Selected Title s i n Thi s Serie s

204 A . M . Vinogradov , Cohomologica l analysi s o f partial differentia l equation s an d Secondary Calculus , 200 1 203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d coding , 200 1 202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r nonlinea r PDE . I , 2001 201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1 200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1 199 Shigeyuk i Morita , Geometr y o f characteristic classes , 200 1 198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology , 200 1 197 Kenj i Ueno , Algebrai c geometr y 2 : Sheaves an d , 200 1 196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes , 200 1 195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1 194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1 193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1 192 Yu . P . Solovyo v an d E . V . Troitsky , C*-algebra s an d ellipti c operator s i n differentia l topology, 200 1 191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f information geometry , 200 0 190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces , 200 0 189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e phenomena , 200 0 188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f random variable s and rando m processes , 200 0 187 A . V . Fursikov , Optima l contro l o f distributed systems . Theor y an d applications , 200 0 186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r theor y 1 : Fermat's dream , 200 0 185 Kenj i Ueno , Algebrai c Geometr y 1 : From algebrai c varietie s to schemes , 199 9 184 A . V . Mel'nikov , Financia l markets , 199 9 183 Hajim e Sato , Algebrai c topology : a n intuitiv e approach , 199 9 182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d conservatio n laws fo r differentia l equation s o f mathematical physics , 199 9 181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . Par t 2 , 199 9 180 A . A . Milyuti n an d N . P . Osmolovskii , Calculu s o f variations an d optima l control , 1998 179 V . E . Voskresenskii , Algebrai c group s an d thei r birationa l invariants , 199 8 178 Mitsu o Morimoto , Analyti c functional s o n th e sphere , 199 8 177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8 176 L . M . Lerraa n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e Hamiltonia n systems wit h simpl e singula r point s (topologica l aspects) , 199 8 175 S . K . Godunov , Moder n aspect s o f linear algebra , 199 8 174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d ellipti c systems, 199 8 173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l propertie s o f distributions o f stochastic functionals , 199 8 172 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . Part 1 , 199 8 171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8 170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals , 199 7

For a complet e lis t o f titles i n thi s series , visi t th e AMS Bookstor e a t www.ams.org/bookstore/ . This page intentionally left blank 10.1090/mmono/204

Translations o f MATHEMATICAL MONOGRAPHS

Volume 20 4

Cohomological Analysis o f Partial Differential Equation s and Secondar y Calculus

A. M. Vinogradov

American Mathematica l Societ y I? Providence , Rhod e Islan d EDITORIAL COMMITTE E AMS Subcommitte e Robert D . MacPherso n Grigorii A . Marguli s James D . Stashef f (Chair ) ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Preidlin (Chair ) A. M . BtraorpaAO B KoroMOJiorH^iecKHH aHajiH 3 ^HCp^epeHixpiajibHBi x ypaBHenH H B MaCTHBI X npOH3BO^HBI X H BTOpHMHO e HC^THCJieHU e

Translated fro m th e origina l Russia n manuscrip t b y Josep h Krasil'shchi k

2000 Mathematics Subject Classification. Primar y 35A30 , 37K10 ; Secondary 37Jxx , 58J10 .

ABSTRACT. Thi s boo k deal s wit h th e principle s o f a ne w theory , whic h play s th e sam e rol e i n general (nonlinear ) system s o f partia l differentia l equation s a s algebrai c geometr y doe s i n sys - tems o f algebrai c equations . Startin g wit h th e fundamental s o f th e classica l geometr y o f partia l differential equation s i n th e spiri t o f S . Lie , rewritte n i n term s o f moder n theor y o f jet spaces , and incorporatin g som e important impulse s fro m th e theor y o f integrable systems , the expositio n eventually arrive s at first element s o f Secondary Calculus . A systematic us e o f the coordinate-fre e language o f differentia l calculu s ove r commutativ e algebra s allow s u s t o revea l basi c underlyin g structures o f the develope d theor y an d i s a characteristic featur e o f the presente d approach . The boo k i s intende d fo r mathematician s (pur e an d applied) , specialist s i n mathematica l an d theoretical physics , mechanic s o f continuou s media , etc. , intereste d i n moder n approache s t o partial differentia l equations . Th e tex t i s als o comprehensibl e t o advance d graduat e student s i n these fields .

Library o f Congres s Cataloging-in-Publieatio n Dat a Vinogradov, A . M . (Aleksand r Mikhailovich ) Cohomological analysi s o f partia l differentia l equation s an d secondar y calculu s / A.M . Vino - gradov. p. cm . — (Translation s o f mathematical monographs , ISS N 0065-928 2 ; v. 204 ) Translated fro m th e origina l Russia n manuscript . Includes bibliographica l reference s an d index . ISBN 0-8218-2922- X (alk . paper ) 1. Differentia l equations , Nonlinear . 2 . Geometry , Differential . 3 . Homolog y theory . I . Title . II. Series .

