A Fermat’s Principle of Least Time Variational Principle: Of all conceivable happenings, nature chooses the one of least ‘effort’. Principle of Least Action ( Pierre-Louis Moreau de Maupertuis , 1746 ) Hamilton’s Principle History topic: The brachistochrone problem The brachistochrone problem was posed by Johann Bernoulli in Acta Eruditorum in June 1696. He introduced the problem as follows:- I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise. The problem he posed was the following:-
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Transcript
A
Fermat’s Principle of Least Time
Variational Principle:
Of all conceivable happenings, nature chooses the one of least ‘effort’.
Principle of Least Action ( Pierre-Louis Moreau de Maupertuis , 1746 )
Hamilton’s Principle
History topic: The
brachistochrone problem
The brachistochrone problem was posed by Johann Bernoulli in Acta
Eruditorum in June 1696. He introduced the problem as follows:-
I, Johann Bernoulli, address the most brilliant mathematicians in the
world. Nothing is more attractive to intelligent people than an
honest, challenging problem, whose possible solution will bestow
fame and remain as a lasting monument. Following the example set
by Pascal, Fermat, etc., I hope to gain the gratitude of the whole
scientific community by placing before the finest mathematicians of
our time a problem which will test their methods and the strength of
their intellect. If someone communicates to me the solution of the
proposed problem, I shall publicly declare him worthy of praise.
The problem he posed was the following:-
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.
Perhaps we are reading too much into Johann Bernoulli's references
to Pascal and Fermat, but it interesting to note that Pascal's most
famous challenge concerned the cycloid, which Johann Bernoulli
knew at this stage to be the solution to the brachistochrone
problem, and his method of solving the problem used ideas due to
Fermat.
Johann Bernoulli was not the first to consider the brachistochrone problem. Galileo in
1638 had studied the problem in 1638 in his famous work Discourse on two new
sciences. His version of the problem was first to find the straight line from a point A
to the point on a vertical line which it would reach the quickest. He correctly
calculated that such a line from A to the vertical line would be at an angle of 45
reaching the required vertical line at B say.
He calculated the time taken for the point to move from A to B in a straight
line, then he showed that the point would reach B more quickly if it travelled along
the two line segments AC followed by CB where C is a point on an arc of a circle.
Although Galileo was perfectly correct in this, he then made an error
when he next argued that the path of quickest descent from A to B
would be an arc of a circle - an incorrect deduction.
Returning to Johann Bernoulli he stated the problem in Acta
Eruditorum and, although knowing how to solve it himself, he
challenged others to solve it. Leibniz persuaded Johann Bernoulli to
allow a longer time for solutions to be produced than the six months
he had originally intended so that foreign mathematicians would
also have a chance to solve the problem. Five solutions were
obtained, Newton, Jacob Bernoulli, Leibniz and de L'Hôpital solving
the problem in addition to Johann Bernoulli.
Now Johann Bernoulli and Leibniz deliberately tempted Newton with
this problem. It is not surprising, given the dispute over the calculus,
that Johann Bernoulli had included these words in his challenge:-
...there are fewer who are likely to solve our excellent problems,
aye, fewer even among the very mathematicians who boast that
[they]... have wonderfully extended its bounds by means of the
golden theorems which (they thought) were known to no one, but
which in fact had long previously been published by others.
According to Newton's biographer Conduitt, he solved the problem
in an evening after returning home from the Royal Mint. Newton:-
... in the midst of the hurry of the great recoinage, did not come
home till four (in the afternoon) from the Tower very much tired, but
did not sleep till he had solved it, which was by
four in the morning.
Newton sent his solution to Charles Montague, the
Earl of Halifax, who was an innovative finance
minister and the founder of the Bank of England.
Montague was the principal patron and lifelong
friend of Newton and, in addition, the 'common-
law' husband of Newton's niece. He was President of the Royal
Society during the years 1695 to 1698 so it was natural that Newton
send him his solution to the brachistochrone problem. However, as
he wrote afterwards, the episode did not please Newton:-
I do not love to be dunned [pestered] and teased by foreigners
about mathematical things ...
The Royal Society published Newton's solution anonymously in the
Philosophical Transactions of the Royal Society in January 1697. His
solution was explained to Montague as follows:-
Problem. It is required to find the curve ADB in which a weight, by
the force of its gravity, shall descend most swiftly from any given
point A to any given point B.
Solution. From the given point A let there be drawn an unlimited
straight line APCZ parallel to the horizontal, and on it let there be
described an arbitrary cycloid AQP meeting the straight line AB
(assumed drawn and produced if necessary) in the point Q, and
further a second cycloid ADC whose base and height are to the base
and height of the former as AB is to AQ respectively. This last
cycloid will pass through the point B, and it will be that curve along
which a weight, by the force of its gravity, shall descend most
swiftly from the point A to the point B.
The May 1697 publication of Acta Eruditorum contained Leibniz's
solution to the brachistochrone problem on page 205, Johann
Bernoulli's solution on pages 206 to 211, Jacob Bernoulli's solution
on pages 211 to 214, and a Latin translation of Newton's solution on
page 223. The solution by de L'Hôpital was not published until 1988
when, nearly 300 years later, Jeanne Peiffer presented it as
Appendix 1 in [1]. Johann Bernoulli gave the solvers, saying:-
... my elder brother made up the fourth of these, that the three
great nations, Germany, England, France, each one of their own to
unite with myself in such a beautiful search, all finding the same
truth.
Johann Bernoulli's solution divides the plane into strips and he
assumes that the particle follows a straight line in each strip. The
path is then piecewise linear. The problem is to determine the angle
of the straight line segment in each strip and to do this he appeals
to Fermat's principle , namely that light always follows the shortest
possible time of travel. If v is the velocity in one strip at angle a to
the vertical and u in the velocity in the next strip at angle b to the
vertical then, according to the usual sine law
v/sin a = u/sin b.
In the limit, as the strips become infinitely thin, the line
segments tend to a curve where at each point the angle
the line segment made with the vertical becomes the angle the
tangent to the curve makes with the vertical. If v is the velocity at
(x, y) and a is the angle the tangent makes with the vertical then
the curve satisfies
v/sin a = constant.
Now, Galileo had shown that the velocity v satisfies
v = √(2gy)
(where g is the acceleration due to gravity) and substituting for v
gives the equation of the curve as
√y/sin a = constant or y = k sin2a
Use y' = dy/dx = cot a and sin2a = 1/(1+cot2a) = 1/(1+y'2) to get
y(1+y'2) = 2h
for a constant h (= 1/(2k2)).
The cycloid x(t) = h(t - sin t), y(t) = h(1 - cos t) satisfies this
equation. To see this note that
y' = dy/dx = dy/dt . dt/dx = -(sin t)/(1 - cos t)
so
y(1+y'2) = h(1 - cos t)(1+sin2t/(1-cos t)2)
= h(1 - cos t + sin2t/(1-cos t))
= h((1-cos t)2+ sin2t)/(1-cos t)
= h(2-2cos t)/(1-cos t) = 2h
Now Huygens had shown in 1659, prompted by Pascal's challenge
about the cycloid, that the cycloid is the solution to the tautochrone
problem, namely that of finding the curve for which the time taken
by a particle sliding down the curve under uniform gravity to its
lowest point is independent of its starting point.
Johann Bernoulli ended his solution of the brachistochrone problem
with these words:-
Before I end I must voice once more the admiration I feel for the
unexpected identity of Huygens' tautochrone and my
brachistochrone. I consider it especially remarkable that this
coincidence can take place only under the hypothesis of Galileo, so
that we even obtain from this a proof of its correctness. Nature
always tends to act in the simplest way, and so it here lets one
curve serve two different functions, while under any other
hypothesis we should need two curves ...
Despite the friendly words with which Johann Bernoulli described his
brother Jacob Bernoulli's solution to the brachistochrone problem
(see above), a serious argument erupted between the brothers after
the May 1697 publication of Acta Eruditorum. It was Jacob Bernoulli
who now challenged his brother. Returning to Galileo's original
question regarding the time to reach a vertical line rather than a
point he asked:-
Given a starting point and a vertical line, of all the cycloids from the
starting point with the same horizontal base, which will allow the
point subjected only to uniform gravity, to reach the vertical line
most quickly.
