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85RESONANCE ⎜ September1999

Wings of Fire

H S Mukunda

The book Wings of Fire, the autobiography ofA P J Abdul Kalam constitutes an extraordinaryreading for the young and old, for a commonman and a professional including managementpundits, and for the uninitiated and the technicallytrained alike. There is something that everybodycan extract from this book. Only the very youngreaders of Resonance may find it beyondthem.

The writing style is very gripping, and thecontents are very unusual in their frankness atseveral levels while not being caustic or cynicaland provide a brief panoramic view of the spaceand missile technology arena of the country.Books of this kind are very rare in our country;and we are prone to idolize and eulogizescientists and other personalities with individualexcellence with almost no appreciation for themilieu in which creativity is possible. We arealso hesitant to record truth, if we take interestin recording at all. The book is the story of a manwho did not quite acquire ‘power’ by climbingthrough academic corridors, an approach withall its penchant for placing or replacing idols inits own sanctum-sanctorum of the academiesor other positions is practised assiduously inthe country. Even in the absence of this power

(or, is it because of its absence), he wovetogether large number of groups, challengedthem to do the ‘impossible’, promoted them towork towards a common goal, and throughsuccesses, raised their self esteem enormously.

Apart from the very indicative title, the chaptersof the book are also touching –orientation,creation, propitiation, and contemplation.Orientation contains 32 years of his early life –days as a child, going through adolescenceand getting into rocketry. His description of thepeople who shaped his life, interweaving religionand education, is a charming part of the bookthat almost nobody would miss reading. One isreminded of R K Narayan while flippingthrough the text. The difficulties involved ingetting education are very reminiscent of theera – the late thirties through the forties – whichis full of examples of people of lower middleclass (to which most of the society belonged)living largely in villages struggling hard to geteducation in cities/large towns with littleinfrastructure and financial support. Many inthe current era of significant urbanization maynot even be able to visualize the trauma that thegeneration of those times went through. Thefirst chapter captures it succinctly.

Chapter 2 entitled ‘Creation’ describing the nextseventeen years till 1980 covers his struggle atISRO going from one-engineer-amongst-manyto the successful project director of SLV bringingpride to the nation through the technologicalachievement of putting a satellite in the orbit. Heforgets not recording the contributions of themany – both high profile as well the bravebackroom ‘boys’. There are several

Wings of Fire

A P J Abdul Kalam

Universities Press

pp.180, Rs. 200

RESONANCE ⎜ September 199986

BOOK ⎜ REVIEW

technological aspects discussed here and thesewould interest the technically minded a greatdeal. The description of Sarabhai, the earlyleader for space technology, his vision, modeof working is historically important. Kalam’schoice as the Project Director by Dhawan asChairman, ISRO and Brahm Prakash asDirector, VSSC has been left unexplained, butperhaps may not require one; they saw in himqualities of bringing together others and gettingthings done with least interpersonal pain whileall the time thinking of possible problem areasrequiring attention. His agony when the firstSLV flight failed to reach the orbit and thematurity of the management in retaining him inthe same position are brought out. There areseveral attractive observations, poems andpersonal details of his family interspersedbetween the hard technical descriptions thatkeep the reader glued to the book. This chapterends in the description of his transition fromISRO to DRDO as the next director of DRDL.

The next ten years are set out in chapter 3.These constitute the outstanding accomplish-ments at DRDL. The way he transformed thelaboratory from one which had a weak heartwith little confidence to one which felt a senseof strong self-esteem and could feel proud bycontributing developed missile systems to theservices, is a remarkable saga. The methodsby which he accomplished this were three.First, generate first rate techno-logicallycontemporary projects and hold them aschallenges to the technical commu-nity.Second, bring together academics andscientists from R & D institutions for reviews, a

somewhat scandalizing event in tradition bounddefence laboratories. Third, hold a fair numberof internal meetings with members drawn fromdifferent projects and disciplines so that latentdifficult issues come up for discussion andlearning from one another was caused. Themethodology that he adopted here was drawnfrom his earlier experience in ISRO.