QA377.V54 200 1 515'.353—dc21 200104608 7

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying, or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed to the Assistant t o the Publisher, America n Mathematical Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionQams.org. © 200 1 b y th e America n Mathematica l Society . Al l right s reserved . 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 this 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 6 0 5 04 0 3 02 0 1 Contents

Foreword vii

Introduction i x

Chapter 0 . Fro m Symmetrie s o f Partial Differentia l Equation s to Secondar y Calculu s 1 0.1. Wha t ar e symmetries o f partial differentia l equations , and wha t ar e partial differentia l equation s themselves ? 2 0.2. Jet s 3 0.3. Higher-orde r contac t structure s 5 0.4. Differentia l equation s ar e diffietie s 6 0.5. Wha t ar e symmetries o f partial differentia l equations ? 8 0.6. Infinitesima l symmetrie s o f partial differentia l equation s are secondar y quantized vecto r fields 1 0 0.7. Digression : o n symmetries o f partial differentia l equation s 1 2 0.8. Secondar y ("quantized" ) function s 1 4 0.9. Higher-orde r scala r secondar y ("quantized" ) operator s 1 6 0.10. Secondar y ("quantized" ) differentia l form s an d C-spectra l s 1 9 0.11. Ho w does the C-spectra l sequenc e work ? 2 1

Chapter 1 . Element s o f Differential Calculu s in Commutative Algebra s 2 5 1.1. Genera l preliminarie s 2 5 1.2. Adjoin t operator s 3 1 1.3. Spence r complexe s an d the Gree n formul a 3 8 1.4. Quadrati c Lagrangian s an d the Eule r operator 4 4 1.5. Conservatio n law s in the linea r theor y 4 7 1.6. Automorphism s an d th e linea r Noethe r theore m 5 1 1.7. Concludin g remark s 5 5

Chapter 2 . Geometr y o f Finite-Order Contac t Structure s and th e Classica l Theor y o f Symmetrie s of Partial Differentia l Equation s 5 7 2.1. Necessar y fact s fro m th e geometr y o f jet space s 5 7 2.2. Th e structure o f ^/-transformation s 6 6 2.3. Infinitesima l automorphism s o f the Carta n distributio n 7 2 2.4. Th e structure o f automorphisms o f the Carta n distributio n on the manifold s J k(E,m), k < o o 8 0 2.5. Classica l theory o f symmetries o f partial differentia l equation s 8 6 2.6. Concludin g remark s 9 3 vi CONTENT S

Chapter 3 . Geometr y o f Infinitely Prolonge d Differentia l Equations an d Highe r Symmetrie s 9 5 3.1. Geometr y o f infinitely prolonge d equations , and relate d differentia l calculu s 9 5 3.2. Horizontalizatio n operation , th e structur e o f Cartan submodule s and o f C-differential operator s 10 0 3.3. Highe r infinitesima l symmetrie s o f differential equation s 10 7 3.4. Th e structure o f C-transformations 11 3 3.5. Example s o f diffieties 12 2 3.6. Concludin g remark s 12 5 Chapter 4 . C-spectra l Sequenc e an d Som e Applications 12 7 4.1. Spence r complexe s an d the Gree n formul a i n the C-theor y 12 7 4.2. Nonlinea r Lagrangia n formalis m 13 3 4.3. Th e C-spectral sequenc e 13 7 4.4. Th e C-spectral sequenc e o f infinitely prolonge d equation s 15 5 4.5. Application s to the Lagrangia n formalis m wit h constraint s and to the theory o f conservation law s 17 2 4.6. Characteristi c classe s 18 2 4.7. Concludin g remark s 18 7 Chapter 5 . Introductio n t o Secondar y Calculu s 18 9 5.1. Firs t ide a o f Secondary Calculu s 18 9 5.2. Secondar y linea r algebr a 19 6 5.3. Secondar y s 20 3 5.4. Secondar y linea r differentia l operator s 21 0 5.5. Cy-spectra l sequenc e 21 3 5.6. Secondar y multivector-value d differentia l form s 22 1 5.7. Th e secondar y Frolicher-Nijenhui s formalis m 22 7 5.8. Concludin g remark s 23 3

Bibliography 237 Foreword

This book originates from a text I wrote at the beginning o f the 1980s , when the existence of Secondary Calculus became evident to me. Th e subsequent exploratio n of a possibilit y t o publis h a polishe d versio n o f thi s tex t a s a boo k i n my Sovie t homeland showe d that i t would be hardly possible in a reasonably near future. Thi s is why th e tex t wa s divide d int o severa l part s publishe d a s separate papers . Th e fact that the y ar e parts o f one construction wa s lost to a greater extent durin g thi s operation. After m y mov e t o Ital y i n 1990 , I wa s strongl y an d permanentl y presse d b y Natasha, m y wife , t o publis h th e ol d manuscrip t i n English . Curren t researc h prevented i t fo r a lon g time , til l th e momen t whe n Josep h Krasil'shehi k becam e her all y an d propose d t o translate the text int o English . After th e translation wa s finished an d I wrote th e necessar y ne w chapte r containin g recen t development s i n the field, it has become rather clea r that th e whole text mus t be reelaborated. Thi s is finally done , an d I a m ver y gratefu l t o Josep h fo r hi s patienc e durin g thi s no t very pleasant proces s o f repeated modifications whic h required his additional effort s to kee p the text smooth . My pleasant obligation is to acknowledge that this work was partially supporte d by the Italian Ministero dell 'Universitd e della Ricerca Scientifica e Tecnologica and by Istituto Nazionale di Fisica Nucleare. Bot h th e autho r an d th e translato r wer e also supported b y the INTA S grant 96-0793 . This page intentionally left blank Introduction

1. Thi s book deals with principles o f a theory playing the same role for genera l systems o f (nonlinear ) partia l differentia l equation s a s algebraic geometr y doe s fo r algebraic equations . Th e object s studie d i n algebrai c geometr y ar e varietie s o f al l solutions t o a give n syste m o f algebrai c equations . Th e history , fro m Descarte s to Grothendieck' s schemes , o f what shoul d b e calle d th e "variet y o f al l solutions " shows that i t i s not a ver y simpl e problem . I t become s eve n mor e complicate d i f one tries to answe r the question ho w the "variet y o f all solutions to a given syste m of partial differentia l equations " shoul d b e formalize d properly . A specifi c featur e of thi s proble m i s t o understan d ho w t o match , i n a reasonabl e way , solution s with differen t domains . Thi s aspec t i s o f crucia l importanc e i n physics , wher e a possibility t o localiz e th e theor y mus t b e guaranteed . I t i s quite clea r tha t thes e requirements canno t b e satisfie d b y construction s buil t wit h standar d mean s o f functional analysis , topology , etc . On e o f th e goal s o f thi s boo k i s t o introduc e the concep t o f a diffiety (differential variety), a ne w mathematica l object , whic h plays th e sam e rol e i n th e theor y o f (nonlinear ) partia l differentia l equation s a s affine algebraic varieties d o i n th e theor y o f algebrai c equations . Rathe r length y preparations ar e neede d t o organiz e properl y preliminar y materia l comin g fro m various areas , but afte r tha t th e results prove d brin g inevitabl y t o this notion . Diffleties are , generally infinite-dimensional, manifold s carrying a structure that may b e calle d th e infinite-order contact structure, an d ar e locall y equivalen t t o infinite prolongation s o f differentia l equations . Du e t o thi s structure , a ver y spe - cial kin d o f differentia l calculus , calle d Secondary Calculus, ca n b e develope d o n a diffiety . Variou s natura l characteristic s o f a diffiet y and , consequently , o f th e corresponding system s o f partia l differentia l equations , ar e expresse d i n term s o f Secondary Calculu s and vice versa. Secondar y Calculu s i s distinguished b y its com- pletely cohomologica l nature . I n othe r words , i t operate s wit h quantitie s tha t ar e certain cohomolog y classe s o f differentia l complexe s togethe r wit h th e operator s connecting them . Fro m th e poin t o f vie w o f categor y theory , th e object s i n Sec - ondary Calculu s are homotopy types o f special differentia l complexe s over diffletie s while the morphisms are suitable homotopy classes of differential cochai n mappings. From th e technica l poin t o f view , the principa l ingredient s o f Secondar y Calculu s are a rather specifi c mixtur e o f commutative an d homologica l algebr a wit h differ - ential geometry . A t firs t glance , i t look s rather surprisin g tha t a natural languag e one needs to dea l with system s o f nonlinear partia l differentia l equation s i s o f thi s kind. However , no w we can be sure that quantu m physic s and, abov e all, quantu m field theor y an d it s generalization s fin d thei r natura l mathematica l backgroun d i n Secondary Calculus .

IX x INTRODUCTIO N

We us e th e adjectiv e "secondary " t o emphasiz e tha t th e Secondar y Calculu s is a natura l languag e programme d i n term s o f th e classical , i.e. , primar y Calcu - lus. Havin g i n min d it s relationship wit h quantu m physics , i t coul d b e als o calle d "Quantized", o r "Quantum" , Calculus . Thes e words , however , wer e to o abuse d recently t o b e mentione d onc e agai n i n a completel y differen t context . Neverthe - less, w e shal l d o i t just onc e to stat e tha t th e mai n goa l o f thi s boo k i s to buil d a provisiona l bridg e betwee n classica l an d quantu m mathematics . Th e autho r rec - ognizes al l deficiencie s o f the presente d constructio n bu t hope s that i t ha s a t leas t one merit: t o serv e as a mean fo r reasonabl y courageou s adventures i n crossing the rifts.

2. Thi s boo k consist s o f th e mai n tex t precede d b y introductor y Chapte r 0 . The basi c tex t i s written i n a standard mathematica l style , whic h leave s n o roo m for motivations and informal arguments . Bu t this is hardly justified i n the situatio n the boo k deal s with an d a compensation i s given i n the introductory chapter . We d o no t touc h tw o importan t aspect s o f Secondar y Calculu s i n th e book . One o f them concern s nonlocal quantities, i.e. , the quantities that depen d no t onl y on original unknown function s an d their arbitrary derivatives , but als o on variable s obtained b y applyin g inversion s o f differentia l operation s (fo r instance , integrals) . An introduction t o this topic i s given in the survey article by Krasil'shehik an d th e author Nonlocal trends in the geometry of differential equations [74 ] (se e also [19]) . The centra l concep t o f th e nonloca l theor y i s that o f a covering i n th e categor y of differentia l equations . Th e notio n wa s introduce d b y th e autho r i n [122 ] an d is discussed quit e briefl y i n Chapte r 5 o f this book . Anothe r aspec t i s the grade d (or "super" ) versio n o f Secondary Calculus . Th e firs t result s i n this directio n wer e obtained b y Krasil'shchi k an d ar e presented i n hi s joint boo k with Kerste n [70] . The theor y presente d i n this boo k ha s a wide range o f applications. Th e sim - plest o f them ar e relate d t o computation s an d th e subsequen t us e o f highe r sym - metries and conservatio n law s o f a given system o f partial differentia l equations . A relatively shor t introductio n t o thi s subjec t ca n b e foun d i n [127] , whil e th e col - lection [139 ] show s the correspondin g computationa l technique s "i n action" . Th e book Symmetries and Conservation Laws for Differential Equations of Mathemat- ical Physics [19 ] contain s a systemati c expositio n o f thes e matter s an d ma y b e regarded a s a collectio n o f examples illustratin g th e genera l theory expose d i n ou r book. I t is strongly recommended to the readers interested in seeing how this theory is used i n actual applications . It shoul d b e stresse d fro m th e ver y beginnin g tha t bot h differentia l operator s and al l natura l construction s relatin g the m ar e treate d i n th e mai n tex t i n a n algebraic coordinate-fre e manner . Thi s simplifie s computation s considerabl y an d makes i t possibl e t o singl e ou t som e fundamenta l structure s o f the theory , whic h could hardl y b e observe d i n th e framewor k o f th e standar d approach . Althoug h adaptation t o thi s nontraditiona l approac h wil l probabl y requir e som e additiona l efforts fro m th e readers , they wil l be rewarde d fo r thei r wor k eventually . Our expositio n i s self-containe d i n th e sens e tha t al l preliminar y fact s an d constructions ar e introduced . Furthe r detail s concernin g th e preliminar y materia l can b e foun d i n th e boo k [71] , while the boo k [19] , as alread y mentioned , ca n b e viewed a s a collectio n o f particula r example s illustratin g th e genera l theory . A n elementary introductio n t o differentia l calculu s ove r commutativ e algebra s ca n b e found i n [95] . INTRODUCTION x i

3. Th e text i s organized a s follows . Chapter 0 i s a n informa l introductio n t o th e basi c concept s o f th e moder n theory o f partial differentia l equations . Ou r mai n goa l there i s to sho w why thes e fundamentals ar e unavoidable fro m both mathematical and physical points o f view. In the firs t chapte r al l necessary prerequisites o f differential calculu s in commu- tative algebra s ar e presented . Furthe r expositio n i s based o n thi s language , whic h is absolutely indispensabl e t o revea l basi c structures o f the theory . Thi s i s exactl y the poin t a t whic h ou r approac h start s t o diverg e rapidl y fro m th e on e commonl y adopted i n mathematical physics . Fundamentals o f linear Lagrangian formalism ar e then developed using this lan- guage. Thi s allows us to discover a natural cohomological structure o f the formalis m itself an d som e relate d constructions . Amon g th e latter , conservatio n law s ar e t o be mentioned first . Th e related cohomolog y come s from som e multianalogs o f Diff - Spencer complexes . A versio n o f this kin d o f cohomolog y wil l pla y a n importan t role i n Chapters 4 and 5 in the contex t o f Secondary Calculus . Naturalness o f a purely algebrai c framewor k fo r Lagrangia n formalis m give n in this chapter i s well confirmed b y the fac t that i t i s generalized straightforwardl y t o any reasonabl y noncommutativ e situatio n (se e [116]) . By contrast , th e secon d chapte r i s o f purel y geometri c natur e an d deal s wit h finite-order je t space s an d genera l system s o f (nonlinear ) partia l differentia l equa - tions interpreted a s submanifolds i n these spaces. Expositio n i s concentrated ther e around variou s kind s o f automorphism s o f higher-orde r contac t structure s natu - rally interprete d a s symmetrie s o f partia l differentia l equations . I t i s show n tha t the symmetry grou p i s not enlarge d whe n the order o f jets, i.e. , o f involved deriva - tives, increases . Thi s mean s that th e classica l symmetr y theory , originate d b y Li e and develope d late r b y Backlun d an d others , ha s n o chanc e t o incorporat e ne w "experimental data " emergin g i n th e theor y o f integrabl e systems . A t th e sam e time, thi s compel s u s to pas s t o infinit e jets i n searc h fo r a suitabl e extensio n fo r the concep t o f a symmetry o f nonlinear partia l differentia l equations . This ste p i s rewarde d late r i n Chapte r 3 where , amon g othe r constructions , the theory o f higher symmetries fo r general systems o f partial differential equation s is developed . Thi s i s don e b y studyin g th e geometr y o f infinite-orde r je t space s and infinitel y prolonge d differentia l equations . Thes e ar e the simples t example s o f diffleties. I t turns out that infinite-orde r infinitesima l contac t transformation s for m a much larger Lie algebra than the classical ones. Th e group of infinite-order contac t transformations als o becomes larger, but t o much smaller exten t than infinitesima l transformations. Eve n more , i n som e situation s i t doe s no t chang e a t all . This , at firs t sigh t disappointing , discrepancy , whic h wa s mysterious fo r a long time, ha s a natura l explanation : infinite-orde r infinitesima l contac t transformation s are , b y their nature , cohomolog y classe s o f vecto r fields . Thes e cohomolog y classe s ar e exactly th e secondary vector fields a s explaine d i n Chapte r 5 . But , similarl y t o quantum particles , secondar y vecto r field s d o no t hav e trajectories . Thi s i s th e main philosophica l lesso n o f Chapter 3 . In thi s chapter , element s o f th e differentia l calculu s agre e wit h th e infinite - order contac t structur e (C-differentia l operators , etc. ) ar e als o elaborated. Th e ad - vantages o f the algebrai c setting fo r differentia l calculu s given in Chapter 1 become evident whe n w e pas s t o th e infinit e je t space . I t allow s u s t o avoi d difficultie s typical fo r infinite-dimensiona l situations , an d als o improper temptations . xii INTRODUCTIO N

Foundations o f the highe r symmetr y theory , a s presente d i n Chapte r 3 , wer e developed b y th e autho r i n 1976-7 7 (se e [121 , 137 , 124 , 142] ) an d appeare d later i n a numbe r o f publications (see , fo r instance , [54 , 96 , 146] ) presente d i n a coordinatewise way . Computationa l aspect s o f this theor y were , fo r th e firs t time , tested extensivel y i n the collectio n [139] . Th e boo k [19 ] contains a n update d an d detailed expositio n o f the basi c theor y togethe r wit h th e recen t ramification s an d applications. I t i s very impressiv e that th e first attemp t t o interpret infinite-orde r contact transformations a s a natural extension o f Lie's theory was done by Backlund in [11 ] about 10 0 years ago . The centra l cohomologica l constructio n fo r th e theor y o f partia l differentia l equations, th e C-spectral sequence associate d t o a give n syste m o f equations , i s introduced earlie r i n Chapter 4 . Th e firs t ter m o f this sequenc e i s most important , since Lagrangian s (actions) , conservatio n laws , symplecti c structure s i n th e field theory, etc. , ar e parts o f it. I t allow s on e to answe r suc h question s a s what i s th e complete list o f conservation law s admitted b y a given system o f partial differentia l equations, or could such a system be obtained fro m variational principles, and s o on. When th e numbe r o f independen t variable s become s equa l t o zero , th e C-spectra l sequence reduces to the standard d e Rham complex . Thi s is one of the reasons why later, i n Chapter 5 , we call its elements secondary differential forms. It i s mor e difficul t t o characteriz e th e secon d ter m o f the C-spectra l sequenc e in the standard language . Fo r instance , it s element s migh t b e calle d characteristi c classes o f cobordisms compose d o f solutions o f the equation i n question. I n partic - ular, variou s well-know n characteristi c classe s ca n b e obtaine d i n this way , i.e. , a s elements o f the secon d ter m o f the C-spectra l sequenc e associate d wit h som e uni- versal equations . Alternatively , they may be called secondary de Rham cohomology classes, sinc e i n th e cas e whe n th e numbe r o f independen t variable s equal s zero , the constructio n give s th e standar d d e Rha m cohomology . Classica l ("primary" ) mathematics give s no hints o n ho w to cal l higher terms o f the C-spectral sequence . The main technical problem discusse d i n Chapter 4 is how to compute the first term o f the C-spectra l sequence . A genera l metho d base d o n a direc t us e o f th e Spencer typ e cohomolog y i s develope d there . I t give s a n exhaustiv e answe r fo r infinite-order je t spaces , an d a rathe r efficien t computationa l algorith m fo r infin - itely prolonge d equation s i s constructe d o n it s basis . I n particular , i t allow s on e to describ e th e domai n i n th e standar d (p , g)-diagram wher e al l nontrivia l term s are locate d (th e 2 - and p-Lin e Theorems) . A formall y equivalen t bu t muc h mor e manageable metho d i s discussed i n Chapter 5 . As regards th e secon d ter m o f the C-spectra l sequence , a n analog y wit h usua l vector fields an d differentia l form s allow s on e t o sugges t som e simpl e homotop y techniques tha t wor k wel l fo r a larg e clas s o f equations . O n th e othe r hand , thi s analogy lead s to the first ide a o f Secondary Calculu s a s i s explained i n Chapter 5 . Some application s t o th e theor y o f conservatio n laws , Lagrangia n formalis m with constraints an d othe r problem s ar e discusse d a t th e en d o f the chapter. Eve n in standard situations, the results obtained there reveal new important features. Fo r instance, the method o f "generatin g functions " develope d with computational pur - poses show s that highe r symmetrie s an d conservatio n law s o f a (nonlinear ) syste m of partial equation s ar e solution s o f two mutuall y adjoin t C-differentia l equations . These equations coincid e fo r Euler-Lagrange equations . Thi s clarifies the nature o f the Noethe r theore m an d allow s u s to generaliz e i t t o a muc h large r clas s o f equa - tions. Th e computationa l technique s fo r conservatio n law s o f concret e equation s INTRODUCTION xii i can b e foun d i n [127] . Th e boo k [19 ] i s once agai n strongl y recommende d t o th e readers intereste d i n these aspects . The result s containe d i n Chapte r 4 wer e obtaine d b y th e autho r mainl y i n 1976-77 and publishe d i n two short note s [129 ] and [121] . However , i t turned ou t to be impossible to publish a detailed exposition o f these results i n the USSR then. They appeare d later , wit h almos t si x yea r delay , i n th e US A [134] . Meanwhil e Tsujishita publishe d a detailed descriptio n o f a simplifie d versio n o f the C-spectra l sequence calle d th e variational bicomplex [112] . Tsujishita' s approac h require s a fibered structure o f the equation i n question, which furnishes th e second differentia l coming fro m th e correspondin g Leray-Serr e spectra l sequence . Fo r thi s reason , problems suc h a s tha t o f minima l surfaces , etc. , canno t b e treate d directl y wit h this method . Moreover , hi s approac h appeal s t o a coordinat e descriptio n rathe r