Lie s invariant theory of contact transformations and its extensions.

Transcription

1 Les Ivaratetheore der Berührugstrasformatoe ud hre Erweterug, from the Jahresbercht der deutsche Mathematker-Veregug, v. 5, H. Lebma ad F. Egel, De Berührugstrasformatoe: Geschchte ud Ivaratetheore, B. G. Teuber, Lepzg, 94. Le s varat theory of cotact trasformatos ad ts extesos. By FRIEDRICH ENGEL Gesse ) Traslated by: D. H. Delphech Cotets Page. The blear covarat The tegrato problem of a Pfaffa equato ad a Pfaffa expresso Uos the space of elemets x,, x, p,, p Cotact trasformatos the x, p Dfferetal equatos x,, x, p,, p The varat theory of cotact trasformatos the x, p Other treatmets of the theory of fucto groups. Kator s geeralzato of the problem The varat theory of cotact trasformatos, as carred over to Pfaffa expressos 2 varables.. 4 Appedx. 60 I the year 872, the Proceedgs of the Scetfc Socety of Chrstaa receved a bref barely three pages commucato of Sophus Le: Zur Ivarate-Theore der Berührugstrafromatoe. Ths s oteworthy more tha oe way. Frst, due to the partcularly mportat applcatos that Le made of hs ew theory to the tegrato of partal dfferetal equatos of frst order. Secod, especally for the fact that t treated the varats of a specal fte group, the group of all cotact trasformatos. Before the, there was, at frst, oly oe example of such a varat theory that bega wth Gauss, was further developed by Codazz, Mard, ad Beltram, ad became the varat theory of quadratc dfferetal forms several varables by Rema, Chrstoffel, ad Lpschtz. Fally, oe observes that thrdly, Le s varat theory exsted precsely the same tme perod whch F. Kle developed the geeral deas that he lad dow hs Erlager Programm, so Le had already worked out a mportat example of what Kle had proposed as a program for the future. Ideed, Le kew of these deas, for the most part, from speakg wth Kle, ad had hmself cotrbuted to ts developmet, but the ovelty these deas was that for hm may realms of exstg mathematcs could be descrbed as varat theores of groups, whle, o the other had, ) Talk submtted to the Germa Socety of Mathematcas ad excerpts preseted at ts meetg Vea, September 93.

2 Le s varat theory of cotact trasformatos ad ts extesos 2 the questo of the varat propertes of the trasformed pcture uder a gve group was completely atural to hm. Le s frst presetato of hs varat theory of cotact trasformatos suffered from the fact that he stll was o posto to gve a smple dervato of the formulas for cotact trasformatos, but based t o the geeral theory of Pfaffa problems. He dd that 874 the great treatse volume VIII of the Mathematsche Aale. Soo after, Adolph Mayer gave a relatvely smple, drect presetato of the theory of cotact trasformatos (Gött. Nachr. 874), but Le could stll ot make up hs md whether to accept the Mayer approach outrght. Ths s a pecularty of Le hmself: He made t hs ambto to foud hs ew theory oly such a way that he hmself had thought of, ad he wet out of hs way to avod use of ay smplfcatos that orgated wth ayoe else. By the same ducemet, oe s obvously also led to the remarkable fact that Le made o meto whatsoever of the blear covarats of a Pfaffa expresso that Lpschtz ) had already gve 869 ad had the bee utlzed to great effect by Frobeus 2 ) 877, or dd he use them aywhere. As S. Kator justfably suggested a 90 paper 3 ), t s precsely wth the help of these covarats of the theory of Pfaffa problems, ad especally the Mayer formulato, as well, by whch the theory of cotact trasformatos ca be greatly smplfed, amely, whe oe otces addto that the Posso bracket symbol ca be preseted as a form plae coordates that s covarat to these blear covarats. Sce the presetato of the aalytcal theory of cotact trasformatos that was gve the secod volume of Trasformatosgruppe was also strogly affected by Le s aforemetoed pecularty, t does ot seem superfluous to me to preset the theory of cotact trasformatos ad the assocated varat theory as t s ow possble to do by usg the moder tools. I geeral, may thgs that are kow wll also have to be reterated. However, t wll yeld that the geeralzato of the theory that Le hmself already had md to the case where oe does ot base everythg o the cotact trasformatos for a specfc ormal form for the Pfaffa expresso s ot slghtest more dffcult. I have the aforemetoed paper of S. Kator ad aother oe that was publshed the meatme 4 ) to thak for may essetal spratos. Furthermore, I caot, by ay meas, clam that the sghts of S. Kator were my ow. Smlarly, I caot help but stress that some of the very elevated clams that Kator made seemed completely ujustfed to me. Alog wth the may good deas that oe fds the papers o how to approach matters, oe also fds a large umber of flawed or outrght false oes, ad the lack of orgazato the presetato has the effect that overall, t suffers from a lack of clarty that the reader of the paper does ot fd edfyg. ) I the paper: Utersuchuge Betreff der gaze homogee Fuktoe vo Dfferetale. Crelles Joural, v. 70, pp. 72, et seq. 2 ) Über das Pfaffsche Problem, Crelles Joural, v. 82, pp ) Über ee eue Geschtspukt der Theore des Pfaffsche Problemes, der Fuktoegruppe ud der Berührugstrasformatoe. Weer Berchte, Math.-aturw. Klasse, Bd. CX, Abt. IIa, December 90, pp. 47, et seq. 4 ) Neue Grudlage für de Theore ud Weteretwckluge der Lesche Fuktoegruppe, bd., Bd. CXII, Abt. IIa, July 903, pp. 755, et seq. Obvously, the paper that follows ths oe mmedately o pp ( Über ee eue Klasse gemschter Gruppe ) belogs wth ths oe, as well.

3 Le s varat theory of cotact trasformatos ad ts extesos 3. The blear covarat. Frst, I must brefly dscuss the blear covarat of a Pfaffa expresso. Let: () D = α ( x,, x) dx be a arbtrary Pfaffa expresso, ad the same expresso, whe formed aother system of dfferetals δx would be deoted by f: (2) x = ϕ (y,, y r ) ( =,, ), where the ϕ are completely arbtrary fuctos, ad thus D s coverted oto a ew Pfaffa expresso the r varables y,, y r : (3) D = α ( x,, x) dx = β ( y,, y ) dy = ad oe thus obtas, by meas of (2): r k = k (4) d δ D = ( dαδ x δαdx ) + α( dδ x δ dx ) = = r r = ( dβkδ yk δβkdyk ) + βk ( dδ yk δ dyk ). k= k = O the other had, however, from (2), oe gets: dx = so, as oe easly sees: (5) d δx δ dx = from (3), t the follows that: ϕ r dyk, k = yk r ϕ ( dδ yk δ dyk ) ; y k = k r α ( dδ x δ dx) = = k = βk ( dδ yk δ dyk ), such that oe ca deduce from equato (4) that: (6) ( dαδ x δαdx ) = ( dβkδ yk δβkdyk ). = r k =

