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Physics as a Model

Lucas describes a recent realization in physics.

Lucas describes a recent realization in physics.

I don’t know quite how I started thinking along these lines, but recently, I challenged myself to summarize the “biggest lesson” I learned from each course I’ve taken at Carleton thus far. Honestly a kind of dumb experiment, I think, because no class can truly be boiled down to a single sentence or idea; indeed, many courses are patently about complex and even internally contradicting areas of thought. But it did lead me to an interesting realization: my sentence for my current physics course, Analytical and Computational Mechanics, came immediately: “Physics can be modeled in multiple ways”. A sentence I probably would have agreed with, and thought I understood, last year, but one that means quite a bit more to me after this trimester in particular.

I think I’ve outlined this before, but basically, my current physics class serves as a second look at mechanics. Where the first course focused on Newtonian mechanics and often allowed us to ignore factors such as air drag, friction, etc., this course looks precisely at these more complicated examples, and where mathematical/computational and Lagrangian tools can come in handy. By mathematical and computational tools, I simply mean that we’ve sometimes found ourselves in situations where solutions to certain equations can’t be found analytically. Instead, we then turn to our computers, where we can use software such as Wolfram‘s Mathematica to run some blunt force calculations in order to obtain approximate numerical answers.

“Lagrangian” tools are a little harder to explain. Basically, Newtonian mechanics encompasses a certain framework within which one can solve various mechanics problems (mechanics problems usually means you’re trying to figure out how to predict or understand the motion of certain bodies or systems). Fascinatingly, Lagrangian mechanics is an entirely different framework through which one can find the exact same solutions to the exact same problems. It’s pretty amazing.

In Newtonian mechanics, you frequently solve problems by drawing a free body diagram of the body in question and work out what happens from Newton’s laws of motion. With Lagrangian mechanics, however, you come up with this equation where the body’s kinetic energy minus its potential energy equals zero, and from there you use differential equations to figure out the body’s equation(s) of motion. So that’s what I mean by different models: Newtonian mechanics is all about identifying the different forces at play and figuring out how they contribute to the overall acceleration of the system, whereas Lagrangian mechanics is a more conceptual (at least in my opinion) application of the principle of least action. Sometimes one approach makes more sense, and sometimes the other does, but the incredible thing is that both are completely equivalent in output yet entirely dissimilar in form.

Personally, I think Newtonian mechanics’ strength lies in how, at least at the introductory level, it forces students to more specifically think through the forces at play. However, in complicated situations, the math of this can get so nasty that you’re almost guaranteed to make some errors. Lagrangian mechanics, on the other hand, is what I’d call the more elegant model, and it allows one to apply more or less the same systematic approach to a wide array of problems with consistent efficiency. I definitely think it makes sense to start with Newtonian mechanics, but I am honestly surprised I hadn’t heard much about Lagrangian mechanics until this course – it’s a huge deal! (And apparently the Hamiltonian, which you kind of get from the Lagrangian, is really important to proper quantum mechanics.)

I hope the point I’m trying to make is getting through here. I’d always understood conceptually that physics, like many (all?) disciplines, just happens to be a body of the most effective frameworks we’ve yet come up with to tackle various questions. However, the statement “physics is (just) a model” means a lot more to me now that I’ve worked with multiple ones. It’s both a little disappointing and a lot fascinating to realize that the physics I’ve been trusting and relying upon for years now is just one person’s (or set of people’s) preferred mathematical toolbox that happens to yield accurate predictions, and not some objectively authoritative representation of How Things Really Are. Now I’m learning a much less intuitive in theory, yet much simpler to apply in practice, way of “doing physics” that’s completely distinct from how most of my knowledge of the subject has been built so far.

I don’t know. Maybe I sound like a crazy person. But I think this stuff is really, really cool.

Lucas is in his sophomore year at Carleton, bringing with him a passion for all things nerdy and a talent for overthinking and awkwardness (and self-deprecation). He hails from Pasadena, California, and yes, he realizes it gets cold out here. He currently sees himself majoring in Physics, although he hopes to explore Cinema and Media Studies, Chemistry, Economics, and Computer Science (among many other subjects) as well. He misses his bearded dragon. Meet the other bloggers!