This mathematical blog post will be focused on the
Brachistochrone problem. This is a famous problem that dates back to the time
of Newton, Leibniz and Bernoulli. Unlike many of the other problems mentioned
in this blog, this one has been solved, regardless it is still a great
mathematical problem. The first part of the post will cover the history of the problem. Then the second part will be an explanation of Bernoulli’s solution, and finally closing with a part on the geometrical
representation of the problem.
Brachistochrone
comes from two Greek words, meaning, the shortest time. Johannes Bernoulli used
this to describe a problem concerning traveling between two points with the
effect of gravity. The problem can be imagined with two points, A, and B. These
two points are on a plane, point A being slightly elevated above point B. The
problem is, what would be the best past path for an object to roll down form
point A to reach point B? This of course being which path would be the fastest
path, hence the Greek term for shortest time. This is not as straight forward
of a problem as it seems. There are two factors that come into play. The first
being the distance traveled. A straight line has the shortest distance, so it
would take an object less time. The second being the speed of the object. If
there is a curve in the line, it allows for the object to accelerate, and
therefore cover the longer distance faster. Galileo had purposed before
Bernoulli that an arc of a circle would be the best path. The circle is better
than the straight line, but it is still not the best solution to the problem.
The balance lies in between a circle arc, and a straight line. Bernoulli came up
with a solution to the problem, but rather than sharing the solution, he sent out
the problem as a challenge to the other mathematicians at the time. These
mathematicians included Leibniz, Newton, and Johannes Bernoulli’s brother
Jacob. Bernoulli challenged them in order to show that he was the cleverest, or
the smartest mathematician at the time. Some historians speculate that he was
really only trying to rival his brother, but either way he was trying to show
off his mathematical skills. It may have back fired on him though, because
Newton solved it over night, when it took Bernoulli two weeks to solve it.
Bernoulli, though not being able to solve it the fastest, came up with a rather
clever way of looking at the problem.
Bernoulli
came up with a way of explaining the curve with light. Fermat at the time came
up with a principle that light would move with different angles, through
different mediums, to maximize the speed of the light. This can be proven with
Snell’s Law. The important part of Snell’s law that needs to be understood with
relation to this post is that light will take the best path when passing through
different mediums. If light passes through multiple layers of a medium it
starts to curve in line segments. If the number of mediums increases, the curve
becomes smoother. If the number of mediums approaches infinite the curve for
the best path of our object becomes apparent. This was Bernoulli’s solution,
and it is correct.
Johannes
Bernoulli also saw some geometry in the curve. He recognized that at any point
on the curve, the sin of the angle between the tangent line at that point, and
the vertical line at the point, divided by the square root of the distance from
the point, to the top of the start of the curve, is constant. He recognized
that to be what is called a cycloid. A cycloid is the path of a rolling wheel
along it radius. Imagine a wheel, and on the edge of the wheel an expo marker
was taped so that it was flush with the radius and did not affect the roll of
the wheel. If the wheel was rolled next to a white board a cycloid would be draw
with the marker. However, the relation between the cycloid, and the discovery
made by Bernoulli, is not as obvious as Bernoulli made it seem. At first the
sin(theta)/sqrt(y) does not seem to have any relation to a rolling wheel.
However, when looking at the wheel, and using some geometry it can be proven
that the relation is directly proportional to the diameter of the wheel, this
meaning since the diameter is constant the ratio of the sin(theta)/sqrt(y) is
constant as well.
(cycloid)