Minimize the Time

The concept behind optimal space trajectories (e.g. the cheapest way to get into orbit) came to us from the Bernoulli brothers. In the seventeenth century, James and John Bernoulli amused and challenged each other by inventing mathematical puzzles for the other to solve. It was typical sibling rivalry carried to extremes. The game reached its zenith when one of the brothers proposed what is now called the "Brachistochrone problem." (Consider it the brontosaurus of mathematics.) The name derives from the Greek roots that appear in brachiopod and chronology, so it translates to the "shortest time" problem.

The problem is very similar to that of launching a spacecraft into orbit. Because the propellant is burning at a furious rate, we burn the least if we get into orbit by the quickest path.

To understand the Bernoulli problem, let us think of a roller coaster. Suppose we have a coaster with a hundred-foot drop over a hundred-foot range. What shape would the track have to be so that the roller coaster gets to the bottom in the shortest time? This is the Brachistochrone problem. Suppose we build a straight-line track—that is, an inclined plane—that runs from one hundred feet high down to the bottom, one hundred feet along the ground. It would be on a 45-degree angle. Wouldn't this be the fastest path?

The answer, surprisingly, is no. It's the shortest length but not the fastest path. (And these people want to get down there as quickly as possible—they want some thrills in their lives.) It turns out that you can satisfy your thrill seekers and get them to the bottom faster if you have the track drop more steeply at first. It will seem to your riders that they are dropping almost straight down on the first plunge. (Serves them right—these people really bring out the sadist in the amusement-ride designer.) A steeper drop in the beginning gives them greater speed in the beginning (gets their hearts pumping) and this extra speed more than makes up for the longer track (compared with the soporific straight-line incline). Make it too steep, however, and the total trip time gets to be longer (and therefore boring).

The problem the Bernoulli brothers invented stumped the great mathematicians of Europe for six months. (The problem is to find a mathematical function to describe the exact shape.) A wonderful account is given in Eric Bell's Men [sic] of Mathematics (which does discuss some female mathematicians) and is an inspiring read. Even Isaac Newton worked on the problem, but unlike everyone else, he solved it in a single day. Both James and John came up with their own solutions.

Then an amazing thing happened. The scientists of Europe discovered that the equation that solved the Brachistochrone problem is the same equation that governs the basic laws of physics—the laws of motion, of the planets, of everything!

It seemed that Mother Nature was an optimizer herself. Nature finds the shortest time path for a beam of light traveling through air, water, and glass. Because light travels slower in glass, it avoids spending too much time in the glass and takes a more distant path through the air where it travels faster (than it does in glass). This is Fermat's principle of optics. It is the same strategy a life guard follows to save a drowning man. If the drowning man is located on a diagonal path across the water, she will run as fast as she can along the beach and only enters the water at a point where she will get to the drowning man the quickest. She knows that she swims a lot slower than she can run. She is an optimizer.

The principle of the shortest time (and related optimization problems) is ubiquitous in nature and in human behavior. It applies to quantum physics, general relativity, optics, space trajectories, traffic control, aircraft design, game theory, and population dynamics. All of these theories and applications came from the original

Chapter 36 Minimize the Time work of the Bernoullis who fought like brothers but thought like rocket scientists.

So how can we apply the principle to ordinary life? Perhaps Benjamin Franklin said it best in his advice to a young tradesman: "Remember that time is money."

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