4 Le s varat theory of cotact trasformatos ad ts extesos 4 Wth ths, oe shows that the expresso: (7) ( dαδ x δαdx ) = =, α α dx δ x x x s covarat to the Pfaffa expresso, ad deed s ot merely a troducto of ew varables, sce uder ay substtuto (2) the fuctos ϕ may or may ot be depedet of each other. I all of the represetatos that are kow to me, oe costructs the blear covarat of () by regardg the expressos dδx ad δdx the expresso d δd as equal, ad oe must the verfy the covarace property by computato. Here, ths property emerges as a mmedate cosequece of the smple fact that from (5) the expressos dδx δdx are cogredet to the dx ad δx. Also, t s mportat that the covarace property of (7) s ot assured merely by the troducto of ew depedet varables, but for ay arbtrary substtuto of the form (2). It must be further remarked that equato (5), whch follows from (2), yelds: F ( dδ x δ dx ) = x = r F ( dδ yk δ dyk ), y k = k whe F deotes a arbtrary fucto of x,, x. If oe uderstads dy k ad δy k here to mea the creases that the y k experece uder two arbtrary ftesmal trasformatos Y f ad Y 2 f y,, y k the oe mmedately recogzes that dδy k δdy k s the crease that y k expereces uder the ftesmal trasformatos, whch wll be represeted by the bracket expresso: (Y Y 2 ) f = Y Y 2 f Y 2 Y f. Equatos (5) thus express the kow fact that ths bracket expresso represets a ftesmal trasformato that s covarat to both ftesmal trasformatos. 2. The tegrato problem of a Pfaffa equato ad a Pfaffa expresso. Le s frst paper o the theory of partal dfferetal equatos (loc. ct.) was the oe whch he orgally posed the questo of Pfaff oce more, whch cossted the questo of whether ay mafold the space of x,, x o whch the Pfaff equato s fulflled could be regarded as a tegral mafold of a Pfaffa equato: α ( x,, x) dx = 0. =

5 Le s varat theory of cotact trasformatos ad ts extesos 5 If oe thks of a m-fold exteded mafold as represeted by m depedet equatos: Φ k (x,, x ) = 0 (k =,, m) the t s a tegral mafold whe ad oly whe ay system of values x,, x, dx,, dx that satsfes the equatos Φ k = 0, dφ k = 0 also satsfes the equato α dx = 0. O the other had, f oe thks of the mafold as represeted wth the help of m depedet varables u,, u m the form: x = ϕ (u,, u m ) ( =,, ) the t s a tegral mafold whe ad oly whe the equato α dx = 0 becomes a detty uder the substtuto x = ϕ. It s coveet to troduce, a correspodg way, the oto of a tegral mafold of a Pfaffa expresso α dx, ad uderstad ths to mea a mafold o whch the expresso α dx becomes a complete dfferetal. ) If oe recalls the kow theorem that a Pfaffa expresso s a complete dfferetal whe ad oly whe ts blear covarat vashes detcally the oe mmedately recogzes that a mafold: x = ϕ (u,, u m ) ( =,, ) represets a tegral mafold of the Pfaffa expresso α dx whe ad oly whe ts blear covarat vashes detcally uder the substtuto x = ϕ. However, ths, tur, yelds that a system of equatos: Φ k (x,, x ) = 0 (k =,, m), represets a tegral mafold whe ad oly whe ths blear covarat vashes for all systems of values x, dx, δx that satsfy the equatos Φ k = 0, dφ k = 0, δφ k = 0. Had Le troduced the oto of a tegral mafold of a Pfaffa expresso the he would have bee able to represet a o-trval part of hs vestgatos much more coveetly. 3. Uos the space of elemets x,, x, p,, p. The tegrato of those partal dfferetal equatos z, x,, x whch the ukow fucto z tself does ot appear subsumes the problem of fdg the -fold exteded tegral mafolds of the Pfaffa expresso: ) I have already bee usg ths formulato my lectures for may years. Oe also fds, moreover, as I have remarked after the fact that t was already used by S. Kator : Über ee Klasse gemschter Gruppe, loc. ct., Bd. CXII, Abt. IIa, July 903, pp. 72 o. 5.

6 Le s varat theory of cotact trasformatos ad ts extesos 6 (8) = p dx the space x,, x, p,, p that satsfy oe or more gve equatos betwee the x,, x, p,, p. We must therefore frst say a few thgs o the tegral mafolds of (8). We the brefly call a system of values x, p a elemet. It s clear that ay famly of elemets s a tegral mafold of the Pfaffa expressos (8). Thus, tegral mafolds of (8) always go through ay two eghborg elemets x, p ad x + dx, p + dp. O the cotrary, f we demad that the tegral mafold stll cludes a secod ftely close elemet x + δx, p + δp the, from 2, the codto: (9) ( dxδ p δ xdp ) = 0 = must be fulflled. Whe two of the elemets x, p are ftely close ad fulfll (9), we would lke to say that they are uted, ad accordgly, we would lke to brefly say that the tegral mafolds of (8) are uos. A uo ow cludes the elemet 0 x, 0 p ad m ftely close elemets 0 x + dx, 0 p + dp (k =,, m) that belog to o mafold of dmeso less tha m, for whch ot all m-rowed determats of the matrx: (0) dk x dk xdk p dk p (k =,, m) ecessarly vash. Therefore, oe must frst fulfll the equatos: () m ( dk xd j p dk pd jx ) = 0 (k, j =,, m), = ad secodly, all of the elemets x, p that are ftely close elemets of the uo must satsfy the m mutually depedet equatos: (2) m ( dk xdp dk pdx ) = 0 (k =,, m). = O the other had, sce (2) possesses the m learly depedet solutos: dx = d k x, dp = d k p, (k =,, m), oe the demads that oe must have m, such that there are o uos of more tha elemets. However, f m the there s always a uo of m elemets that cludes the elemet x, p ad the m gve ftely close elemets. Namely, f oe sets: 0 0 d k x = a k δt, d k p = b k δt

7 Le s varat theory of cotact trasformatos ad ts extesos 7 the oe has: x = x +, p = 0 m p k = m 0 akuk k = + b u ( =,, ), k k whe oe cosders u,.., u m to be depedet varables of such a uo. ) From the prevous statemets, t emerges that ay uo wll be represeted by equato of the form: x = Φ (v,, v m ), p = X (v,, v m ) ( =,, ), where m, ad amog the 2 fuctos Φ, X, m of them are mutually depedet. Amog the fuctos Φ,, Φ, let exactly l m of them be mutually depedet, so oe ca represet the equatos x = Φ the form: (3) x = ϕ (u,, u l ) ( =,, ), of whch, l of them ca be solved for u,, u l. Now, should the expresso: pdx = = = p dϕ be a complete dfferetal, the t could obvously clude o other depedet varables tha just u,, u l, so oe must have: pdϕ = dω(u,, u l ), = a equato that ca be subdvded to the followg oes: (4) ϕ p = = u k Ω u k (k =,, l). However, t s clear that equatos (3) ad (4) collectvely always represet a uo of elemets whe oe chooses the fuctos ϕ,, ϕ ad Ω completely arbtrarly ad cares oly whether the l fuctos ϕ,, ϕ are mutually depedet. Lkewse, oe demads that all uos of elemets ca be foud ths maer. O the other had, each uo of m elemets (l m ), amog whose equatos of them of the form (3) ca be foud, must belog to a uo that s determed by equatos of the form (3) ad (4), ad ay famly of elemets t that s cluded a uo must obvously defe a uo ts ow rght, ad wth ths, the determato of all possble uos s acheved. Oe obtas all of them whe oe chooses the fuctos ϕ,, ϕ, Ω the ) Cf., G. Kowalewsk, Lepz. Ber. 900, pp. 96, et seq.

