One of the difficult aspects of mathematics is that it often scales poorly with the complexity of the problem. Math is great when there are relatively few variables and nice functions (for instance, continuous and differentiable functions) involved. But in the real world, there can be so much going on in a system that chugging through the math is incredibly inefficient, and not very enlightening.
An example where the math still shines through, however, is in ice skating. I first learned about this problem from a fellow student I worked with over the summer on a research project. He knew that I liked physics, and mentioned that he had read some articles about mathematically modeling ice skating. The mechanics involved are very tricky, but it is based on the same classical mechanics that all of us learn here at Stevens, and often apply to more complex problems.
The ice skating modeling problem started with a desire to understand optimal movements for ice skaters. One of the main strengths of physics is to solve these optimality problems in a very straightforward way. The method is to find a function that describes a certain quantity of the system—typically energy is of interest. The energy of a system will depend on the position and velocity of its constituent parts. In this case, we have one large system, the body of an ice skater, but its position and velocity are just six variables altogether.
Then, physicists will consider varying that position and velocity by looking at different possible paths the system can take—in our case, the path and movements of an ice skater. This can be tricky, but there are specific conditions we can set so that energy is optimized; we can find a path and set of movements where the skater, or any system, for that matter, can use the least amount of energy.
This method, known as the Lagrangian formalism of mechanics in honor of its creator, Joseph-Louis Lagrange, also spits out equations of motion so that we can exactly model the evolution of the system. Sometimes though, these equations can be very complex, and it’s still difficult to find a solution for the position and velocity of the system at every point. This is what separates what are called “integrable” systems (meaning the equations can be exactly solved) and “non-integrable systems” (we can at best approximate them with computers).
Amazingly, the equations of motion that come from applying the Lagrangian formalism to the ice skating problem are integrable! So, we can write down equations that exactly give the position of a skater at every point in a desired path. This was done by researchers at the University of Alberta, and you can read their paper for the more technical details.
What was coolest about this for me is that one of the researchers was herself an ice skater, and wanted to study this problem since it was a passion of hers. It is a great example of finding your own specific problem to solve based on your interests. I hope that you can find these problems in your studies at Stevens and beyond—it can be difficult with the prescribed problems assigned in class, but there’s always the opportunity to pursue some work of your own on the side, or dedicate your senior project to it. Then, an exciting outcome is all the more fulfilling.
As for mathematics, it’s a rare case where it’s scaled well with the complexity of the problem. The Lagrangian formalism, even when it doesn’t do this, has provided a lot for physics over the years. It is the basis of modern physics research in both small-scale physics (electrons, atoms, and fundamental particles) and large-scale physics (galaxies and black holes). So, if you’re able to apply the Lagrangian formalism to that above problem you’re passionate about, try it! There’s a good chance it’ll give a promising outcome.
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