than th e algebrai c conceptua l languag e an d thu s hide s basi c natura l structure s of th e theor y and , i n particular , th e perspectiv e o f Secondar y Calculus . Fo r th e same reasons , Tsujishita' s approac h become s mor e popula r amon g specialist s i n mathematical physics . O n the other hand, i n the sam e paper Tsujishit a discovere d important link s t o th e theor y o f characteristi c classes . Perhaps , th e difficultie s related to a systematic us e o f the natural algebrai c language o f differential calculu s in the context unde r consideration gav e rise to some works o f a coordinate-oriente d nature (see , fo r instance , book s [3 ] and [146]) . Th e are a o f mathematics tha t wa s formed aroun d th e variationa l bicomple x wa s calle d b y Tsujishit a [110 ] "forma l geometry o f differential equations" . Sinc e then, a flavor o f formality o f the subjec t was maintained b y severa l authors . U p to a certain extent , thi s poin t o f vie w ca n be justified, bu t fo r Secondar y Calculu s i n genera l (se e Chapte r 5 ) i t i s no longe r true and , eve n more , woul d b e counterproductive i n a strategic perspective . The concludin g Chapte r 5 i s o f a syntheti c nature . It s mai n goa l is , firs t o f all, t o sho w ho w the result s o f the precedin g chapter s ar e self-organize d naturall y into a ne w Calculu s calle d Secondary , t o presen t furthe r structure s o f Secondar y Calculus an d t o provid e som e perspectives . Proof s o f the result s reporte d i n thi s chapter are , a s a rule, omitted . The notio n o f a diffiet y introduce d earlie r i n Chapte r 3 i s used systematically . Infinitely prolonge d equation s ar e the simples t example s o f this concept . Diffietie s may als o b e calle d "secondar y manifolds " an d a s suc h ar e th e stage s wher e th e performances o f Secondary Calculu s take place. The n w e pass to "secondar y vecto r bundles" an d "secondar y modules " ove r "secondar y smoot h functio n algebras " o n which "secondar y differentia l operators " act . Al l these objects ar e homotopy type s of special differential complexe s over the base diffiety O. Fo r example, the secondary smooth functio n algebr a o n O i s the homotop y typ e o f the algebr a o f grade d dif - ferential operator s actin g o n the horizonta l d e Rham comple x ove r O. Homotop y classes o f the column s o f th e zer o ter m o f the associate d C-spectra l sequenc e ar e secondary differentia l forms . Secondar y differentia l operator s ar e define d a s suit - able homotop y classe s o f differentia l cochai n mapping s o f complexe s representin g secondary modules , etc. When w e tak e fo r th e bas e diffiet y al l covering s ove r a n infinitel y prolonge d equation, th e cohomolog y o f the correspondin g complexe s contain s exhaustiv e in - formation concernin g th e origina l equation . Thi s i s wh y Secondar y Calculu s i s much mor e tha n jus t a forma l theor y (se e above) . Thus , th e proble m o f efficien t computations o f the cohomolog y become s centra l i n Secondar y Calculus . I t i s re- markable tha t th e C-spectra l sequenc e metho d ca n b e extended quit e naturall y t o xiv INTRODUCTIO N general secondar y modules . Thi s generalizatio n i s described togethe r wit h a ver y important improvemen t o f th e origina l author' s metho d (Chapte r 4 ) du e t o th e works b y M . Marvan , T . Tsujishita , D . Gessler , an d A . Verbovetsky , wher e th e compatibility complex wa s used instea d o f the relate d Spence r typ e cohomology . Another topi c to whic h considerable attentio n i s paid i n Chapter 5 is the "sec - ondarization problem" , i.e. , th e proble m o f finding secondar y counterpart s fo r al l ingredients o f the standar d (primary ) Calculus . I t i s a rather delicat e proble m a s one coul d se e fro m th e precedin g discussion , an d o f rather grea t importanc e a s a whole. I n spit e o f the fac t tha t a n almos t algorithmi c procedur e ca n b e suggeste d to solv e i t now , a detaile d elaboratio n o f nonstandar d cohomologica l technique s and, no t infrequently , remakin g o f well-know n piece s o f the standar d Calculu s i n order t o brin g the m t o a secondarizable for m i s required. Suc h a n enterpris e goe s far beyon d th e bound s o f this book . Fo r thi s reaso n w e restrict ourselve s t o a de- scription o f secondary differential operators , some elements o f the secondary Hamil - tonian formalis m and , i n more details, o f secondary multivector-value d differentia l forms includin g the secondary Schouten-Nijenhui s an d Frolicher-Nijenhuis bracke t formalism. Thes e concrete constructions ar e sufficiently instructiv e an d lea d to nu - merous interestin g application s a s well . A s a n illustration , w e mention her e tha t the Eule r operator , associatin g t o a give n Lagrangia n th e correspondin g Euler - Lagrange equations , turn s ou t t o b e a first-order secondar y differentia l operato r while its forma l coordinat e descriptio n present s i t a s an infinite-orde r operator . It shoul d als o b e stresse d tha t Secondar y Calculu s ha s variou s level s corre - sponding t o th e secon d an d highe r orde r term s o f the C-spectra l sequence . W e d o not touc h upo n thes e aspect s here .