On page 122 in this chapter, he discusses thelogic for the choice of the Project Directors forthe five integrated guided missile developmentprojects (all of these were his masterly creation).The three paragraphs provide an insightfulreading even for a management man, comingas it were from a successful leader. Hisremarks on T N Seshan (the well-known retiredChief Election Commissioner) are so apt thatnobody would disagree. The fact that the bookcarries several truthful asides like this onpersonalities known to many including thepresent reviewer makes it authentic and worthyof reading.

Kalam has pet themes – one of these is the factthat unless we become strong there is nolikelihood of us being respected as a superpower (in today’s terminology, a country whichholds the ability to destroy some other countryfrom a distance). He has used the phrase“strength respects strength” several times in hislectures. This assessment is indeed correct.The second theme is that India should becomea developed nation. In fact, the last sentenceof the book is along these lines. The plan tomake India an economically strong nation isvital for India to be rated well along with the

BOOK ⎜ REVIEW

87RESONANCE ⎜ September1999

super powers. For this to happen, many otheraspects – the enormous moral decadenceleading to corruption in higher levels of society,the inability to prevent this from affecting thecommon man’s life, the inability to improve thequality of life of 70% of the population residingin the villages, the inability to reduce the paceof urbanization and the inability to reduce theinequity between the rich and the poor need tobe accounted for. These are of course beyondthe preview of one worthy citizen whose life isdescribed in this book.

The last chapter entitled ‘Contemplation’contains a condensation of ideas and thoughtsarising out of his colourful life, the awards that

he received (the highest honour, namely BharataRatna that he received does not find a mentionin this book) and some messages for the futuregeneration.

The book is worthy of being read by everyIndian who can read!

H S Mukunda, Department of Aerospace

Engineering, Indian Institute of Science, Bangalore

560 012, India.

Ordinary Differential

Equations

Shiva Shankar

“Space and time are commonly regarded as theforms of existence of the real world, matter asits substance. A definite portion of matteroccupies a definite part of space at a definitemoment of time. It is in the composite idea ofmotion that these three fundamental conceptionsenter into intimate relationship”. So declaredHermann Weyl, and this declaration of the

Ordinary Differential Equations

V I Arnold

(reprinted by Prentice-Hall of India from

the MIT Press translation by R

Silverman of Obyknovennye

Differentsial’nye Uravnenia), Price: Rs.95

supreme importance of the study of motion isat the same time also a declaration of thesupreme position that the theory of differentialequations holds in mathematics and physics.For, since Newton, it has been clear that the“shape of motion” is determined by local data,encapsulated in what we call a differentialequation, and the study of motion is thussynonymous with the study of differentialequations. But the shape of motion equallyclearly depends on the global shape of (phase)space, and the study of differential equationsmust therefore be geometric in its methods.This geometric nature of the study of motionwas of course clear to Newton (as even asuperficial browsing of the Principia willconfirm), but had to be reaffirmed by Poincaréafter a 200 year interregnum, perhapsinevitable, during which time Lagrange could

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BOOK ⎜ REVIEW

even exult that he had banished figures from thestudy of mechanics!

The book under review is a first level text in thegeometric spirit of Newton and Poincaré andcontains 259 illuminating figures. But how isone to start such a study of differentialequations? After all, people observed things inmotion for many thousands of years beforethey realized that these trajectories weresolutions to differential equations. Naturallythen does Arnold start with the formal propertiesof solutions or phase curves, viz. the conceptof a one-parameter group of diffeomorphismsof phase space. This is the collection of allsolutions considered (and pictured in one’smind) as a single geometric object. Thedifferential equation itself can then be picturedas a collection of vectors (arrows), at everypoint tangent to the phase curves (i.e. as asection of the tangent bundle, or a vector field).Thus given a differential equation (i.e. a vectorfield), to solve it is to fill up phase space withcurves, all tangent to the vector field.