8 Le s varat theory of cotact trasformatos ad ts extesos 8 most geeral maer for all possble values of l (0 l ), ad thus adds to equatos (3), the most geeral maer, the equatos: (5) p = χ (u,, u l, u l+,, u m ) ( =,, ) (l m ), such that equatos (4) are fulflled detcally. The uo of elemets, o whch, as we have see, all uos must le, must the be cosdered partcular. Let the equatos: (6) Φ (x,, x, p,, p ) = 0 ( =,, ) 0 0 be mutually depedet, ad let x, p be a system of values that satsfes (6) ad that does ot make all of the -rowed determats of the matrx: (7) Φ Φ Φ Φ ( =,, ). x x p p vash. Whe does the totalty of all elemets that le a certa eghborhood of the elemet x, p ad satsfy equatos (6) represet a uo of elemets? 0 0 From 2, t s ecessary ad suffcet that for all systems of values x, p, dx, dp, δx, δp that satsfy the equatos Φ = 0, dφ = 0, δφ = 0 the equato: (9) ( dxδ p dpδ x ) = 0 = s fulflled. Sce we restrct ourselves to such elemets that le a certa eghborhood 0 0 of the elemets x, p, we ca, ad would prefer to moreover, cosder oly such elemets x, p that fulfll (6) ad also do ot make all of the -rowed determats of (7) vash. For each elemet x, p, the equatos dφ = 0 represet depedet equatos for the dfferetals dx, dp. If the systems of values d k x, d k p (k =,, ) are learly depedet systems of solutos to the equatos dφ = 0 the the equatos: (8) ( dk xδ p dk pδ x ) = 0 (k =,, ) = are learly depedet of each other, ad sce these equatos must be satsfed for all systems of values dx, dp that satsfy the equatos δφ = 0, ths demads that the system of equatos (8) must be equvalet to the system δφ = 0. I ths fact, oe fds that the expressos: Φ µ Φµ (9) dx = λµ dt, dp = λµ dt p x µ = µ =

9 Le s varat theory of cotact trasformatos ad ts extesos 9 wth the arbtrary parameters λ,, λ, represet the most geeral system of values that satsfy the equatos dφ = 0. If we therefore set, wth the employmet of the Posso bracket symbol: (20) ϕ χ ϕ χ = p x x p = (ϕ χ), the the expressos: dφ = µ = λ ( Φ Φ ) dt µ µ must vash for arbtrary λ µ ; that s, all expressos (Φ µ Φ ) must vash for the system of values x, p that s cosdered here. Ths codto s ow ot merely ecessary, but also suffcet. Namely, f t s fulflled the obvously for arbtrary λ equatos (9) represet a system of values that satsfes the equatos dφ = 0, ad deed, the most geeral system of values of ths type; however, by meas of (9), oe wll have: ( dxδ p dpδ x ) = = λµ δφ µ dt, µ = whch, due to the equatos δφ = 0, must vash. Thus, we have the well-kow theorem: 0 0 If x, p s a elemet that satsfes the depedet equatos (6) ad does ot make all of the -rowed determats the matrx (7) vash the the mafold of 0 0 elemets that s represeted by (6) the eghborhood of the elemet x, p s a uo whe ad oly whe all of the expressos (Φ µ Φ ) (µ, =,, ) also vash for each elemet x, p that fulflls (6) ad les a certa eghborhood of brefly, whe the equatos (Φ µ Φ ) = 0 are a cosequece of (6). x, 0 p, or, more Sce we are stuck wth the Posso bracket symbol here, t s lkewse advsable to add the mportat relatoshp that exsts betwee ths symbol ad the blear covarat of the Pfaffa expresso, a relatoshp that lkewse seems to have frst bee establshed by S. Kator, or somethg close to t. Namely, f oe terprets the quattes dx, dp as homogeeous pot coordates a plae space R 2 of 2 dmesos, ad f oe defes homogeeous plae coordates by the equato: ( udx + vdp ) = 0 the the covarat form that belogs to the blear alteratg form: = 0

10 Le s varat theory of cotact trasformatos ad ts extesos 0 reads, plae coordates: = ( dx δ p dp δ x ) = ( vu uv ). Now, sce the dervatves of two fuctos ϕ ad χ wth respect to x, p are othg but two such systems of coordates, the Posso bracket expresso s smply the covarat costructed from the plae coordates to the blear covarat of the Pfaffa expresso. Furthermore, sce the equato: (9) ( dxδ p dpδ x ) = 0 = represets a lear complex R 2, the demad that the equatos Φ = 0 should determe a uo obvously says that all of the les the ( )-fold exteded mafold: (2) Φ Φ dx + dp = x p = 0 ( =,, ) R 2 belog to ths complex. However, ths meas the same thg as sayg that the plaar (2 2)-fold exteded mafolds R 2 whose tersecto s (2) fulfll the equatos (Φ µ Φ ) = 0. The mportace of ths relatoshp betwee the blear covarats ad the Posso bracket expressos rests especally upo the fact that, wth o further assumptos, t ca be carred over to ay Pfaffa expresso 2 varables that cludes the ormal form p dx + + p dx. We have see that a system of depedet equatos: (6) Φ (x,, x, p,, p ) = 0 ( =,, ) represets a uo of elemets whe ad oly whe all expressos (Φ µ Φ ) vash due to (6). Sce each system of equatos that s equvalet to (6) represets the same uo, t must possess ths property, ad, partcular, t must the be true for every system of equatos that follows by solvg (6). We would lke to assume that (6) ca be solved for exactly m of the p, so t ca take o the form: p + ( x,, x, p,, p ) = 0 (µ =,, m), ϕ µ µ m+ χ k (x,, x ) = 0 (k =,, m), whch we uderstad,, to mea ay permutato of the umbers,,. If ths ow lets us derve a relato betwee just the x,, x from the equatos χ m k = 0 perhaps:

11 Le s varat theory of cotact trasformatos ad ts extesos x + ω ( x,, x ) = 0, m m the the equatos of the uo ca take o such a form that the two equatos p m + ϕ m = 0, x m + ω = 0 emerge. However, from the equatos of the uo, the left-had sde of the expresso: ( m p + ϕ m, x + ω) = m must the vash, whch s mpossble. Thus, the quattes x m+,, x caot be elmated from the m equatos χ k = 0, ad the equatos of our uo or, deed, ay uo of elemets ca the be put to the form: (22) p + ϕ ( x,, x,,, ) 0 p m p µ µ m = + x + χ (,, ) 0 m k x x + = m (µ =,, m; k =,, m). Here, however, the bracket expressos o the left-had sdes are fuctos of oly the x,, x, p m m+,, p ad must the be detcally zero, sce they must vash due to (22). Whe two fuctos ϕ ad χ make the bracket expresso (ϕ χ) vash detcally, oe says that they le voluto. We ca the also express our result the form: The equatos of a uo of elemets ca always be put to such a form: Ω (x,, x, p,, p ) = 0 ( =,, ) that the fuctos Ω,, Ω le parwse voluto. Ths shows that there are systems of depedet fuctos of x, p that le parwse voluto. If X,, X s such a system the the equatos: (23) X (x,, x, p,, p ) = a ( =,, ) always represet a uo of elemets whose values are also gve by a,, a. Oe the has a famly of uos ad elemets, ad t s clear that the space of 2 elemets x, p s dvded to uos of elemets by meas of equatos (23), such that ay elemet x, p belogs to oe, ad geerally oly oe, of these uos. Coversely, f oe kows that equatos (23), whch the X are depedet fuctos, represet othg but uos the oe ca fer that the X le parwse voluto f the expressos: (X µ a µ, X a ) = (X µ X ) must always vash for arbtrary a, due to (23), whch s possble oly whe they vash detcally.

12 Le s varat theory of cotact trasformatos ad ts extesos 2 Oe ca, moreover, easly form the most geeral system of equatos (23) that represets uos of elemets. Oe eeds oly to choose the fuctos ϕ ad Ω equatos (3) ad (4) to be fuctos of l varables u,, u l ad parameters a,, a the most geeral way that makes equatos (3), (4) soluble terms of u,, u, a,, a. If equatos (23) represet uos of elemets for arbtrary choces of the costats a, ad f Φ,, Φ m (m < ) are arbtrary fuctos that are depedet oly of each other ad the X the obvously the equatos: X = a,, X = a, Φ = b,, Φ m = b m also represet uos for arbtrary values of the a, b, ad deed, uos of m elemets. Thus, there are certa systems of equato of the form: (24) F (x,, x, p,, p ) = a ( =,, + m, 0 m ) that determe uos for arbtrary values of the values a. The space of 2 elemets x, p wll be subdvded by such a system of equatos to a famly of +m uos of m elemets such a way that each elemet x, p belogs to oe, ad geerally oly oe, of these uos. Let ψ,, ψ m be fuctos of the x, p that are depedet of each other ad the F,, F +m. If we the set: (25) ψ k (x,, x, p,, p ) = u k (k =,, m) the equatos (24), (25) may be solved for the x, p, ad we obta a ew represetato of our +m uos from ths soluto: (26) x = Φ( u,, u m, a,, a+ m) p = X ( u,, u m, a,, a+ m) ( =,, ), whch the u,, u m are to be regarded as depedet varables. The system of equatos (26) may thus be obvously solved for the u ad a, ad thus aga delvers equatos (24) ad (25). Sce equatos (24) represet uos for arbtrary values of the a, the expresso X dφ represets a complete dfferetal the varables u, so oe has: (27) Φ Ω( u,, u, a,, a ) du m m m + m X duk = k= uk k = uk k. If we make the substtuto a = F, u k = y k ths detty, whch we would lke to suggest by eclosg them square brackets, ad mage that [Φ ] x, [X ] p the we obta a detty the varables x, p of the form:

13 Le s varat theory of cotact trasformatos ad ts extesos 3 Φ + m pdx p df = = = a + m Ω d[ Ω] df = a, or, whe we set: (28) a detty: (29) [ Ω ] = ω( x,, x, p,, p) Φ Ω p = f ( x,, x, p,, p ), = a a + m f df = pdx + dω, = whch clearly expresses the fact that equatos (24) represet a famly of +m uos. Here, the fucto Ω s determed by (27) up to a arbtrary, addtve fucto ϑ of a,, a +m, so oe ca replace ω wth ω +ϑ(f,, F +m ), from whch (29) assumes the form: m ϑ f + df pdx + d(ω + ϑ). = F = It s also easy to see that ths we have foud the most geeral form for ths detty the form of (29). Namely, f: the: + m = + m = f df p dx + dω = ( f f ) df d( ω ω) s therefore equal to a complete dfferetal, ad sce the F are depedet fuctos of the x, p, ω ω s a fucto ϑ of F,, F +m ; hece: ϑ ( F,, F ) f f = + m F ( =,, +m). If oe has foud ω by the aforemetoed quadrature, to whch certa elmatos must geerally be added, the oe fds the f from the 2 lear equatos: F f = p + x + m = F f = p + m = ω x, ω p,

14 Le s varat theory of cotact trasformatos ad ts extesos 4 to whch (29) separates. Amog these equatos, whch are certaly compatble wth each other, there are exactly + m mutually depedet oes, due to the depedece of the F. Thus, f equatos (24) represet uos of m elemets for a arbtrary choce of the a the there always exsts a detty of the form (29), where ω must be foud by a quadrature, whle the f are determed after dscoverg ω by lear equatos. ) If we costruct the blear covarats from the two sdes of the detty (29), whch are furthermore detcally equal, for self-explaatory reasos, the we obta the ew detty: (30) I ths, f we set: + m ( dfδ F δ f df ) ( dpδ x dxδ p ). = = ϕ ϕ δx = δ t, δp = δ t, p x whch ϕ s uderstood to mea a arbtrary fucto, the ths yelds: + m dϕ, = (3) (( ϕf ) df ( ϕ f ) df ) ad from ths, oe further obtas by the substtuto: the detty: χ χ dx = dt, dp = dt, p x + m (32) (( ϕf )( χ f ) ( ϕ f ) ( χ F )) { } = dϕ. Coversely, f (32) s true detcally for all fuctos ϕ ad χ the obvously (30) s also fulflled detcally, ad there thus lkewse exsts a detty of the form (29). If we ow assume, partcular, that m = 0 the we cosder the case whch equatos (24), or, as we would lke to ow wrte them: (24 ) X (x,, x, p,, p ) = a ( =,, ), represet a famly of uos of elemets the all (X µ X ) 0. If we the wrte the detty (29) the form: (29 ) P dx p dx + dω = = ) Cf., Le, Math. A., Bd. XI, pp. 465, et seq.

