symmetries, on the other hand , ar e formal symmetry vector fields whose coefficient s
depend o n derivative s o f arbitrar y order . The y wer e introduce d b y Emm y Nothe r
in her famou s theore m relatin g symmetrie s o f variational problem s t o conservatio n
laws, an d the y hav e rise n t o prominenc e i n th e recen t pas t i n th e contex t o f com -
pletely integrabl e evolutio n equations , throug h hierarchie s o f commuting flows. Al l
of this materia l i s of course well known, an d ca n b e foun d i n a number o f excellen t
texts, [Olvl] , [Ovs] , [BK] . Ou r expositio n i s therefore quit e brief .
In Chapte r 3 , we review th e concep t o f a n interna l symmetr y o f a differentia l
equation, whic h wa s first formulate d i n it s ful l generalit y b y Eli e Carta n i n term s
of infinitesima l automorphism s o f exterio r differentia l systems . Th e interpla y be -
tween thes e thre e type s o f symmetr y ca n b e fairl y subtle , an d i s nicely illustrate d
by th e cas e o f a n under-determine d ordinar y differentia l equatio n first studie d b y
Hilbert, [Hilb] . Thi s equatio n wa s als o considere d b y Eli e Cartan , [Cal5] , wh o
showed tha t i t admit s th e non-compac t rea l form o f the 14-dimensional exceptiona l
Lie algebr a $2 a s it s Li e algebr a o f interna l symmetries . Muc h o f thi s chapte r
is devote d t o a detaile d stud y o f thi s interpla y fo r variou s classe s o f differentia l
equations, [AKO] .
In Chapte r 4 , w e stud y th e Laplac e transformatio n o f linea r hyperboli c par -
tial differentia l equation s i n th e plane , togethe r wit h it s geometri c counterpar t fo r
surfaces admittin g a parametrizatio n i n whic h th e secon d fundamenta l for m i s di-
agonal. Eve n thoug h thi s transformatio n wa s originall y introduce d a s a metho d
of closed-for m integratio n fo r som e specia l classe s o f hyperbolic partia l differentia l
equations, it now plays an important rol e in the study of integrable systems, notabl y
through the Darboux-Crum transformatio n o f Sturm-Liouville operators, [DT] , and
also i n th e contex t o f th e Tod a field theories , [Val] . Th e mai n referenc e fo r thi s
Chapter i s the beautifu l classica l treatmen t o f Darboux , [D] , and th e mor e recen t
expositions give n i n [Chi] , [KTl] , [T] .
Chapter 5 is devoted t o th e higher-dimensiona l analogu e o f the Laplac e trans -
formation, [KTl] . A t the geometrical level , this corresponds to a transformation o f
a specia l clas s o f submanifolds o f n-dimensional rea l projectiv e space , first studie d
by Chern , [Ch2] , [Ge] . Analytically , i t give s ris e t o a transformatio n an d closed -
form integratio n metho d fo r a clas s o f over-determine d linea r partia l diferentia l
equations i n n independen t variables .
The clas s o f strongly hyperboli c Hamiltonia n systems , [DN] , which w e review
in Chapte r 6 , provides a n interestin g aren a o f application fo r man y o f the concept s
introduced above . W e wil l se e tha t th e over-determine d syste m o f linea r partia l
differential equation s whic h govern s th e conserve d densitie s fo r suc h a syste m i s
precisely o f th e typ e amenabl e t o th e higher-dimensiona l Laplac e transformation ,
provided tha t th e give n syste m i s ric h i n conservatio n law s i n th e sens e o f Deni s
Serre, [Se] , o r semi-Hamiltonia n i n th e sens e o f Tsarev , [Tsa] . A s a result , w e
obtain a geometri c metho d fo r constructin g familie s o f suc h integrabl e systems ,
whose commutin g flows can b e determine d explicitly , [KT2] , [Fe2] .
Chapter 7 consists i n a brief introductio n t o th e variationa l bi-complex , [Al] ,
[A2], [Tsu] , [Vi] . Thi s i s a bi-grade d comple x o f differentia l form s o n th e je t
bundle of infinite order , which arises from th e de Rham comple x by a decomposition
induced b y th e underlyin g contac t structure . Th e variationa l bi-comple x provide s
a natura l framewor k fo r th e stud y o f variationa l principle s an d conservatio n laws ,
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