Wandering about in a dark labyrinth:
The role of mathematics in the
sciences and engineering
Gangan Prathap
C-MMACS & JNCASR
Bangalore 560037 and 560064
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1
Galileo Galilei
(1564-1642)
2
Philosophy is written in this
grand book - I mean the universe which stands continuously open to
our
gaze,
but
it
cannot
be
understood unless one first learns
to comprehend the language and
interpret the characters in which
it is written. It is written in
the
language
of
mathematics...
without
which
it
is
humanly
impossible to understand a single
word of it. Without these one is
wandering
about
in
a
dark
labyrinth.
Galileo Galilei
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Mathematics, rightly viewed,
possesses not only truth, but
supreme beauty - beauty cold and
austere, like that of sculpture,
without appeal to any part of our
weaker
nature,
without
the
gorgeous trappings of painting or
music, yet sublimely pure, and
capable of a stern perfection such
as only the greatest art can show.
The true spirit of delight, the
exaltation, the sense of being
more
than
Man,
which
is
the
touchstone
of
the
highest
excellence, is to be found in
mathematics
as
surely
as
in
poetry.
BERTRAND RUSSELL
Study of Mathematics
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THE LAW OF THE LEVER
GIVE ME A PLACE TO STAND AND I WILL MOVE
THE EARTH
A remark of Archimedes quoted by Pappus of
Alexandria in his "Collection" (Synagoge, Book
VIII, c. AD 340 [ed. Hultsch, Berlin 1878, p.
1060]).
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THE LAW OF THE LEVER
The two wonders:
1. The phenomenological or empirical law
Ancient Greeks such as Aristotle knew about the
principle of the lever (or "law of the lever")
very early on in history, but they had trouble
proving their theories.
2. The proof from 1st principles
Archimedes, a Greek mathematician who lived from
287-212 B.C., made a statement about when levers
are in equilibrium.
"The law states that a lever is in
equilibrium when the product of the applied force
and the distance from the from the point of
application to the fulcrum equals the product of
the resisting force and the distance from it's
point of application to the fulcrum."
L1
F1
L2
F2
F1L1 = F2L2
How did Archimedes derive this law?
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More on the law of the lever
The concept of bending moment about a point or
the equilibrium of such moments was not known
till the time of Stevinus, i.e. nearly 19 centuries
later!
Archimedes was the first to use the principle of
virtual work.
d1
d2
L1
L2
F1
F2
The principle of virtual work:
d1F1 = d2F2
From Euclid’s geometry:
d1/L1 = d2L2
Therefore,
F1L1 = F2L2
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More on the law of the lever
F1
F2
d1
L1
θ
L2
d2
Kinematics
ìïd1 üï éL1 ù
í ý=ê
ú {θ }
d
L
ïî 2 ïþ ë 2 û
i.e. {d} = [T ] {θ }
Kinetics
ìF1 ü
ý = {M}
î 2þ
ëL1 − L 2 û íF
i.e. [T ]T {F} = {M}
Why does this very interesting relationship emerge?
In kinematics, we have considered pure deformation, without worrying
about forces.
In kinetics, we have considered forces at equilibrium, without considering
the deformation at all.
Yet, they are inter-linked through a very interesting relationship. Note
that a purely verbal language would have never been able to show this
awe–inspiring form. Yet the language of mathematics grasps the poetry of
the relationship so elegantly.
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THERE IS A story about two friends, who were
classmates in high school, talking about their
jobs. One of them became a statistician and was
working on population trends. He showed a reprint
to his former classmate. The reprint started, as
usual, with the Gaussian distribution and the
statistician explained to his former classmate
the meaning of the symbols for the actual
population, for the average population, and so
on. His classmate was a bit incredulous and was
not quite sure whether the statistician was
pulling his leg. "How can you know that?" was his
query. "And what is this symbol here?" "Oh," said
the statistician, "this is pi." "What is that?"
"The ratio of the circumference of the circle to
its diameter." "Well, now you are pushing your
joke too far," said the classmate, "surely the
population
has
nothing
to
do
with
the
circumference of the circle."
The Unreasonable Effectiveness of
Mathematics in the Natural Sciences
Eugene Wigner
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WHAT IS PHYSICS?
The physicist is interested in
discovering the laws of inanimate
nature.
What is
nature"?
the
concept,
"law
of
Schrodinger has remarked, that it
is a miracle that in spite of the
baffling complexity of the world,
certain regularities in the events
could be discovered. Being able to
notice this and express this as an
empirical or phenomenological law
is the 1st wonder I talked about.
One such regularity, discovered by
Stevinus/Galileo,
is
that
two
rocks, dropped at the same time
from the same height, reach the
ground at the same time.
The laws of nature are concerned
with such regularities.
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Galileo's regularity is a prototype of
a large class of regularities.
It is a surprising
three reasons:
regularity
for
1. It is surprising that it is true
not only in Pisa, and in Galileo's
time, it is true everywhere on the
Earth, was always true, and will
always be true. This property of the
regularity is a recognized invariance
property
and,
without
invariance
principles similar to those implied in
the
preceding
generalization
of
Galileo's observation, physics would
not be possible.
2. The regularity is independent of so
many conditions which could have an
effect on it. It is valid no matter
whether it rains or not, whether the
experiment is carried out in a room or
from the Leaning Tower, no matter
whether the person who drops the rocks
is a man or a woman. It is valid even
if
the
two
rocks
are
dropped,
simultaneously
and
from
the
same
height, by two different people.
