SIAM Past President Douglas Arnold was in Philadelphia earlier this week delivering talks as part of Drexel University’s Distinguished Lecture Series. In the first of the annual series of lectures held on May 11, “Mathematics that swings: the math behind golf,” Arnold described the many aspects of the game of golf that can be understood and improved by mathematical modeling and analysis. Using several models ranging from simple algebra to advanced computational techniques, he demonstrated the relevance of mathematics on a golf course, reinforcing that math is indeed everywhere.
Arnold immediately captured the attention of the packed auditorium with a shot from a Masters tournament commercial of golfer Phil Mickelson swinging his club; barely discernible but very accurate mathematical equations floated from the swing, with the voiceover emphasizing that “math and science are everywhere.”
A good golfer such as Mickelson can accelerate a golf ball from 0 to 120 miles in a quarter of a second, said Arnold, giving an idea of the speeds involved in a deceptively pastoral sport. Justifying his engagement with golf to merely the theory behind the game, he quoted the famous baseball player Yogi Berra, “Baseball is 90% mental, the other half is physical,” drawing laughter from the audience.
Arnold explained the math models behind three aspects of golf: the swing, impact of the club and ball, and flight of the ball.
The precise motion of a golfer’s swing starts with an acute angle between his arms and the shaft, and ends with a 180 degree angle at the point where the club hits the ball, he said. The two arms form a triangle, the bisector of which can be thought of as a pendulum, which swings around a fulcrum created between the shoulders. The second pivot at the wrist connected to the rigid golf club forms another pendulum, allowing the evolution of this dynamical system to be modeled as a double pendulum equation, where the two angles created are between the vertical and the arms and the shaft and the arms.
Thus using his math model to simulate the swing, Arnold explained that the torque on the shoulder is the most influential factor determining the speed of the ball. “Forces imparted by the wrist don’t help. Almost any torque on the wrist only ends up lessening the speed of impact,” he said.
He next focused on the impact of the club on the ball. During the moment of impact, momentum and energy of the clubhead are transferred to the ball and the idea is to get the ball as fast as possible so that it goes as far as possible, he said. Mathematical models predict that the ball could go at speeds as high as twice the club speed. “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,” said Arnold.
But while the ball cannot leave the club at any higher than double the club speed, it is actually a little less than that because of energy expenditure, he went on to explain. While total momentum—or mass times velocity— of the club-and-ball system is conserved before and after collision, and the overall energy remains the same, some of the kinetic energy is lost in the form of potential, heat, chemical energy and so forth, resulting in ball speeds slightly less than double the club’s acceleration.
Other factors such as the flexibility of the shaft, property of material used to create the club, movement of the ball on the turf and surface of the ball can also affect speed and movement.
Moving on to the third aspect, the flight of the ball, Arnold explained that a lot of interesting math and physics has been developed to model this part of game. The trajectory of the ball is mainly influenced by the initial speed and launch angle.
In addition, there are forces acting on the ball in flight. Besides gravity, there are forces of air on the ball, which tend to decompose as lift and drag. Lift is the effective air resistance that pushes the ball upwards. Drag is caused by the friction between the surface of the ball and the air, and the difference in pressure ahead of and behind the ball. The drag on a ball can be determined by the Reynold’s number, which factors in the density and viscosity of air, in addition to the size and speed of the ball.
The drag is also affected by the surface of the ball, Arnold went on to say. Very early on, golfers started noticing that rubber golf balls went further and faster when they got old and scratched up, which ultimately led to golf balls being created with bumps, and later, with dimple patterns on them. The reason for this is that the rougher surface has a better aerodynamic performance because of a steep decrease in air resistance or drag. Once it became known that roughening a ball helped it go further, optimization of “roughening” became a very complex mathematical problem, explained Arnold. A wind range of variations could be designed based on number, size, depth, shape and distribution of dimples.
Arnold reiterated the significance of this topic by referring to the previous day’s New York Times article explaining USGA rules on dimple patterns. There are thousands of patents filed for dimple patterns, he said.
Because of the innumerable possibilities, it would be hard to optimize dimple patterns by hand analysis or trial and error, said Arnold. But computational models could be the answer. Illustrating a computer model of a golf ball developed by a combined team from Arizona State University and the University of Maryland, Arnold explained that the model provides fantastic detail, allowing researchers to zoom in and look at the effect of individual dimples and ask very detailed questions. This model works on spin of the golf ball, he said, introducing another very important aspect of a golf ball’s movement. It is important for a golf ball to spin on a horizontal axis, as opposed to a vertical axis, he explained, as the former is very important for lift.
There are still some limitations to the model, such as its low resolution and lack of movement, Arnold admitted. “A real golf ball spins at 3,600 rpm; incorporating spin of the golf ball in these simulations is important and complex,” he said. “We are far from being where we can give an optimum pattern for dimples but with computational simulation we are getting closer to it.”
Arnold reinforced the importance of mathematical modeling in sports by referencing another sport that extensively uses optimization. Citing the irony that Switzerland, a land-locked country, was the first European nation to win the America’s Cup sailing competition, he explained that their victory was mostly owing to remarkable design and efficient mathematical modeling in boat construction.
Arnold concluded by emphasizing the importance of computer modeling with a quote from the President’s Information Technology Advisory Committee (PITAC):
“Computational science now constitutes the third pillar of the scientific enterprise, a peer alongside theory and physical experimentation.”