4. Som e general remark s concernin g the actua l statu s o f the theor y presente d in this boo k ar e to b e added . , i.e. , the genera l theory o f systems o f (nonlinear ) algebrai c equations, i s on e o f th e mos t respecte d an d "noble " area s o f th e moder n pur e mathematics. O n the contrary , accordin g t o the 199 1 AMS Subjec t Classification , a simila r theor y concernin g (nonlinear ) partia l differentia l equation s doe s no t ex - ist a t all . Moreover , th e commo n opinio n i s that a substantiv e genera l theor y o f (nonlinear) partia l differentia l equation s eve n canno t exist . Thus , th e stud y o f a concrete system o f (nonlinear ) differentia l equation s o f a particular applie d interes t is left t o experts i n the correspondin g specia l fields o f mechanics, mathematica l o r theoretical physics , etc. It i s also commonl y forgotte n tha t a t th e en d o f the 19t h centur y an d th e be - ginning o f the 20t h the situation wa s completely opposite . Th e great symphon y b y Sophus Lie , who laid the firs t foundatio n stone s i n the buildin g o f the general the - ory o f nonlinea r partia l differentia l equations , wa s highl y recognize d a s a "noble " part o f pur e mathematics . I t attracte d attentio n o f man y distinguishe d mathe - maticians o f that epoch . But , quit e surprisingly , a t leas t a t th e first glance , thi s glorious period suddenl y ende d afte r th e World War I , which seemingly completel y destroyed th e grea t nonlinea r cultur e o f ol d masters . Fro m Lie' s symphony , onl y the score s o f Li e groups an d Li e algebras wer e extracted, subsequentl y elaborate d and develope d i n thousands o f works. Dynamic s an d cause s o f these events are ye t to be analyzed. Tw o o f them are clear, however. First , a very attractive temptatio n to lineariz e rea l lif e wa s satisfie d b y functiona l analysi s an d becam e dominan t i n almost everythin g that concern s partial differentia l equations . Second , a consisten t INTRODUCTION x v interaction o f man y mathematica l structure s i s indispensable whe n studyin g suc h equations, eve n linear . A lo t o f thes e structures , an d especiall y thos e relate d t o the genera l homologica l cultur e hav e not eve n been discovere d then . Th e resultin g underestimation an d underdevelopmen t o f the genera l theor y o f nonlinea r partia l differential equation s i s i n th e origi n o f a difficul t an d tortuou s progres s i n man y key areas o f mathematical natura l sciences . Secondary Calculus is a natural continuation o f the opera by Sophus Lie and his followers, enriche d b y the result s o f the forma l theor y o f partial differentia l equa - tions. I t i s worth emphasizin g that cohomologica l methods i n the general theory o f differential equation s appeare d fo r th e firs t tim e exactl y i n the contex t o f the for - mal theory due to D. Spencer and Goldschmid t (see , for instance, [44 , 45, 103] ) a s Spencer's resolution , Spencer' s cohomology , etc . Thes e author s discovere d a dee p cohomological natur e o f the ol d forma l theor y develope d b y P . Riquier , M . Jane t and others . Sinc e th e beginnin g o f th e 1970 s th e Mosco w schoo l i n geometr y o f partial differentia l equation s ha s brought th e abov e theories to a natural synthesi s on the basis o f the algebraic "perestroika " o f Calculus [136 , 133] , and this reveale d the firs t structure s o f Secondary Calculu s a t th e beginnin g o f the 1980s . At present, i t i s clear that wit h Secondar y Calculu s the general theory o f (non - linear) partia l differentia l equation s find s it s soli d natura l basis . Eve n more , Sec - ondary Calculu s showed itsel f to be a unifying theory , whose potential application s range fro m arithmeti c algebrai c geometr y t o moder n theorie s o f elementary parti - cles. Thi s features coul d be hardly expected fro m th e ver y beginning. O f course, a substantial systemati c preliminar y wor k ha s to b e don e t o ge t a realization o f th e opened perspectives. Thus , the author woul d be very satisfied i f this book drew the interest o f young mathematicians an d maybe physicists towards this ne w nonlinea r mathematics. Since the works by Elie Cartan summarized i n his book [29 ] (se e also [24 ] for a modern formulatio n o f Cartan's approach) , there exist s a tradition t o treat partia l differential equation s a s exterior differentia l system s (EDS ) [79 , 99] . Thi s point o f view is useful fo r a variety o f problems. ED S are geometric structures ver y clos e to PDE, an d fo r thi s reaso n man y topic s considere d i n this boo k ca n b e brought int o the contex t o f EDS. The reade r wil l fin d a n instructive exampl e o f this kin d i n th e papers b y Bryant an d Griffith s [25 , 26] , as wel l as in [27 , 28]. This page intentionally left blank This page intentionally left blank Bibliography