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3. The preceding two points, though highly
significant from the point of view of the
philosopher, are not the ones which surprised
Galileo most, nor do they contain a specific law
of nature. The law of nature is contained in the
statement that the length of time which it takes
for a heavy object to fall from a given height is
independent of the size, material, and shape of
the body which drops. In the framework of
Newton's second "law," this amounts to the
statement that the gravitational force which acts
on the falling body is proportional to its mass
but independent of the size, material, and shape
of the body which falls.
The preceding discussion is intended to
remind us, first, that it is not at all natural
that "laws of nature" exist, much less that man
is able to discover them.
There is a succession of layers of "laws of
nature," each layer containing more general and
more encompassing laws than the previous one and
its discovery constituting a deeper penetration
into the structure of the universe than the
layers recognized before. However, the point
which is most significant in the present context
is that all these laws of nature contain, in even
their remotest consequences, only a small part of
our knowledge of the inanimate world. All the
laws of nature are conditional statements which
permit a prediction of some future events on the
basis of the knowledge of the present, except
that some aspects of the present state of the
world, in practice the overwhelming majority of
the determinants of the present state of the
world, are irrelevant from the point of view of
the prediction. The irrelevancy is meant in the
sense of the second point in the discussion of
Galileo's theorem.
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Tempus in quo aliquod spatium a mobili conficitur
latione ex quiete uniformiter accelerata, est
aequale
tempori
in
quo
idem
spatium
{10}
conficeretur ab eodem mobili motu aequabili
delato, cuius velocitatis
gradus subduplus sit
ad summum et ultimum gradum velocitatis prioris
motus uniformiter accelerati.
The time in which any space is traversed by a
body starting from rest and uniformly accelerated
is equal to the time in which that same space
would be traversed by the same body moving at a
uniform
speed whose value is the mean of the
highest
speed
and
the
speed
just
before
acceleration began.
t = s/v = s/([v0 + v1]/2)
or
s = [v0 + v1]t/2
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Let us represent by the line AB the time in which the
space CD is traversed by a body which starts from rest at
C and is uniformly accelerated; let the final and highest
value of the speed gained during the interval AB be
represented by the line EB drawn at right angles to AB;
draw the line AE, (Condition 2/00-th-00-dialog1) then all
lines drawn from equidistant points on AB and parallel to
BE will represent the increasing values of the speed,
beginning with the A. Let the point F bisect the line EB;
draw FG parallel to BA, and GA parallel to FB, thus
forming a parallelogram AGFB which will be equal in area
to the triangle AEB, since the side GF bisects the side
AE at the point I; for if the parallel lines in the
triangle AEB are extended to GI, then the sum of all the
parallels contained in the quadrilateral is equal to the
sum of those contained in the triangle AEB; for
those in the triangle IEF are equal to those contained in
the triangle GIA, while those included in the trapezium
AIFB are common. Since each and every instant of time; in
the time-interval AB has its corresponding point on the
line AB, from which points parallels drawn in and limited
by the triangle AEB represent to increasing values of the
growing velocity, and since parallels contained within
the rectangle represent the values of a speed which is
not increasing, but constant, it appears, in like manner,
that the momenta (momenta) assumed by the moving body may
also be represented, in the case of the accelerated
motion, by the increasing parallels of the triangle AEB,
and, in the case of the uniform motion, by the parallels
of the rectangle GB. For, what the momenta may lack in
the first part of the accelerated motion (the deficiency
of the momenta being represented by the parallels of the
triangle AGI) is made up by the momenta represented by
the parallels of the triangle IEF. (Condition Aristotspace-prop) Hence it is clear that equal spaces will be
traversed in equal times by two bodies, one of which,
starting
from
rest,
moves
with
uniform
acceleration, while the momentum of the other, moving
with uniform speed, is one-half its maximum momentum
under accelerated motion. Q. E. D.
t = s/v = s/([v0 + v1]/2)
or
s = [v0 + v1]t/2
14
FROM CLASSICAL TO QUANTUM PHYSICS
The principal purpose of the preceding
discussion is to point out that the
laws of nature are all conditional
statements and they relate only to a
very small part of our knowledge of
the world. Thus, classical mechanics,
which is the best known prototype of a
physical theory, gives the second
derivatives
of
the
positional
coordinates of all bodies, on the
basis
of
the
knowledge
of
the
positions, etc., of these bodies. It
gives no information on the existence,
the present positions, or velocities
of these bodies.
It should be mentioned, for the
sake of accuracy, that we discovered
about thirty years ago that even the
conditional
statements
cannot
be
entirely precise: that the conditional
statements are probability laws which
enable us only to place intelligent
bets on future properties of the
inanimate
world,
based
on
the
knowledge of the present state.
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FROM PHYSICS TO ENGINEERING
As regards the present state of
the world, such as the existence
of the earth on which we live and
on
which
Galileo's
experiments
were performed, the existence of
the
sun
and
of
all
our
surroundings, the laws of nature
are entirely silent. It is in
consonance with this, first, that
the laws of nature can be used to
predict future events only under
exceptional circumstances - when
all the relevant determinants of
the present state of the world are
known. It is also in consonance
with this that the construction of
machines, the functioning of which
he can foresee, constitutes the
most spectacular accomplishment of
the physicist. In these machines,
the physicist creates a situation
in
which
all
the
relevant
coordinates are known so that the
behavior of the machine can be
predicted.
Radars
and
nuclear
reactors are examples of such
machines.
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Wandering about in a dark labyrinth:
Mathematics is the lamp that lights
up the way.
Gangan Prathap
C-MMACS & JNCASR
Bangalore 560037 and 560064
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