Other Titles in This Series

4 V . V. Prasolov, Intuitiv e topology, 199 5 3 L . E. Sadovskii and A. L. Sadovskii, Mathematic s and sports , 199 3 2 Yu . A. Shashkin, Fixe d points , 199 1 1 V . M. Tikhomirov, Storie s about maxim a and minima , 199 0

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Mathematical World • Volum e 1

Stories abou t Maxima

and Minim a V. M. Tikhomiro v

Translated from th e Russian by

Abe Shenitzer

http://dx.doi.org/10.1090/mawrld/001

B. M . TMXOMMPO B

PACCKA3M O MAKCHMYMA X M MHHHMYMA X

«HAYKA», MOCKBA , 198 6

Translated fro m th e Russia n b y Ab e Shenitze r

1991 Mathematics Subject Classification. Primar y 00A07 , 00A30, 00A35 , 01-01 , 46-01 , 49-01 , 49-03 , 49J9 9

Library o f Congres s Cataloging-in-Publicatio n Dat a

Tikhomirov, Vladimi r M . (Vladimi r Mikhallovich) , 1934 — Stories abou t maxim a an d minima/V . M . Tikhomirov . p. cm.—(Mathematica l world , ISS N 1055-9426 ; 1 ) ISBN 0-8218-0165- 1 1. Maxima an d minima . 2 . Calculu s o f variations . 3 . Mathematica l optimization .

QA306T55 199 0 90-2124 6 51T.66—dc20 CI P

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To the Memor y of My Dear Friend ,

V. M. Alekseev

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Table o f Content s

Introduction i x

Part One . Ancien t Maximu m an d Minimum Problem s

The first story Wh y D o W e Solv e Maximu m an d Minimu m

Problems? 3

The second story Th e Oldes t Problem—Dido' s Proble m . . . . 9

The third story Maxim a an d Minim a i n Nature (Optics ) . . . 1 9

The fourth story Maxim a an d Minim a i n Geometry 2 7

The fifth story Maxim a and Minima in Algebra and in Analysis 3 7

The sixth story Kepler' s Proble m 4 7

The seventh story Th e Brachistochron e 5 5

The eighth story Newton' s Aerodynamica l Proble m 6 5

Part Two. Method s o f Solution o f Extrema l Problem s

The ninth story Wha t i s a Function? 8 1

The tenth story Wha t i s an Extrema l Problem ? 9 3

The eleventh story Extrem a o f Function s o f On e Variabl e . . . . 9 9

v i i

Vl l l CONTENTS

The twelfth story Extrem a o f Function s o f Man y Variables .

The Lagrang e Principl e 10 9

The thirteenth story Mor e Proble m Solvin g 11 9

The fourteenth story Wha t Happened Later in the Theory of Extremal

Problems? 14 3

The last story Mor e Accurately , a Discussion 17 9

Bibliography 187

Introduction

In daily life i t is constantly necessar y to choose the best possible (optimal ) solution. A tremendous numbe r o f such problem s arise in economics and i n technology. I n suc h cases i t i s frequently usefu l t o resor t t o mathematics .

In mathematics , th e stud y o f maximu m an d minimu m problem s bega n a ver y lon g tim e ago , i n fact , twenty-fiv e centurie s ago . Fo r a lon g tim e there wer e n o unifor m way s o f tacklin g problem s fo r finding extrema . Th e first general methods of investigation and solution of extremal problems were created abou t 30 0 year s ago , a t th e tim e o f th e formatio n o f mathematica l analysis.

Then i t becam e clea r tha t certai n specia l optimizatio n problem s pla y a crucial rol e in the natural sciences . Specifically , i t was found tha t man y laws of natur e ca n b e derived fro m so-calle d "variationa l principles. " Accordin g to thes e principles , give n an y collectio n o f admissibl e motions , wha t distin -guishes th e actua l motio n o f a mechanica l system , o r o f light , electricity , a fluid, a gas, and s o on, i s that i t maximize s o r minimizes certai n quantities . Some concret e extrema l problems , whos e conten t derive s fro m th e natura l sciences (the brachistochrone problem , Newton's problem, and others) , were posed a t th e en d o f th e seventeent h century . Th e nee d t o solv e these , a s well as many other problems of geometry, mechanics , and physics , led to the creation o f a new branch o f mathematica l analysi s that came to be known as the calculus of variations .

The intensiv e developmen t o f th e calculu s o f variation s continue d fo r about tw o centuries . Man y o f th e finest scientist s o f th e eighteent h an d nineteenth centurie s too k par t i n thi s process , and , b y th e beginnin g o f thi s century, i t seeme d a s if they ha d exhauste d th e topic.

But i t turne d ou t tha t thi s wa s no t th e case . Th e need s o f practica l life , especially in economics and technology, gave rise to new problems that could not b e solve d b y th e ol d methods . On e ha d t o advance . I t wa s necessar y to creat e a ne w field of mathematica l analysis , know n a s "convex analysis, " involving the stud y o f convex function s an d conve x extrema l problems .

ix

X INTRODUCTION

The need s o f technology , an d i n particula r th e exploratio n o f space , gave rise t o ye t anothe r serie s o f problem s tha t wer e likewis e unsolvabl e b y th e methods o f th e calculu s o f variations . Thus , anothe r ne w theory, know n a s optimal contro l theory , wa s created . Th e fundamenta l metho d o f optima l control theory was worked out i n the 1950 s and 1960 s by Soviet mathemati -cians, namely L . S . Pontryagin an d hi s colleagues. Thi s provided a new and powerful impuls e fo r furthe r investigation s i n th e theor y o f extrema l prob -lems.

This book aims to acquaint the reader with this whole circle of ideas. How -ever, this is not the author's only purpose. Throughou t th e history of mathe-matics, maximum an d minimu m problem s have played an importan t rol e in its evolution. Durin g thi s time man y beautiful , important , brilliant , an d in -teresting problem s i n geometry , algebra , physics , an d s o on, hav e appeared . The greates t scientist s o f th e past—Euclid , Archimedes , Heron , Tartaglia , Johann and Jakob Bernoulli , Newton, and many others—took par t i n the so-lution of these concrete problems. Th e solutions stimulated the development of the theory and , as a result, techniques were elaborated tha t made possibl e the solution o f a tremendous variety o f problem s by a single method .

The author would like the reader to understand how and why a mathemat-ical theory i s born. I n Par t One , th e reade r wil l ge t t o kno w man y concret e problems, and i n the course of the discussion o f their solution s he will come in contact wit h the creative work o f som e of the best mathematician s o f th e past. Thi s i s not onl y o f historica l interest . Fo r the mos t part , the ideas and methods created by eminent mathematician s i n connection with the solution of problem s d o no t di e an d ar e certai n t o b e reborn , give n enoug h time . That i s why to fatho m th e conception s o f grea t me n i s always a n enrichin g experience.

The nee d t o solv e a large numbe r o f varied problem s establishe s th e pre -conditions fo r th e creation o f a general theory . I n Par t Tw o I will introduc e a method fo r solvin g maximum and minimum problems that originated with Lagrange. Th e basi c conceptio n o f thi s metho d ha s endure d fo r ove r tw o centuries. It s content has varied constantly , but it s key thought has remained unchanged. I t i s not a simple matte r t o understand th e reasons fo r thi s uni -versality o f Lagrange' s idea . O n th e othe r hand , i t i s no t a t al l difficul t t o learn t o use Lagrange's principle fo r th e solution o f problems. A t the end of Part Tw o al l problem s discusse d i n Par t One , problem s marke d b y th e dis -similarity o f their solutions , are investigated an d solve d by means of a single general method , i n a standard way , using one and th e same scheme.

The autho r ha s trie d t o sho w ho w th e analysi s o f divers e fact s give s ris e to a genera l idea , ho w thi s ide a i s transformed , ho w i t i s enriched b y ne w content, an d ho w i t remain s th e same unde r al l changes .

With th e exceptio n o f th e concludin g par t o f th e fourteent h story , thi s book i s primaril y aime d a t hig h schoo l students . Bu t I woul d ver y muc h

INTRODUCTION XI

like its readers to include college students intereste d i n mathematics and , of course, teachers . Th e last stor y i s addressed abov e al l to them. I t impinge s on the question of how and why to teach. I think that the content of the book supplies materia l tha t i s ideally suite d fo r a discussion o f this topic , a topic that i s bound t o concern u s for many year s to come. Thus , I hope tha t thi s book wil l also be read by my colleagues who study mathematics and teach it to their students .

I wish to thank al l those who read the manuscript an d commented o n it. This refers , abov e all , to Andrei Nikolaevi c Kolmogorov , Nikola i Borisovi c Vasil'ev, Ivan Penkov , and Georgii Georgevi c Magaril-Il'yaev .

I am grateful t o Prof. E . Barbeau fo r a number o f valuable remark s tha t have bee n include d i n th e Englis h translatio n o f m y book. I als o wis h t o express my deep appreciation to Prof. A . Shenitzer for his work as translator.

V. M. Tikhomirov

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Cambridge, 1952 . 8. J . Kepler , Nova stereometria doliorum vinariorum. In : Johanne s Keple r Gesammelt e

Werke, Munich , 1937 . 9. A . Koestler , The sleepwalkers, Macmillan , Ne w York , 1968 .

10. The mathematical papers of Isaac Newton, edite d b y D. T . Whiteside , Cambridg e Univ . Press, Cambridge , 1967 .

11. I . Niven, Maxima and minima without calculus, Dolcian i Mathematica l Exposition s No . 6, 1981 .

12. H . Rademache r an d O. Toeplitz, The enjoyment of mathematics, Princeto n Univ . Press , Princeton, 1966 .

13. I . M . Yaglo m an d V . G . Boltyanskii , Convex figures. Holt , Ne w York , 1961 . 14. H . Zeuthen, Geschichte der Mathematik imAltertum und Mittelalter, Copenhagen , 1896 . 1R. B . M . AjieKceeB , B . M . THXOMHPOB , C . B . OOMHH , OnTHMajibHo e ynpaBJiemie.—M. :

HayKa, 1979 . 2R. B . M . AjieKcee B , 3 . M . TajieeB , B . M . THXOMHPOB , C6OPHH K 3a^a n n o onTHMH3auHH. —

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