Too Nice to Ignore, Gravity and Bernoulli's Lemniscate

 

From a 2009 post with additional material added.

*Wik

Two stories intersect here, one a famous event from the history of calculus that most folks are familiar with, and one that seems not to make it much into classrooms and that I only learned about today. 
Almost every student of mathematics will sooner or later come across the beautiful problem of the Brachistochrone, which in Greek means "shortest time." It is the path that will carry a point-like body from one place to another in the least amount of time under the force of constant gravity. Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve. The curve does not depend on the body's mass or on the strength of the gravitational constant.

The story goes that Johann Bernoulli posed the problem to readers of Acta Eruditorum in June, 1696. He published his solution along with four other solutions from Newton, Jakob Bernoulli, Gottfried Leibniz, and Ehrenfried Walther von Tschirnhaus. ( l'Hôpital also seems to have had a correct solution, but it was not published).

Newton historians claim that Newton received the problem in the mail one afternoon after returning from his job at the mint (so this would be after he was older). The story goes that he solved it overnight, and posted it the next morning. Since it took weeks for some of the others to solve it, we may assume that Newton was still a pretty good mathematician well after his known prime.

A footnote to this story, in the epic novel Moby Dick, "Ishmael thinks about them while cleaning the try-pots (giant cauldrons in which whale blubber is rendered) on the deck of the Pequod.  It was in the ...trypot with the soapstone diligently circling around me, that I was first struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, ...,will descend from any point in precisely the same time."  

How would Melville's modest education bring him this incredible math fact.  A possible answer.....Joseph Henry, you know, the guy for whom the unit of inductance is named.  

It is almost certain that the limited public school education of Melville would not include this fact.  Most high school students today would never be introduced to it.  But in Melville's brief time at the Albany Academy it was said that Herman excelled in "ciphering" and won the school prize.  Perhaps his interest in geometry and such was inspired by an outstanding teacher, and former alumni of the Albany Academy, young Joseph Henry.

(Once Upon A Prime by Sarah Hart)  

Ok, so that's the one everybody knows about. But I was just fixing up some notes on the life of Gian Francesco Malfatti, whose date of death was, well, today, in 1807. Now Malfetti did some nice stuff, too. He did some really important work on fifth degree equations; but he is best known for a geometry problem about three mutually tangent circles inscribed in a triangle. He posed the problem of as how to
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*Mathworld

carve three circular columns out of a triangular block of marble, using as much of the marble as possible. He thought the solution was described by the triangle problem. The Geometry problem of constructing three circles each tangent to each other and two sides of the triangle is now generally known as Malfatti's problem, even though he didn't do it first, Japanese geometer Chokuen Ajima beat him too it.  (Both the earliest solutions, I'm told, used a combination of geometry and algebra, but it seems Steiner did show how to do it with pure geometric methods). What's worse is that it wasn't even the solution to the physical problem of the columns he was trying to show. Around 1930 someone proved that it wasn't always the best solution, and then in the 60's, M. Goldberg showed it was NEVER the best solution.

Then while checking some dates on his Wikipedia entry, I noticed something else he had done, the something I had never heard of, and this time he was right. He was working with the lemniscate (it means ribbon) first named and described in 1694 by Jakob Bernoulli, and he noticed an interesting gravitational relationship about it. If you draw a chord (pick a chord, any chord... ok; sorry) through the center and any other point on the lemniscate, then a point acting under the influence of gravity will reach that point of intersection at the same moment, whether it travels down the chord, or around the lemniscate. Now that just seems to nice to have ignored in math and calculus classrooms. If you are a teacher, maybe the next time you talk about the brachistochrone, (or maybe tomorrow when they need a diversion) you should point out this little beauty, also.

The actual lemniscate  is actually a figure eight balanced on the center of intersection. One way to write the equation is r^2 = a^2 sin(2t).  This would produce the oblique lemniscate below.  If you replace sin with cosine, the lemniscate would be horizontal.  In the example below a = 2.  

 

This one seems to have it's "highest" point at theta = pi/3.  looking at it made me wonder if a particle placed at this point would actually trace the entire figure eight under the force of gravity in the same way as Bernoulli's half lemniscate if a tiny force bumped it off equilibrium. 

It can be fun asking students to figure out what happens if the 2 in sin(2t) is replace with another number.  When they are sure they have it figured out, ask them abut a fraction like 3/2 or 5/3, or 1/3 without their calculators, and of course, does it matter if you use negative 2t.