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adjoint complex , 4 1 of bordisms, 2 4 adjoint operator , 32 , 3 3 Pontryagin, 183 , 18 7 in the C- theory, 12 8 characteristic covector , 3 0 left, 21 4 characteristic difnety , 18 2 right, 21 5 regular, 18 3 admissible submanifol d o f a variationa l characteristic mapping , 18 2 problem, 13 3 regularization, 18 3 algebra o f symbols, 2 9 C-homotopy equivalence , 20 3

alternative Lagrangians , 2 2 C(/c)-field, 11 5 classical contact structure , 5 Bohr correspondenc e principle , 19 5 classical situation, 2 6 bordisms o f geometric structures , 18 6 Cy-spectral sequence , 21 6 characteristic classes , 18 6 vanishing theorem s brackets V-fc-Line Theorem , 22 1 Frolicher-Nijenhuis, 22 6 V-Two Lin e Theorem, 21 9 higher Jacobi , 7 5 cobweb o f coverings, 1 4 Poisson, 10 , 30, 10 6 common equation , 8 7 Richardson-Nijenhuis, 22 5 compatibility complex , 170 , 22 0 Schouten-Nijenhuis, 22 6 Diff-compatibility, 23 1 bundle o f jets, 5 8 Jet-compatibility, 23 1 Cartan distribution , 5 , 62, 96, 19 3 conformally selfadjoin t equation , 17 3 Cartan plane , 9 6 conic equation, 17 2 Cartan submodule , 77 , 9 9 connection i n a module , 19 9 category DE , see categor y o f differentia l conservation law , 23 , 5 0 equations linear, 4 8 category o f differential equations , 9 , 12 1 local, 13 6 homotopic morphisms , 17 1 nonlocal, 19 2 homotopy, 17 1 proper, 17 3 morphisms, 12 1 rigid, 2 3 objects, 12 1 rough,173 Poincare lemma , 17 2 contact structur e C-complete equation, 8 8 classical, 5 C-complex, 20 3 generalized, 6 1 C-conneetion, 20 6 infinite-order, 19 3 c-density, 47 , 48, 13 6 k-th order , 5 , 6 2 equivalence, 5 0 correspondence principle , 19 5 nonlinear, 5 0 covering, 1 3 C-differential operator , 10 0 cobweb, 1 4 C-field, 72 , 77 , 10 7 in the categor y DE , 19 2 C-filtration, 138 , 21 6 C-spectral sequence, 13 8 C-general equation, 8 8 of a diffiety , 2 0 C-Green formula , 13 0 vanishing theorem s characteristic, 76 , 8 2 fc-Line Theorem fro m th e Bottom , 23 2 characteristic classes , 18 2 One Lin e Theorem, 22 7 Euler, 18 7 Two Lin e Theorem , 161 , 171 of 3-webs , 18 4 C-transformation, 11 8 244 INDEX degree o f a functor , 10 1 functorial fibe r bundle , 6 8 density o f a linea r conservatio n law , see c- fundamental isomorphism , 22 7 density de Rham complex , 2 8 ^-connected domain , 6 7 DifT-compatibility complex , 23 1 general diffiety , 19 3 diffeomorphism o f domains i n J°°, 11 8 generalized contac t structure , 6 1 differential calculu s functors , 2 7 generalized Noethe r theorem , 13 7 degree, 2 7 generalized Schwar z formula , 18 2 differential complex , 19 6 generalized symmetry , 1 3 differential grade d cochai n mapping , 19 7 generating function , 1 0 differential linea r mapping , 197 of a conservation law , 2 3 differential linea r structure , 19 7 of a Li e field , 81 , 84 diffiety, 7 , 12 2 of an evolutionar y derivation , 1 0 C-fields, 9 generating operato r o f a secondar y contact fields , 9 differential operator , 1 8 C-spectral sequence , 2 0 geometric filtere d module , 9 7 elementary, 19 1 geometric modules , 2 8 general, 19 3 geometric structure , 122 , 18 2 horizontal d e Rha m cohomology , 1 5 scalar differentia l invariant , 12 3 horizontal differential , 1 5 type of , 12 2 horizontal differentia l forms , 1 5 geometrization functor , 2 8 secondary differentia l forms , 2 0 gluing operator , 2 7 symmetries, 9 Green ^-formula , 4 6 symmetry algebra , 9 Green formula , 4 1 trivial contac t fields , 9 C-formula, 13 0 diffiety dimension , 7 , 19 2 for a differentia l operator , 4 2 diffiety morphisms , 19 2 general, 4 2 divergence, 3 3 C-formula, 4 6 divided loca l chart, 4 Hamiltonian formalis m i n the C-theory , 10 6 elementary diffiety , 19 1 Hamiltonian homomorphism , 21 2 equation normal with respect to an operator , Hamiltonian operator , 21 2 173 higher extrinsi c infinitesima l symmetry , 10 9 equivalent c-densities , 5 0 higher infinitesima l symmetry , 10 7 equivalent equations , 9 2 higher Jacob i bracket , 7 5 equivalent fli t rat ions, 12 1 homogeneous splitting , 4 2 Euler operator , 46 , 136 , 14 5 homotopic morphism s i n DE, 17 1 Euler-Lagrange equation , 46 , 13 6 homotopy equivalenc e o f differentia l Euler-Lagrange operato r fo r a problem wit h complexes, 19 7 constraints, 18 1 homotopy i n the categor y DE , 17 1 Euler-Lagrange theorem , 13 6 horizontal de Rham cohomology o f a diffiety , evolutionary derivation , 10 , 7 4 15 exceptional case , 6 2 horizontal d e Rham complex , 10 0 exterior differentiatio n operator , 2 8 horizontal differentia l for m o n a diffiety , 1 5 exterior produc t o f differentia l forms , 2 8 horizontal differentia l o n a diffiety , 1 5 extrinsic infinitesima l symmetry , 8 7 horizontal element , 10 1 extrinsic symmetry , 8 7 horizontal Jet-Spence r comple x wit h coeffi - cients i n a module , 23 2 factorization, 12 2 horizontal module , 20 7 FG-eategory, 9 8 horizontalization operation , 10 0 filtered differentia l operator , 9 7 filtered homomorphism , 9 7 infinite-order contac t structure , 19 3 first variatio n formula , 13 5 infinitesimal automorphis m o f the Carta n ^-symmetry, see nonloca l symmetr y distribution, 10 7 flat C-connection , 20 7 infinitesimal Li e transformation, 8 1 flat connection , 20 0 infinitesimal Stoke s formula , 150 , 15 1 formally integrabl e equation , 6 2 infinitesimal symmetr y o f a density , 13 6 FP-equation, 4 7 infinitesimal symmetr y o f a Lagrangia n Frolicher-Nijenhuis bracket , 22 6 density, 5 2 INDEX 245

inner product , 8 4 V-extension o f a differentia l operator , 21 5 insertion, 8 4 V-fc-Line Theorem , 22 1 integrable C-field , 12 1 V~Two Lin e Theorem, 21 9 integral manifol d o f the Cartan distribution , natural S-module , 199 , 20 7 8, 6 2 natural