. This is referred to as computational modeling. Computational modeling is also used to show weathercasts.

He showed a graph comparing this technology with the algorithms that have been contributed by mathematicians and physicians over a 35-year period.

“A very important message of what people don’t understand is that the math speed-up to solving these important scientific problems is at least as great as, or in this case quite a bit larger than the hardware speed-up,” Arnold said to end the lecture.

According to Perline, both lectures of the series consisted of multidisciplinary concepts.

“He’s the guy who can talk across fields. By that I mean there are engineers that really can’t talk to mathematicians and mathematicians that really can’t talk to engineers. And he’s right there in the middle,” Perline said.