REVISITING A CLASSIC
LEAST TIME PROBLEM
Ben Szapiro
The Department of Physics
The University of the South, Sewanee


Abstract

The brachistochrone problem, that is, "find the path of shortest time of a particle moving between two points on a vertical plane", was proposed, solved erroneously, and studied experimentally by Galileo, and solved mathematically by Jacques Bernoulli's variational calculus methods in 1697. We will revisit the brachistochrone problem from the prospective of an undergraduate student: we will analyze the "default" answer (straight line = least time), the possibility of breaking down the motion in a succession of straight lines, the subsequent time optimization strategy, friction and rolling ball effects, the 3-D extension, and its connection with Huygens' cycloidal pendulum. Finally, a demonstration of the apparatus will be presented and compared "live" with a simulation of the motion.


Extremum problems provide wonderful material for teaching thinking, inventiveness, flexibility, creativity... But the only way to teach thinking is with concrete special problems... But we still have to show the existence of general principles and laws in science...But we also want to transmit science as part of our cultural heritage...But the notation and methods look so different....But...(


"BRACHISTOCHRONE"

BRACHIS = SHORT

CHRONOS = TIME


From Acta Eroditorum, the first scientific journal (Vol. 15):

Statement of the Problem:

"Let two points A and B be given in a vertical plane. Find the curve that a point M, moving on a path AMB must follow such that, starting from A, reaches B in the shortest time under its own gravity"


RESPONSES TO THE CHALLENGE:

1) Johann Bernoulli (Of course; solved Fermat's way)
2) Gottfried Wilhelm von Leibniz ("Splendid problem")
3) Jakob Bernoulli (Johann's brother, legal counsel says OK)
4) de l'Hospital
5) Isaac Newton (as "an anonymous englishman", but: "ex ungue leonem")

(The first Dream Team)


"If one considers motions with the same initial and terminal points
then the shortest distance between them being a straight line ,
one might think that the motion along it needs least time.

It turns out that this is not so."

Galileo Galilei
Discourses on Mechanics (1588)



Johann Bernouilli

(1667-1748)





 


Lets look at the solution


Galileo Galilei (1629-1695) and Christiaan Huygens (1564-1642)





Straight from A -> B (distance s)



First Down to C, then Horizontal to B: