As Scholars are Wont to do: Poetry of the brachristochrone problem

Time is an important dimension to consider when tracing the path of a media-represented concept. This is a theme that has stuck with me since reading Brenda Laurel’s “The Six Elements and Causal Relations Among Them” and coming across this brilliant bit of prose:

As scholars are wont to do, I will blame the vagaries of translation, figurative language, and mutations introduced by centuries of interpretation for this apparent lapse and proceed to advocate my own view. – Brenda Laurel, “The Six Elements and Causal Relations Among Them”

Which, should be noted, already inspired a previous post.

The particular idea that Laurel was referring to was one first explored by Aristotle over 2000 years prior, and I was struck with the eerie feeling of being transported into a conversation spanning millennia as well as minds. Like Michelangelo chipping away at a block of marble to reveal the sculpture hidden within, poets and philosophers carve away at the layers of representation in an attempt to reveal the essence of a concept encased within.

It occurred to me that the same process is occurring in the sciences as well, although usually the bricks of representation1 are mathematical symbols rather than linguistic. In a moment of serendipity, while I was musing over the conversation of this morning’s class, I opened a book on Nonlinear Geometric Control Theory that, a few weeks back, I had checked out of the library on a whim. If the title means little to you, rest assure, it does me as well. This is not a topic I am too familiar with, but for a reason I can not fully explain, ever since I was introduced very briefly to differential forms in a real analysis class I suspected that the language of differential geometry had the potential for elegant representations of optimal control problems. Low and behold, I opened this book and immediately saw a paper in which the authors, Sussmann and Willems, explored several representations of the brachistochrone problem, concluding with a differential-geometric approach that they claimed as the most elegant thus far.

The brachristochrone problem

In 1696, Johann Bernoulli asked a question that has become a classical problem used as motivating example for calculus of variations and rears its head again in optimal control2:

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time. – The bracristochrone problem

The brachistochrone (shortest time) problem

The problem itself is older than Bernoulli: Galileo had conducted his own exploration in 1638 and incorrectly deduce the solution to be the arc of a circle.3 Bernoulli’s revitalization of the question is what led to the first correct answer, that the solution is a cycloid.

Sussmann and Willems summarize the various solutions to the problem as follows:

  1. Johann Bernoulli’s own solution based on an analogy with geometrical optics,
  2. the solution based on the classical calculus of variations,
  3. the optimal control method,
  4. and, finally,

  5. the differential-geometric approach

– Sussmann & Willems

[place holder for a more in-depth summary…ran out of time tonight]

They demonstrate how each successive method refines the solution space and eliminates unnecessary assumptions to approach what could be considered the essence of the problem itself. They end with the differential-geometric approach with the claim that it is thus far the best at elegantly capturing the problem, but it hardly seems like this will be the final word on such a well traveled challenge.

So it seems the path the poet takes in exploring the nature of life is not all that dissimilar from the path the mathematician, scientist or engineer takes, only the tools differ, and even then, the difference is often over stated. They are all just different colors of bricks1.

Footnotes and References

1 David, write about your brick analogy so I have something to link to!

2 worthy of an entirely separate post: through my digging around for more details about the brachistochrone problem I discovered a paper discussing it as a Large Context Problem, an approach to education that is of extreme interest to me that I now have a name for!

3 The brachristochrone problem

Sussmann, Hector J. and Willems, Jan C. The brachistochrone problem and modern control theory