# Variational Principles and the Story of Your Life

As I mentioned a few days ago, a colleague asked me if I’d be interested in doing a guest lecture for a class on science fiction. She suggested that a good way to go might be to pick one story to have the class read, and talk about that.

Kicking ideas around with Kate, I latched onto the Ted Chiang story “Story of Your Life,” from the Starlight 2 anthology (and also his collection Stories of Your Life and Others), because it’s got a lot of great stuff in it– linguistics, physics, math, really alien aliens, and fantastic human characters and interactions. If you haven’t read it, it’s a great story– here’s a spoiler-laden review by a linguist, if you’d like to get more of the idea.

Of course, reading the story got me to thinking about Fermat’s Principle and the calculus of variations in general. Which, as usual, led to the realization that I don’t understand the subject as well as I ought to. I’m not sure that Chiang’s presentation works, but explaining my reasoning involves some math and spoilers, which I’ll put below the fold.

The central conceit of the story is that the alien race whose language the narrator is trying to learn have a way of looking at the universe that treats variational principles as more fundamental than the sort of dynamics we usually think about. The example given is the least-action formulation of Snell’s Law for refraction, in which light is always found to take the fastest path between two points, and this can be used to find the optimal angle of refraction at an interface between two positions.

In the context of the story, this is presented as requiring knowledge of both the start and end points in advance. The aliens view this formulation of physics as fundamental because this is how they see the world– they know what’s going to happen in advance, and this has profound effects on their language, and the mind of the human linguist learning to write it.

The thing is, when I try to think about the variational approach, this explanation ends up seeming a little arbitrary, in a manner similar to the ever-popular anthropic principle. You can use variational principles to calculate the optimal path between two points, but the choice of points is essentially arbitrary. It’s true that if you know a given light ray will be at point A and then at point B, you can find the path from A to B using variational principles, but there’s nothing inevitable about point B. Fermat’s Principle doesn’t tell you that a light ray starting at point A will necessarily reach point B, it just tells you what path it will take from A to B if it happens to go through point B. There are an infinite number of light rays emanating from point A that never pass through point B at all.

If you know points A and B in advance, the variational calculus will give you all the points in between, which seems really impressive from point B. But people arriving at point C will be equally impressed. It’s the same problem as with the “anthropic principle” arguments about the values of fundamental constants– if the constants of nature had slightly different values, life as we know it would be impossible, which seems really awesome if looked at in a certain way (being stoned helps, I hear). But there’s no reason there couldn’t be another universe out there in which beings radically different from ourselves write long philosophical tracts marveling at how well-suited their universe is for their form of life.

Knowing point A doesn’t inevitably determine point B, unless you provide enough extra information that you would’ve been able to determine point B using non-variational methods, as well. Which undercuts the whole premise of the story a little bit. It’s still a powerful piece of work, but the implicit inevitability of those events seems a little dubious.

In optics and quantum physics, of course, the variational principle can be justified using something like Huygens’s Principle, in which waves emanate out from every point in all directions, and interfere with each other constructively only along the extremal path predicted by the variational principle. In a sense, the optimization happens because the waves really do take every possible path between A and B (and other points as well), but the non-optimal paths cancel each other out. That formulation makes a great deal more sense to me, and doesn’t require advance knowledge of point B in the same way that Chiang’s presentation does. It’s not nearly as magical, which makes it less fun for stories, but more satisfying as science.

(Yeah, this is really going to bring in the blog traffic… I should go back to ranting about unions…)