Vll l Preface all plana r close d curve s o f fixed length , enclose s th e maximu m area . But sh e ha d th e choic e o f territory wit h a flat coastlin e an d cleverl y chose a semi-circula r region , wit h th e cord' s endpoint s o n th e shore - line. Thi s gives more area and i s actually the mathematically optima l solution. A change o r variation o f the shap e o f the cor d canno t giv e a ne w regio n o f greater area . Calculus of Variations arise s when one differentiates, i n the sens e of th e calculu s o f Newto n an d Leibniz , a one-paramete r famil y o f such variations. Thi s firs t occur s i n the work s by P.L.M . d e Mauper - tuis (1698-1759) , G.W . Leibni z (1646-1716) , Jako b Bernoull i (1654 - 1705), Johan n Bernoull i (1667-1748) , L . Eule r (1707-1783) , an d J.L . Lagrange (1736-1813) . I t ha s historicall y largel y bee n th e stud y o f optimal paths , for exampl e as a geodesic curve in a space or as a pat h of least action in space-time. Se e the nice presentation i n Chapter 1 of The Parsimonious Universe: Shape and Form in the Natural World by S . Hildebrandt an d A . Tromb a (Copernicus , Ne w York , 1996) . In modern language , th e birth o f the calculus of variations occur s in th e transitio n fro m th e stud y o f a critica l poin t o f a functio n o n a lin e (a s i n calculus ) t o tha t o f a critica l curv e o r critica l surfac e for a functional , suc h a s lengt h o r area , o n a n infinite-dimensiona l space o f suc h objects . A s discusse d i n Chapte r 1 b y Fran k Jones , the conditio n o f criticalit y fo r thes e object s lead s t o th e importan t partial differentia l equation s o f Euler an d Lagrange . Variou s physica l problems also give rise to natural condition s constraining th e space of admissible objects . On e such constraint involve s a fixed boundary, a s with a classica l vibratin g strin g o r a soa p film spannin g a wire . An - other constrain t i s seen i n Quee n Dido' s problem . He r proble m ma y be equivalently reformulate d a s the isoperimetric problem of finding a curve of minimum lengt h enclosin g a given fixed area. Th e analogou s two-dimensional isoperimetri c proble m o f finding a surfac e o f leas t area enclosin g a give n volum e (o r volumes ) occur s i n soa p bubbl e models. In Chapte r 2 b y Robi n Forman , on e consider s th e connectio n between suc h critica l o r equilibriu m point s an d th e topolog y o r ge - ometry o f th e ambien t spaces . Her e i s a quic k elegan t introductio n to th e simple , bu t subtle , idea s o f Marsto n Mors e fro m th e 1940s .
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