which wil l for m th e theme s o f Chapter s 8 an d 9 . W e wil l presen t th e basi c loca l
and globa l exactnes s result s whic h w e will need i n the nex t tw o lectures .
One o f th e basi c question s tha t on e ca n as k abou t a differentia l operato r i s
whether it can be expressed as the Euler-Lagrange operator of a variational problem .
This i s the mos t restricte d versio n o f what i s known a s the invers e proble m o f th e
calculus o f variations. I n Chapte r 8 , w e cover som e o f basic aspect s o f th e invers e
problem o f th e calculu s o f variations , i n it s local , globa l an d equivarian t versions .
This is a well developed subject, wit h connections to differential geometry , topology ,
and analysis , [Ta] , [Al] , [A2] , [GH] . Ou r treatmen t remain s quit e superficial , i n
the sense that i t deals almost exclusivel y with the obstructions to the solution of the
most restricte d for m o f the invers e problem, bu t i t serve s to illustrat e th e powe r of
the variationa l bi-comple x a s a general too l fo r tacklin g thes e questions . Th e mai n
references w e have used fo r thi s Chapte r ar e [Ta] , [Al] , [A2] .
In Chapte r 9 , w e conside r th e propert y o f Darbou x integrabilit y fo r scala r
second-order hyperboli c partia l differentia l equation s i n two independent variables .
Darboux integrabilit y i s a geometri c integrabilit y propert y whic h i s base d o n th e
behavior of the characteristic systems associated to the partial differential equation .
In particular , Darbou x integrabl e equation s ar e integrabl e b y ordinar y differentia l
equation methods , an d w e will show tha t the y admi t non-trivia l conservatio n law s
of al l degrees , whic h ar e reflecte d th e cohomolog y o f th e constraine d variationa l
bi-complex. I t i s a highly non-trivia l proble m t o determine a workable set o f neces-
sary an d sufficien t condition s fo r a hyperboli c equatio n t o b e Darboux-integrable .
The constraine d variationa l bi-complex , obtaine d b y restrictin g th e tautologica l
variational bi-comple x introduce d i n Chapte r 7 to jet s o f solution s o f th e partia l
differential equation , provide s a settin g fo r tacklin g thi s questio n i n term s o f loca l
differential invariant s o f the equation . ( A more intrinsic settin g i s provided b y th e
characteristic cohomolog y o f exterio r differentia l systems , [BG1], [BGH1]. ) W e
will review some recent wor k o n this problem , base d o n a non-linear generalizatio n
of the metho d o f Laplace. W e will also show how to compute the conservatio n law s
of a genera l hyperboli c partia l differentia l equatio n i n this geometri c context . Th e
results appearin g i n this chapte r ar e take n fro m [AK1], [AK2] , an d [AK4] .
In Chapte r 10, whic h i s th e las t lectur e o f thi s series , w e giv e a brie f intro -
duction t o on e of the mos t significant recen t development s i n the stud y o f exterio r
differential systems , namel y th e characteristi c cohomolog y theor y fo r exterio r dif -
ferential systems . Thi s concep t ha s bee n define d an d studie d i n dept h b y Bryan t
and Griffiths , [BG1], [BG2] , an d by Bryant, Griffith s an d Hsu, [BGH1], [BGH2],
for hyperboli c exterio r differentia l systems . W e begi n wit h a ver y brie f revie w o f
the theor y o f exterio r differentia l systems , includin g th e Cartan-Kahle r Theore m
and Eli e Cartan' s involutivit y test . W e the n presen t som e genera l vanishin g the -
orems fo r th e characteristi c cohomolog y o f exterio r differentia l systems , base d o n
the behavio r o f their Car t an characters .
Niky Kamra n
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