Selected Topic s in the Geometrica l Study o f Differentia l Equations In fron t o f the Founder s Librar y From lef t t o right: crouching : Gloria Mari-Beffa , Pete r Olver , Thierry Robart , Nik y Kamran; stand - ing (fron t row) : Pete r Hydon , Barbar a Shipman , Irin a Kogan , Evelyn e Hubert , Mik e Lacorte , Mohammad Mahmood , Da n Grossman , Pau l Kainen , Elizabet h Mansfield , Sylvi e Desjardins , Maya Chhetri , Joshu a Leslie , Debr a Lewis , Gabrie l Ayine , Zhoud e Shao , Jin g Pin g Wang , Mingxiang Chen , Ami r Maleki ; standin g (bac k row): Stephen Shipman , Mar k Fels , Rafael Heredero , Enrique Reyes , Stanle y Einstein-Matthews , Andr e Wehner , Vladimi r Itskov , Pete r Clarkson , Jeffrey Morton , Austi n Roche , Pete r va n de r Kamp , Al i Ayari, Rober t Milson , Matthe w Biesecker , Clement Lutterodt . http://dx.doi.org/10.1090/cbms/096 Conference Boar d o f the Mathematical Science s CBMS Regional Conference Series in Mathematics Number 9 6 Selected Topic s in the Geometrica l Study o f Differentia l Equations Niky Kamran Published fo r th e Conference Boar d of the Mathematica l Science s by the American Mathematica l Societ y * fkL'' ^ Providence, Rhod e Islan d with support fro m th e National Science Foundatio n CBMS Conferenc e o n the Geometrica l Stud y o f Differentia l Equation s hel d a t Howar d University, Washington , D.C . June 20-25 , 200 0 Partially supporte d b y th e National Scienc e Foundatio n 2000 Mathematics Subject Classification. Primar y 58A20 , 58J70, 58J72 , 58J10 , 58E30 , 58E40, 35L65 , 35L60. Library o f Congres s Cataloging-in-Publicatio n Dat a Kamran, Niky , 1959 - Selected topic s i n the geometrica l stud y o f differentia l equation s / Nik y Kamran . p. cm . — (Conferenc e Boar d o f th e Mathematica l Science s regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 96 ) Includes bibliographica l references . ISBN 0-8218-2639- 5 (acid-fre e paper ) 1. Differentia l equations . 2 . Geometry , Differential . I . Title. II . Regiona l conferenc e serie s in mathematic s ; no. 96 . QA1 .R3 3 no. 9 6 [QA372] 510 s-dc2 1 [515'.35] 2002066482 Copying an d reprinting . 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The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 7 06 0 5 04 03 0 2 Pour Sepi, Adrien et Nicolas. This page intentionally left blank Contents Preface i x Acknowledgments xii i Chapter 1 . Differentia l equation s an d thei r geometr y 1 1. Th e Cauch y proble m fo r first-orde r partia l differentia l equation s 1 2. Hyperboli c equation s integrabl e b y the metho d o f Darboux 3 3. External , interna l an d generalize d symmetrie s 5 4. Th e invers e proble m o f the calculu s o f variations 6 5. Som e important topic s no t covere d i n these lecture s 7 Chapter 2 . Externa l an d generalize d symmetrie s 9 1. Je t bundle s 9 2. System s o f differential equation s 1 2 3. Externa l symmetrie s 1 3 4. Classica l symmetr y reductio n 1 5 5. Contac t transformation s an d Backlund' s Theore m 1 7 6. Generalize d symmetrie s o f differential equation s 1 9 7. Generalize d symmetrie s an d conservatio n law s 2 2 Chapter 3 . Internal , externa l an d generalize d symmetrie s 2 7 1. Interna l symmetrie s 2 7 2. Norma l system s o f ordinary differentia l equation s 2 8 3. Under-determine d system s o f ordinary differentia l equation s 2 9 4. Contac t condition s fo r ordinar y differentia l equation s 3 2 5. Contac t condition s fo r partia l differentia l equation s 3 5 Chapter 4 . Transformation s o f surfaces 3 9 1. Th e metho d o f Laplace 3 9 2. Th e Laplac e transformation fo r surface s 4 2 3. A n applicatio n t o the sf e Tod a fiel d theor y 4 4 Chapter 5 . Transformation s o f submanifolds 4 7 1. A multi-dimensional geometri c Laplac e transformation 4 7 2. Transformation s o f systems o f partial differentia l equation s 5 1 Chapter 6 . Hamiltonia n system s o f conservation law s 5 7 1. System s o f conservation law s and thei r loca l geometry 5 7 2. Strongl y hyperboli c system s ric h i n conservation law s 6 0 3. Th e Laplac e transformation fo r strongl y hyperboli c system s 6 3 Chapter 7 . Th e variationa l bi-comple x 6 7 viii CONTENT S 1. Je t bundle s o f infinite orde r 6 7 2. Th e tautologica l variationa l bi-comple x o n J°°{E) 6 9 Chapter 8 . Th e invers e proble m o f the calculu s o f variations 7 5 1. Th e local , globa l an d equivarian t invers e problems 7 5 2. Edg e complexe s an d obstruction s 7 8 3. Variationa l principle s an d symmetr y reductio n 8 1 Chapter 9 . Conservatio n law s and Darbou x integrabilit y 8 3 1. Scala r hyperboli c partia l differentia l equation s i n the plan e 8 3 2. Th e constraine d variationa l bi-comple x 8 5 3. Conservatio n law s 8 6 4. Th e generalize d Laplac e invariant s 8 9 5. Vanishin g theorem s fo r conservatio n law s 9 2 6. Darbou x integrabilit y 9 3 7. Furthe r application s 9 7 Chapter 10 . Characteristi c cohomolog y o f differentia l system s 9 9 1. Exterio r differentia l system s 9 9 2. Som e existence theorem s fo r integra l manifold s 10 1 3. Involutiv e system s an d th e Cartan-Kahle r theore m 10 3 4. Characteristi c cohomolog y 10 8 Bibliography 111 Preface The geometrica l stud y o f differentia l equation s ha s a lon g an d distinguishe d history, goin g bac k t o th e classica l investigation s o f Sophu s Lie , Gasto n Darbou x and Eli e Cartan. Thei r idea s ar e at th e sourc e o f a number o f developments whic h currently occupy a central position in several areas of pure and applied mathematics, including th e theor y o f completel y integrabl e evolutio n equations , th e calculu s o f variations an d the stud y o f conservation laws . Our objective i n these lectures i s to giv e an overview o f a number o f significan t ideas and result s that hav e been developed ove r the past decad e i n the geometrica l study o f differential equations . I t i s of course impossible in the course often lecture s to cove r al l the importan t advance s that hav e taken plac e i n suc h a broad field o f research. Thi s survey is therefore far from complete, and it does not succeed in doing full justic e t o al l th e idea s tha t i t aim s t o convey . W e hav e chose n t o focu s ou r attention on a number o f topics which we have found to be of particular significance , or i n whic h w e have bee n involve d throug h ou r ow n research .