Johann Bernoulli solved this problem showing that the cycloid which
allows the particle to reach the given vertical line most quickly is the
one which cuts that vertical line at right angles. There is a wealth of
information in the correspondence with Varignon given in [1]. Jacob
Bernoulli posed isoperimetric problems to Johann Bernoulli and a
bitter dispute arose between the two brothers on these problems
which Varignon also became involved in. It was an unpleasant
incident, but one of great value to mathematics for the problems
being argued about led directly to the founding of the calculus of
variations. The quarrel between the Bernoulli brothers is examined
in detail in [10] where as well as the mathematical details the
author studies the psychological side. He argues convincingly that
the bad feeling between them must a started at home with a strict
and unfriendly father.
The methods which the brothers developed to solve the challenge
problems they were tossing at each other were put in a general
setting by Euler in Methodus inveniendi lineas curvas maximi
minimive proprietate gaudentes sive solutio problematis
isoperimetrici latissimo sensu accepti published in 1744. In this
work, the English version of the title being Method for finding plane
curves that show some property of maxima and minima, Euler
generalises the problems studies by the Bernoulli brothers but
retains the geometrical approach developed by Johann Bernoulli to
solve them. He found what has now come to be known as the Euler-
Lagrange differential equation for a function of the maximising or
minimising function and its derivative.
The idea is to find a function which maximises or minimises a
certain quantity where the function is constrained to satisfy certain
constraints. For example Johann Bernoulli had posed certain
geodesic problems to Euler which, like the brachistochrone problem,
were of this type. Here the problem was to find curves of minimum
length where the curves were constrained to lie on a given surface.
Euler, however, commented that his geometrical approach to these
problems was not ideal and it only gave necessary conditions that a
solution has to satisfy. The question of the existence of a solution
was not solved by Euler's contribution.
Lagrange, in 1760, published Essay on a new method of determining
the maxima and minima of indefinite integral formulas. It gave an
analytic method to attach calculus of variations type problems. In
the introduction to the paper Lagrange gives the historical
development of the ideas which we have described above but it
seems appropriate to end this article by giving what is in effect a
summary of the developments in Lagrange's words:-
The first problem of this type [calculus of variations] which
mathematicians solved was that of the brachistochrone, or the
curve of fastest descent, which Johann Bernoulli proposed towards
the end of the last century. The solution was found by considering
special cases, and it was only some time later, in research
isoperimetric curves, that the great mathematician of whom we
speak and his famous brother Jacob Bernoulli gave some general
rules for solving several other problems of the same type. Since,
however, the rules were not sufficiently general, the famous Euler
undertook the task of reducing all such investigations to a general
method which he gave in the work "Essay on a new method of
determining the maxima and minima of indefinite integral
formulas"; an original work in which the profound science of the
calculus shines through. Even so, while the method is ingenious and
rich, one must admit that it is not as simple as one might hope in a
work of pure analysis ....
Lagrange then goes on to describe his introduction of the differential
symbol . He gives:-
... a method which only requires a straightforward use of the
principles if the differential and integral calculus; but I must strongly
emphasise that since my method requires that a quantity be
allowed to vary in two different ways, so as not to confuse these
different variations, I have introduced a new symbol into my
calculations. In this way z expresses a difference of z which is different from dz, but which, however, will satisfy the same rules;
such that where we have for any equation dz = m dx, we can
equally have z = m x, and likewise in other cases.
Article by: J J O'Connor and E F Robertson
February 2002
MacTutor History of Mathematics
A Brachistochrone curve, or curve of fastest descent, is the curve
between two points that is covered in the least time by a body that
starts at the first point with zero speed and passes down along the
curve to the second point, under the action of constant gravity and
ignoring friction.
The brachistochrone is the cycloid
Given two points A and B, with A not lower than B, there is just one
upside down cycloid that passes through A with infinite slope,
passes also through B and does not have maximum points between
A and B. This is the brachistochrone curve. The brachistochrone thus
does not depend on the body's mass or on the strength of the
gravitational constant.
The problem can be solved with the tools from the calculus of
variations.
Note that if the body is given an initial velocity at A, or if friction is
taken into account, the curve that minimizes time will differ from
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