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Classroom puzzles to cosmic insights: Students and professor demystify mathematical theorem

By Hannah Ashton

Central forces that decay as 1/r² are special, as they guarantee that all bound orbits are going to be closed (Bertrand's theorem). Small changes in the power will lead to significantly different kinds of orbits.

More than 150 years ago, Joseph Bertrand stated a mathematical theorem. Proving why this theorem is true hasn’t been a simple endeavor.

Two College of Science alumni, along with professor Patrick De Leenheer, recently published a paper in the SIAM Review pulling back the curtain on Bertrand’s Theorem. Together, they wrote a proof that is accessible to undergraduate mathematics or physics students.

Bertrand’s Theorem states that among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed, Newtonian and Hookean gravitation.

“If we didn’t live in a gravitational field governed by Newtonian gravitation, the world would be very different and far more unpredictable. For example, we would probably not have seasons,” De Leenheer said. “It’s kind of remarkable that gravity operates in this way. Among all the possibilities, truly infinite, this is the one that we live in and that’s just astounding.”

In the simplest terms, the group started by using a process of elimination, by first showing that gravitation must follow a power law. Next, they narrowed down the power laws until only two of them remained. And finally, they checked that both of these had the property they were looking for.

De Leenheer remembers taking his first physics class in high school and questioning the formula, R: F = G(m1m2)/R2. De Leenheer wanted to know why it was R-squared. Why not R cubed or something different? This led him to Bertrand’s Theorem. He couldn’t find a proof of it, leaving him to wonder why it was true.

Headshot of Patrick De Leenheer on campus

Patrick De Leenheer

John Musgrove, ‘20, and Tyler Schimleck, ‘21, heard about the theorem in De Leenheer’s Vector Calculus 2 course and approached him after class. He enthusiastically brought them on board what turned into a five-year project.

“For me, it was my first time jumping into real mathematical literature. Reading papers by other mathematicians working on the same problem and diving into their research was super exciting for me,” Musgrove said.

Having access to undergraduate research helped both of them successfully pursue a postgraduate degree.

Musgrove graduated from Columbia University with his MS in Operations Research and Schimleck is currently a graduate student at UC Santa Barbara in the Department of Mathematics. Schimleck is interested in differential geometry and mathematical physics.

“It can be so scary at first if you try to read mathematical literature. There is a huge gap between a lot of high-level research and what you learn in undergrad,” Schimleck said. “Doing undergraduate research was a massive confidence boost that helps me say, ‘No, it’s okay, I can do this. I may not understand it at first but eventually, I’ll figure it out and it’ll be okay.’”

Working with a faculty member can be equally terrifying but Musgrove and Schimleck said De Lehneer helped every step of the way.

“One of my favorite things about working with Patrick, would be sitting there in the room and we’re staring at equations on the blackboard in silence for a few minutes and then we will have an “Aha” moment and Patrick will actually say “Aha” and go run to the blackboard with the solution,” Musgrove said. “It's labor. Math doesn’t look appetizing but once you are in it, the energy and emotions sustain the whole thing.”

He said that in a typical undergraduate classroom, students don’t get to experience facing a wall because everything is set up with an answer already.

“That’s the difference with doing research, there is nobody who will tell you the answer. You just have to keep looking at it and thinking about it. There are no shortcuts,” he said.

De Leenheer said it feels like serendipity that a question he had as a young adult, he was able to answer years later.

“I got to know two guys here in class and they showed excitement and they had the dedication, they never quit. Even today I am still thinking about it and it baffles me when I talk to people about this result. All together it was very rewarding,” he said.