With this geometric picture in mind,understanding and progress come easy. Onecan visualize how the shape of a vector fieldmust change under the action of a diffeo-morphism (change of coordinates); one caneven convince oneself (without the crutch of aformal proof) the validity of the ‘basic theoremof the theory of ordinary differential equations’,viz. that in some small neighbour-hood of anonsingular point (i.e. a point where the vectorfield does not vanish), one can by adiffeomorphism carry it to a constant vector

field. Even if the vector field vanishes at somepoint, one can by adding the equation dt/dt =1,reduce this situation to the one above. Localproperties of the flow (existence, uniqueness,dependence on a parameter etc.) can now beeasily deduced from the correspondingproperties of the trivial constant vector field. Aformal proof of this basic theorem, including adiscussion on the contraction mapping principleand other related material, is collected in aseparate chapter towards the end of the book– one therefore makes rapid progress simplyassuming these statements.

Arnold then goes on to discuss in detail theimportant and transparent case of a one-parameter group of linear automorphisms ofeuclidean space, i.e. linear differentialequations. Starting with the definition of theexponential of a linear map, he goes on todiscuss topological conjugacy of linearhyperbolic systems, Lyapunov stability,nonautonomous systems, systems with periodiccoefficients and a host of other topics. The bookends with a chapter on differential equations onmanifolds, including a beautiful discussion(leading up to almost a proof) of the Hopf indextheorem for a flavour of the topological natureof the global theory.

Let me illustrate the beauty and the simplicity ofthe geometric methods in Arnold’s book by thefollowing example. The problem is to solve theinitial value problem for the nonhomogeneouslinear equation

t

x

d

d= Ax+h(t), x(t0)=c,

BOOK ⎜ REVIEW

89RESONANCE ⎜ September1999

where x is in Rn and A is a linear map on Rn. Themethod is the variation of constants, whoseusual explanation of replacing constants withfunctions in the solution of the homogeneousequation always left me mystified. But considerArnold’s expla-nation: the solution of thehomogeneous equation

t

x

d

d= Ax, x(t0)=c,

is etAc. Thus acting on Rn at time t by the linearautomorphism e–tA moves this solution of thehomogeneous equation to the constant curvec, i.e. to the solution of

t

x

d

d= 0, x(t0)=c.

As this is a linear change of coordinates, thederivative of this automorphism is again e–

tA. Under it the nonhomogeneous equationbecomes

t

x

d

d= e–tAh(t), x(t0)=c,

whose solution is clearly x(t)=c ∫t

t0e–τAh(τ)dτ.

Returning by the inverse of this automor-phism,

i.e. by etA, yields the solution of the original

equation i.e. e tA c+ ∫t

t0e(t–τ)A h(τ)dτ !! Can one

praise enough this masterly book by a mastermathematician? I cannot. I learnt the subjectreading this book and I prescribe it even toengineering students (for courses on control).This is the definitive introductory book onordinary differential equations; there is noneed to read (or write) another.

Shiva Shankar, Dept. of Electrical Engineering,

Indian Institute of Technology, Mumbai 400076,

India.

ANNOUNCEMENT

An instructional conference on Number Theory will be held at the University of Hyderabad during the

period 1–21 December, 1999. The conference is sponsored by the National Board of Higher Mathematics (NBHM).

The conference is meant for research scholars and college teachers interested in Algebra/Number Theory. The

conference meets the requirements of a course for career advancement. TA/DA will be paid for most participants

though research scholars who have fellowships are encouraged to use their contingency grants. Boarding and

lodging expenses will be met for all selected participants by the NBHM.

The course will cover basic algebraic number theory and some applications. An attempt will be made

to show how Fermat’s Last Theorem motivated the development of the subject.

Applicants may apply on a plain sheet of paper giving the following information:

1) Name, 2) Age, 3) Address, 4) Sex, 5) Educational qualifications, 6) Present position, 7) Institution currently

studying in or serving, 8) Two letters of recommendation, 9) whether accommodation is required (the university

campus is about 15 km outside the city of Hyderabad), 10) Whether the applicant can bear his/her travel expenses,

11) Other information that may be useful to the organising committee. Applications must be sent to: Coordinator

NBHM conference on Number Theory, Department of Mathematics and Statistics, University of Hyderabad,

Hyderabad 500 046. The last date for submitting applications is 1–11–1999.