Role of Mathematics Lecture

8/13/2019 Role of Mathematics Lecture

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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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Galileo Galilei

(1564-1642)


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Philosophy is written in thisgrand book - I mean the universe -which stands continuously open toour gaze, but it cannot be

understood unless one first learnsto comprehend the language andinterpret the characters in whichit is written. It is written inthe language of mathematics...without which it is humanly

impossible to understand a singleword of it. Without these one iswandering about in a darklabyrinth.

Galileo Galilei

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Mathematics, rightly viewed,

possesses not only truth, butsupreme beauty - beauty cold andaustere, like that of sculpture,without appeal to any part of ourweaker nature, without thegorgeous trappings of painting ormusic, yet sublimely pure, andcapable of a stern perfection suchas only the greatest art can show.The true spirit of delight, theexaltation, the sense of beingmore than Man, which is thetouchstone of the highest

excellence, is to be found inmathematics as surely as inpoetry.

BERTRAND RUSSELLStudy of Mathematics


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THE LAW OF THE LEVER

GIVE ME A PLACE TO STAND AND I WILL MOVETHE 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 leversare 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 L2

F1 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:

d1F

1= d

2F

2

From Euclids geometry:

d1/L1= d2L2

Therefore,

F1L1 = F2L2


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More on the law of the lever

F1 F2

d1

L1 L2 d2

Kinematics

Kinetics

Why does this very interesting relationship emerge?

In kinematics, we have considered pure deformation, without worryingabout 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. Notethat a purely verbal language would have never been able to show this

aweinspiring form. Yet the language of mathematics grasps the poetry of

the relationship so elegantly.

{ } { } [ ]{ } Tdi.e.L-

L

d

d

2

1

2

1=

=

{ } [ ] { } { }MFTi.e.MF

FLL T

2

121 ==


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THERE IS A story about two friends, who were

classmates in high school, talking about theirjobs. 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 hisquery. "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 indiscovering the laws of inanimatenature.

What is the concept, "law ofnature"?

Schrodinger has remarked, that itis a miracle that in spite of thebaffling complexity of the world,certain regularities in the eventscould be discovered. Being able tonotice this and express this as an

empirical or phenomenological lawis the 1st wonder I talked about.

One such regularity, discovered byStevinus/Galileo, is that tworocks, dropped at the same time

from the same height, reach theground at the same time.

The laws of nature are concernedwith such regularities.


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Galileo's regularity is a prototype of

a large class of regularities.

It is a surprising regularity forthree reasons:

1. It is surprising that it is true

not only in Pisa, and in Galileo'stime, it is true everywhere on the

Earth, was always true, and will

always be true. This property of theregularity is a recognized invariance

property and, without invarianceprinciples similar to those implied in

the preceding generalization of

Galileo's observation, physics wouldnot be possible.

2. The regularity is independent of somany conditions which could have an

effect on it. It is valid no matter

whether it rains or not, whether theexperiment is carried out in a room or

from the Leaning Tower, no matterwhether the person who drops the rocks

is a man or a woman. It is valid evenif 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 lawof 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 andits 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 alllines 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 inthe 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 Aristot-

space-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 momentumunder accelerated motion. Q. E. D.

t = s/v = s/([v0 + v1]/2)

or s = [v0 + v1]t/2


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FROM CLASSICAL TO QUANTUM PHYSICS

The principal purpose of the preceding

discussion is to point out that thelaws of nature are all conditional

statements and they relate only to a

very small part of our knowledge ofthe world. Thus, classical mechanics,

which is the best known prototype of a

physical theory, gives the second

derivatives of the positionalcoordinates of all bodies, on the

basis of the knowledge of thepositions, 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 theconditional statements cannot beentirely precise: that the conditional

statements are probability laws whichenable us only to place intelligent

bets on future properties of theinanimate world, based on the

knowledge of the present state.


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FROM PHYSICS TO ENGINEERING

As regards the present state ofthe world, such as the existenceof the earth on which we live andon which Galileo's experimentswere performed, the existence ofthe sun and of all oursurroundings, the laws of nature

are entirely silent. It is inconsonance with this, first, thatthe laws of nature can be used topredict future events only underexceptional circumstances - whenall the relevant determinants of

the present state of the world areknown. It is also in consonancewith this that the construction ofmachines, the functioning of whichhe can foresee, constitutes themost spectacular accomplishment ofthe physicist. In these machines,

the physicist creates a situationin which all the relevantcoordinates are known so that thebehavior of the machine can bepredicted. Radars and nuclearreactors are examples of suchmachines.


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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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Role of Mathematics Lecture

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