15 Le s varat theory of cotact trasformatos ad ts extesos 5 the for ϕ = X µ ad ϕ = P µ the detty (3) delvers ths oe: (33) ( P X µ ) dx dx µ = { ( P X ) dp ( P P ) dx } dp = µ µ µ (µ =,, ). From ths, however, t ext follows from the depedece of the X that: (P X µ ) = ε µ, where ε µ = 0 or, accordg to whether µ or µ =, resp., so oe has, however: (P µ P ) 0. Fally, f we replace of the dx, dp (29 ) wth the expressos that we just employed the for ay arbtrary fucto χ, oe has: P ( χ X ) = = p χ + (χ ω). p We the have the well-kow: Theorem: If X,, X are depedet fuctos of x,, x, p,, p that are parwse voluto or, what amouts to the same thg, f the equatos X = a ( =,, ) represet uos of elemets for arbtrary a, the there exsts a detty of the form: (29 ) P dx p dx + dω, = = where ω s foud by a quadrature, whle the P were obtaed by solvg lear equatos. Betwee the fuctos X, P, ad ω there thus exst the relatos: (34) (X X ) = 0, (P X ) = ε, (P P ) = 0, (, =,, ) ad: (35) X ( ωx ) = pµ µ = pµ P ( ωp ) = pµ P µ = pµ ( =,, ), ad addto, there s the detty:

16 Le s varat theory of cotact trasformatos ad ts extesos 6 (32 ) {( P ϕ)( X χ) ( X ϕ)( P χ) } = whch the ϕ ad χ may also be fuctos of the x, p. ) (ϕ χ), From the exstece of the relatos (34), t follows, moreover, that all 2 fuctos X,, X, P,, P are depedet of each other. Namely, f oe forms the square of the fuctoal determat of the X, P relatve to the x, p, whch oe wrtes these determats the two forms: P P X X p p x x X X P P x x p p ad multples the two together the oe obtas a determat that possesses the value, due to (34). Coversely, f there exsts a detty of the form (29 ) the t s clear that the equatos: X = cost.,, X = cost. represet uos. Were the fuctos X,, X ot depedet of each other the each of these uos would cosst of more tha elemets, whch s mpossble, so we ca coclude that X,, X are depedet of each other ad le parwse voluto. The, however, t lkewse follows that equatos (34), (35), ad (32 ) are vald. Fally, f oe s gve 2 fuctos X,, X, P,, P that satsfy the relatos: (34) (X X ) = 0, (P X ) = ε, (P P ) = 0, (, =,, ) the, as we have see, all of these fuctos are depedet of each other. Oe ca, as a cosequece, express ay arbtrary fucto ϕ of x, p by X,, X, P,, P ad obta: ϕ (36) (ϕ X ) = P, (ϕ P ϕ ) =, ( =,, ). X If oe adds yet a secod fucto χ the oe obtas: ) Stragely eough, the detty (32 ) seems to have ot bee otced up to ow.

17 Le s varat theory of cotact trasformatos ad ts extesos 7 (37) χ χ ( ϕ χ) = ( ϕ X ) + ( ϕ P ) = X P ϕ χ ϕ χ =, = P X X P from whch, we ca also wrte:. = (32 ) (ϕ χ) {( Pϕ )( X χ) ( X ϕ)( Pχ) } However, as we recetly remarked, the detty (30) follows from the exstece of (32) for arbtrary fuctos ϕ, χ so t follows from (32 ) that there s a detty: ( dxδ p dpδ x ) = = ( dx δ P dp δ X ). = I ths, oe sees that the two expressos p dx ad P dx have the same blear covarats, so they dffer oly by a complete dfferetal. As a result, there exsts a detty of the form: (29 ) P dx p dx + dω, = = where the fucto ω satsfes the 2 equatos: (35) X ( ω X ) = pµ, µ = pµ P ( ω P ) = pµ P, µ = pµ through whch, ts 2 dervatves are determed. Thus: Theorem: If the 2 fuctos X, P satsfy relatos of the form (34), the they are depedet of each other, ad there exsts a detty of the form (29 ), where the fucto ω satsfes equatos (35). The blear covarats have already show us ther great utlty the dervato of the equatos (34) from the detty (29 ), ad t wll become eve clearer the proof of the coverse that a detty of the form (29 ) ca be deduced from equatos (34). It was precsely ths proof of the coverse that led Le to such rather crcutous cosderatos. ) ) Cf., Trasformatosgruppe, Bd. II, pp

18 Le s varat theory of cotact trasformatos ad ts extesos 8 We refer to ay trasformato: 4. Cotact trasformatos the x, p. (38) x = X (x,, x, p,, p ), p = P (x,, x, p,, p ), ( =,, ) as a cotact trasformato the x, p whe the Pfaffa expresso p dx remas varat up to a complete dfferetal, so there exsts a relato of the form: (39) p dx = = pdx + dω(x, p). = From (39), t follows that due to (38) a equato of the form: (40) ( dp δ x dx δ p ) = = = ( dp δ x dx δ p ) exsts, that, tur, mples oe of the form (39). The cotact trasformatos the x, p ca therefore also be defed as the group of all trasformatos the x, p that leave the blear covarat of the Pfaffa expresso p dx varat. Lkewse, t s clear that our cotact trasformatos take each elemet x, p, alog wth two ftely close oes x + dx, p + dp ad x + δx, p + δp that t s uted wth, to aother such elemet, ad each uo of elemets to a uo, addto. If (38) s a cotact trasformato the x, p the a detty of the form (29 ) exsts, ad t follows that the fuctos X, P, ω are coupled by the relatos (34) ad (35). Coversely, however, from pp.?, the equatos (34) mply the depedetly of the 2 fuctos X, P ad the exstece of a relato of the form (29 ), where ω satsfes equatos (35). Thus, equatos (38) represet a cotact trasformato the x, p whe ad oly whe they satsfy equatos (34). However, as we just sad, the detty (37) follows from (34), ad thus, the equato: ϕ χ ϕ χ = p x x p = ϕ χ ϕ χ = p x x p, or, wrtte more brefly: (4) (ϕ χ) xp = (ϕ χ) x p. The cotact trasformatos the x, p the leave the Posso bracket expresso varat. Coversely, ay trasformato the x, p uder whch the Posso bracket expresso remas varat s obvously a cotact trasformato the x, p. The group of all cotact trasformatos the x, p ca therefore also be defed by the varace of ths bracket expresso, whch, from the relatoshp of ths expresso to the blear covarat, would ot be surprsg. Now, f:

19 Le s varat theory of cotact trasformatos ad ts extesos 9 or: δx = ξ δt, δp = π δt, ( =,, ) Xf = f ξ f + π = x p s a ftesmal trasformato the oe has: so: or: δ( p dx ) = du(x, p) δt, = ( p dξ + π dx ) = du, ( π dx ξ dp ) = d(u p ξ ). = If we the set u = p ξ = U the we wll have: U ξ =, π = p from whch: (42) Xf = (Uf). U x, The fucto U ca obvously be chose arbtrarly here, ad oe has: p. (43) X( p dx ) = d ( pu U ) From the varace of the Posso bracket expresso, t the follows that the fucto U s varatly coected wth the ftesmal trasformato Xf wth respect to ay fte cotact trasformato. We would lke to call U the characterstc that belogs to the ftesmal cotact trasformato ad the remark that U wll be foud by a quadrature from a gve Xf, so oe has:. = (44) du = ( ξ dp π dx ) If we also choose the trasformato (38) to be ftesmal wth the characterstc V the we have: V x = x + t p δ V, p = p t x δ, so for ay fucto f(x, p) oe wll have:

20 Le s varat theory of cotact trasformatos ad ts extesos 20 Now, however, oe has: f = f(x, p ) = f + (Vf) xp δt, f = f (V f ) x p δt. (Uf) xp = (Uf) x p, = (U (V U ) δt, f (V f ) δt) x p = (U f ) x p (U (V U ) x p δt ((V U ) f ) x p δt = (U f) + {(V(U f)) (U(V f)) ((V U)f)} δt, whch the gves the celebrated Jacob detty: (45) (U(V f)) (V(U f)) = ((U V)f), whch s true for arbtrary fuctos U, V, f of the x, p. Ths proof of the detty, whch goes back to Le, obvously caot be replaced wth a coceptually smpler oe. 5. Dfferetal equatos x,, x, p,, p. Now, let there be gve a system of equatos: F k (x,, x, p,, p ) = 0 (k =,, l), ad suppose that all of the uos of elemets that oe fds wll satsfy ths system of equatos, or, more brefly: all of the assocated tegral uos of elemets. The, from pp.?, oe uderstads that these uos all satsfy equatos of the form: (F k F j ) = 0 (k, j =,, l), as well. If oe fds o cotradcto from the costructo of these equatos ad the oes that follow from them the oe ultmately fds that the problem that we posed always comes dow to the other oe, of fdg all tegral uos of elemets for a system of the form: (46) F µ (x,, x, p,, p ) = 0 (µ =,, l), where the (F µ F ) all vash, due to (46). However, oe calls such a system of equatos a m-parameter system voluto. If equatos (46) defe a m-parameter system voluto that sese the ay equvalet system of equatos Φ = 0,, Φ m = 0 has the same property. I fact, the R 2 of the dx, dp the equatos df = 0,, df m = 0 represet a (2 m )-fold exteded plaar mafold E 2 m, ad deed t represets the tersecto of m (2 2)-fold exteded plaar mafolds. If we restrct ourselves ow to such elemets x, p that satsfy (46) the (F µ F ) = 0 (m, =,, m); that s, ay two plaar (2 2)- fold exteded mafolds u, v ad u, v that go through E 2 m always satsfy the

21 Le s varat theory of cotact trasformatos ad ts extesos 2 equato: ( vu uv ) = 0. O the other had, the equatos: dφ = 0,, dφ m = 0 represet the same mafold E 2 m as oly the tersecto of m other (2 2)-fold exteded plaar mafolds; thus, alog wth the assumptos that oe makes o x, p, oe must also requre that all (Φ µ Φ µ ) vash. If we ow thk of the system of equatos (46) the oe fds, as o pp.?, that the soluto ca be obtaed the form: x + ϕ ( x,, x ) 0, λ λ l = + m (46 ) (λ =,, l; k =,, m l), p + χ (,,,,,,,, ) 0, l k k x x l p m p l p m p m+ = where,, mea ay permutato of the umbers,,. Here, however, the bracket expressos o the left-had sdes are free of x,, x,,, p ad must vash detcally, sce, by vrtue of (46 ) they caot vash. (?) Thus, ay m-parameter system voluto ca be brought to the form: (46 ) Ω µ (x,, x, p,, p ) = 0 (µ =,, m), such that the fuctos Ω,, Ω m le par-wse voluto. If oe replaces the system voluto (46) wth ts solved form (46 ) the geerally ay possble tegral uo of elemets that makes the fuctoal determat: l p l+ m D = F F F F x x p p l l + m l l+ m drops away. Sce, however, these tegral uos satsfy the equatos: F = 0,, F m = 0, D = 0, ther determato comes dow to the tegrato of a at least (m + )-parameter system voluto, ad oe ca say, wth o loss of geeralty, that the determato of the tegral uo of elemets of a gve system of equato ca always come dow to the ormal problem: Itegrate a system voluto of the form (46 ) whe the fuctos Ω,, Ω m le par-wse voluto. What ths ormal problem addresses, we would lke to satsfy ourselves here wth provg that t possesses complete solutos so the 2 m elemets that satsfy (46 ) ca always be arraged to m uos of elemets. All tegral uos ca be foud from just such a complete soluto wthout tegrato. It the comes dow to the addto of equatos: Ω m+k (x,, x, p,, p ) = a k (k =,, m)

22 Le s varat theory of cotact trasformatos ad ts extesos 22 to the equatos (46 ), such that the Ω m+k are depedet of Ω,, Ω m ad each other, ad lkewse le voluto wth the Ω,, Ω m ad each other. Now, however, the m equatos: (47) A µ f = (Ω µ f) = 0 (µ =,, m) are obvously depedet of each other, ad from: A µ A f A A µ f = (Ω µ (Ω f)) (Ω (Ω µ f)) = (Ω µ Ω ) f) 0 they defe a m-parameter complete system. Oe the fds a fucto Ω m+ whe oe seeks a soluto of ths complete system that s depedet of Ω,, Ω m, such that oe obtas Ω m+ by determg a soluto of the (m + )-parameter complete system: (Ω f) = 0,, (Ω m+ f) = 0 that s depedet of Ω,, Ω m, ad so forth. We further meto that the tegrato of the system voluto (46 ) the case of l > 0 ca always be coverted to the tegrato of a (m l)-parameter system voluto 2( l) varables. If oe sets: (48) x λ = x λ + ϕ λ (λ =,, l) the oe coverts the Pfaffa expresso p dx to: If oe the sets: (49) l l p dx dϕ p dx λ λ λ λ= k = ( ). l k l k x l + k = x ( k =,, l) l+ k p = p ( λ =,, l) λ λ l ϕµ p p (,, ) l j p l j j m l + = + µ = µ = x l+ j p = p (,, ) m τ = m + τ m+ τ the equatos (48) ad (49) collectvely represet a trasformato uder whch the Pfaffa expresso p dx deed remas varat, ad s certaly a cotact trasformato. Sce the bracket expresso (ϕ χ) remas varat uder t, t s clear that the ew form: x λ = 0, p χ (,,,,,,,, ) l k k x x l p p p l p + + m+ l + = 0 (λ =,, l, k =,, m l),

23 Le s varat theory of cotact trasformatos ad ts extesos 23 whch cludes the volutve system (46 ), s aga a m-parameter system voluto. From ths, however, t follows that the χ k are free of p,, p such that the equatos: (,,,,, ) p + χ x x p p = 0 (k =,, m l), l + k k l+ m+ whch defe a (m l)-parameter system voluto the 2( l) varables x l+ k, p l+ k, whose tegrato ca be ferred from that of the volutve system (46 ). If l = m the the determato of the -fold exteded tegral uo of (46 ) requres o tegrato whatsoever. I the ew varables, (46 ) actually takes o the form: x λ = 0, (λ =,, l). The Pfaffa expresso p dx thus reduces to: l p dx l+ k, l+ k k = ad all that remas s to determe all uos of l elemets the resdual 2 2l varables. However, that s a feasble operato. 6. The varat theory of cotact trasformatos the x, p. We have see that the tegrato of a system of equatos the x, p ca always be coverted to the tegrato of a system voluto, but the, tur, ths ca lead to oe lookg for solutos of a sequece of complete systems. Thus, each of these complete systems has the form: (47) (Ω µ f) = 0 (µ =,, m), where the fuctos Ω,, Ω m are depedet of each other ad le par-wse voluto. If oe ow happes to fd ot merely oe soluto to oe of these complete systems, but several of them, the ths rases the questo of how oe ca best explot ths stuato for the resoluto of the tegrato problem. By the very fact that he posed ths questo, Le was duced to develop hs varat theory of cotact trasformatos. From the form (47) of the complete solutos, ad from the varace of the Posso bracket symbols uder cotact trasformatos the x, p, t emerges that all of the complete systems that appear (47) are varatly lked wth the orgal system of equatos x, p that s to be tegrated by meas of cotact trasformatos. If oe the kows several solutos of a such a system (47) the the questo arses of what propertes the totalty of all the kow solutos of the system (47) mght possess wth respect to all cotact trasformatos the x, p. If oe kows for the system (47), ot just the self-explaatory solutos Ω,, Ω m, but also a umber of other oes u,, u l that are depedet of each other ad the Ω µ the, frst of all, absolutely ay arbtrary fucto of Ω,, Ω m, u,, u l s lkewse a soluto, ad secodly, the Jacob detty:

24 Le s varat theory of cotact trasformatos ad ts extesos 24 ((Ω µ ϕ)f) ((Ω µ f)ϕ) (Ω µ (ϕ f)), shows that alog wth ϕ ad f, (ϕ f) s lkewse always a soluto. That s, fact, the celebrated Posso-Jacob theorem. Therefore, all of the expressos: (Ω µ Ω ), (Ω µ u k ), (u k u j ) are also solutos of the complete system. Of these solutos, clearly the (Ω µ Ω ) ad (Ω µ u k ) are detcally zero, but the other oes (u k u j ) are possbly ew. If oe adds the ew solutos that cluded amog the expressos (u k u j ).e., the oes that are depedet of Ω,, Ω m, u,, u l ad each other to u,, u l, oce aga apples the Posso-Jacob theorem, ad proceeds that maer the oly two cases are possble: Ether oe fds 2 m depedet solutos of (47), ad therefore, all of the oes that are preset, or oe fds so may ew solutos u l+,, u r that deed m + r < 2 m, but all of the (u k u j ) (k, j = l,, r) ca be expressed terms of Ω,, Ω m, u,, u r. I the frst case, the tegrato of the system voluto: Ω = a,, Ω m = a m, requres oly feasble operatos, whch have geerally bee kow for a log tme for m =, but were frst exhbted by Le a theorem, upo whch, the exteso that he gave of the Cauchy tegrato method rests. I the secod case, a umber of solutos of (47) are stll ukow, ad oe the tres to take advatage of the solutos that oe fds as much as s possble; for that, t s eve ecessary to establsh the varat propertes that the totalty of all kow solutos, ad therefore the totalty of all fuctos of Ω,, Ω m, u,, u r, possess uder all cotact trasformatos of the x, p. The system of fuctos Ω,, Ω m, u,, u r that oe arrves at here has the property that the bracket expresso of ay two fuctos of the system s expressble terms of fuctos of the system aloe. However, t s a completely specal system of ths type, because Ω,, Ω m ad u,, u r are voluto wth each other. It s a closely related problem the for us to lkewse cosder completely geeral systems of r depedet fuctos u,, u r of x, p that are arraged so that relatos of the form: (50) (u u k ) = ω k (u,, u r ) (, k =,, r) exst. The totalty of all fuctos of the fuctos u,, u r of such a system s what Le called a r-parameter fucto group Ω,, Ω m, u,, u r. The sgfcace of hs bref paper of 872 that was metoed the troducto cossts of the fact that all varat propertes that such a r-parameter fucto group possesses relatve to the group of all cotact trasformatos were establshed t. Ths s ot the place to develop the varat theory of a r-parameter fucto group, sce that would be essetally a repetto of the presetato that Le gave the secod volume of Trasformatosgruppe. I thus cotet myself wth the followg remarks: The fudametal theorem of the theory s that the r mutually depedet equatos: (5) (u k f) = 0 (k =,, r) defe a r-parameter complete system wth 2 r depedet solutos ad that the totalty of all solutos of (5), ad thus, the totalty of all fuctos of v,.., v 2 r, defe

25 Le s varat theory of cotact trasformatos ad ts extesos 25 a (2 r)-parameter fucto group, amely, the group v,, v 2 r that s recprocal to the group u,.., u r. The two fucto groups of combed fuctos are sde of each group of fuctos of the group that s voluto wth all of the fuctos of the group; they are called the dstgushed fuctos of each group. The umber of parameters r of a fucto group ad the umber m of mutually depedet dstgushed fuctos that the group cludes are the oly two varat propertes of the group uder all cotact trasformatos. Two fucto groups that are both assocated wth the same umbers r ad m ca always be mapped to each other by cotact trasformatos the x, p. The proof of ths theorem led Le to show that ay r- parameter fucto group ca be brought to a caocal form. If we, wth S. Kator, call ay system of r depedet fuctos of a r-parameter fucto group a bass for the fucto group the we ca also say: Oe ca determe a caocal bass for ay r- parameter fucto group, whch s the r depedet fuctos: X X, P P, X + X +m (2 + m = r) that belog to the group ad satsfy the caocal relatos: (52) (X X k ) = 0, (P X k ) = ε k, (P P k ) = 0. Therefore, the fuctos of X +,, X +m are dstgushed fuctos of the group; t the happes that the dfferece betwee the umber of parameters r ad the umber of depedet dstgushed fuctos s always eve. By pursug the varat theory of fucto groups x, p, Le was the a posto to establsh whch varat propertes a arbtrary gve system of fuctos the x, p: ϕ k (x,, x, p,, p ) (k =,, m) possesses uder all cotact trasformatos. Oe ca brefly express the result that oe arrves at as follows: To ϕ,, ϕ m, oe adds all fuctos: (ϕ ϕ k ), ((ϕ ϕ k ), ϕ j ), ((ϕ ϕ k ) (ϕ j ϕ l )),, such that whe oe forms the bracket expresso of ay two gve fuctos, t gves oly those fuctos that ca be expressed terms of oly the gve fuctos. I ths way, oe arrves at a system: ϕ,, ϕ m, ϕ m+,, ϕ r, whose fuctos do ot eed to be mutually depedet, but has the property that all (ϕ ϕ k ) are expressble terms of the ϕ,, ϕ r aloe. All of the varat propertes of the system ϕ,, ϕ m wll the be represeted by the totalty of all relatos that exst betwee the ϕ,, ϕ m ad (ϕ ϕ k ) (, k =,, r). I other words: If χ,, χ m s a secod fucto system the there s a cotact trasformato the x, p that takes the χ,, χ r the sequece to ϕ,, ϕ r f ad oly f the followg requremet s fulflled:

26 Le s varat theory of cotact trasformatos ad ts extesos 26 To the χ,, χ m, oe adds the expressos (χ χ k ), (χ χ k ) χ j ),, the same sequece that oe defes for the ϕ, ad deotes the correspodg umbers as χ m+,, χ r. Therefore, the same relatos must exst betwee χ,, χ m ad all (χ χ k ), (, k =,, r) that exst betwee the correspodg quattes ϕ,, ϕ r ad all (ϕ ϕ k ). 7. Other treatmets of the theory of fucto groups. Kator s geeralzato of the problem. Sce a r-parameter fucto group wth the bass u,, u r cossts of the totalty of all fuctos of u,, u r, ths suggests that stead of defg the group terms of such a bass, oe regards, the (2 r)-parameter complete system whose most geeral soluto s a arbtrary fucto of u,, u r as gve. The dfferece betwee these two vewpots s precsely the same as whe oe, o the oe had, operates wth the roots of a algebrac equato, whle, o the other had, oe regards oly the algebrac equato as gve. I ay evet, t seems desrable to also treat the theory of fucto groups from ths ew vewpot. Le hmself has occasoally assumed ths vewpot. For example, he already showed 877 that whe a arbtrary complete system s preset, oe ca always preset the complete system such a way that ts solutos cosst of all fuctos that are voluto wth the solutos of the gve complete system. ) I partcular, whe a fucto group s defed by a complete system, oe ca the always preset a complete system that defes the recprocal fucto group. By cotrast, Le dd ot geerally go to the questo that he posed more detal aywhere. S. Kator frst placed hmself at the vewpot of the prevously-metoed papers as a foudato, ad defed the fucto groups through complete systems, ad the took that as a excuse to geeralze the etre theory a extraordary way. We must cotet ourselves wth just a few remarks here. It s kow that there exsts a correspodece betwee systems of Pfaffa equatos ad systems of lear, homogeeous, partal dfferetal equatos of frst order. I varables, ay m-parameter system of the frst type s coversely assocated wth a ( m)-parameter of the secod type, ad deed ths s lkewse equvalet to whether the system questo s or s ot a tegrable or complete system. I the space x,, x, p,, p, oe ca add aother type of recprocty to ths correspodece that s determed by the blear covarat: = ( dx δ p dp δ x ) of the Pfaffa expresso p dx, or also through the assocated covarat: ) Neue Itegratos-Methode der Moge-Ampèresche Glechug, Archv for Math. og Naturvd. Bad II, pp. 4.

27 Le s varat theory of cotact trasformatos ad ts extesos 27 plae coordates. I fact, let: = ( v u u v ) (53) ( αkdx + βkdp ) = 0 (k =,, m) = be a m-parameter Pfaffa system, ad let: (54) f f ρ j + σ j = x p = 0 (j =,, 2 m) be the assocated (2 m)-parameter system of lear, partal dfferetal equatos, such that betwee the fucto α, β, ρ, σ, there exst the m(2 m) dettes: (55) ( αkρ j + βkσ j ) = 0 (k =,, m, j =,, 2 m). = If oe ow mages that the ρ j, σ j are trasformed lke the pot coordates dx, dp, the α k, β k, ad the dervatves of f wth respect to the x ad p are trasformed lke the plae coordates u, v the oe recogzes that the form ( vu uv ) of the system (53) s assocated wth a covarat m-parameter system of lear, partal dfferetal equatos: (53) f f βk αk = x p = 0 (k =,, m), ad the form (dx δp dp δx ) of the system (54) s assocated wth a covarat (2 m)-parameter Pfaffa system: (54 ) ( σ jdx ρ jdp ) = 0 (j =,, 2 m). = Lkewse, t s clear that oe also obtas the system (53 ) whe oe subjects the dervatves of ϕ the equato: to the relatos: (54) ϕ f ϕ f = p x x p = 0 ϕ ϕ ρ j + σ j = x p = 0 (j =,, 2 m),

28 Le s varat theory of cotact trasformatos ad ts extesos 28 whle stll regardg these dervatves as arbtrary. Oe also sees, a correspodg way, that the system (54 ) emerges from (53) by the use of the equato (dx δp dp δx ) = 0. That s the geeral recprocty betwee the systems (53) ad (54 ) ad the assocated systems (54) ad (53 ) that S. Kator frst proved. If the system (54) possesses a soluto u such that u s lkewse a tegral fucto of the Pfaffa system (53) the there s a multpler χ k such that: m χ ( α dx + β dp ) du. k k k k = = The, however, oe wll obvously have: m f χ β α f k k k k = = x p (u f); that s, the system (53 ) cludes the equato (u f) = 0. Coversely, f (53 ) cludes a equato of the form (u f) = 0 the u s a tegral fucto of (53), ad therefore a soluto of (54). If the system (54) possesses two solutos u, u 2 the (53 ) cludes the two equatos (u f) = 0, (u 2 f) = 0, ad, whe t s, moreover, complete, t cludes the equato: (u (u 2 f)) (u 2 (u f)) ((u u 2 ) f) = 0. as well. Ths comes from the fact that the solutos of (54), ay case, defe a fucto group whe the recprocal system (53 ) s complete. Should the system (54) defe a m-parameter fucto group partcular, the t must have m depedet solutos u,, u m, so t must be complete, ad, addto, every (u µ, u ) (µ, =,, m) must be a soluto. The, however, the m-parameter recprocal system (53 ) cludes the m depedet equato (u µ f) = 0 (µ =,, m), ad lkewse, every equato: ((u µ u ) f) = (u µ (u f)) (u µ (u f)) = 0 (µ, =,, m), so t s lkewse complete. Oe ca, however, coclude that (54) defes a m-parameter fucto group whe ad oly whe the recprocal systems (54) ad (53 ) are both complete. These crtera were already foud by S. Kator. If (54) has a soluto u ad (53 ) has a soluto v the (53 ) cludes the equato (u f) = 0, ad (54) cludes the equato (v f) = 0, ad oe the has (u v) = 0. Therefore, (53 ) s the system of equatos that Le already taught us to address, whose solutos are all fuctos that are voluto wth (54). Thus, t wll geerally be assumed that (54) s a (2 m)-parameter complete system.

29 Le s varat theory of cotact trasformatos ad ts extesos 29 If the system (54) s complete ad defes a m-parameter fucto group the oe ca, as Le showed, determe a caocal bass X,, X l+h, P,, P l (2l + k = m) for ths fucto group, for whch the caocal relatos: (52) (X X k ) = 0, (P X k ) = ε k, (P P k ) = 0 exst. The complete system (53 ) that the recprocal group defes ca the take the form: (X k+l f) = 0, (P l+h+j f) = 0, (k =,, l; j =,, l h). S. Kator geeralzed ths to the case where the system (54) s completely arbtrary. He called two equatos: f f ρ + σ x p = 0, ρ + σ x f f p = 0 cojugate whe the covarat ( ρσ σ ρ ) vashes. Thus, f the equatos (54) are ot par-wse cojugate the oe ca choose a equato: (56) f f ρ + σ x p = 0 from ths system that s ot cojugate to all equatos of the system, ad ca the determe a equato: (57) f f ρ + σ x p = 0 of the system such a way that oe has ( ρ σ σ ρ ) =. If oe has chose (56) ad (57) that way the oe easly covces oeself that the system (54) cludes precsely 2 m 2 depedet equatos that are cojugate to (56), as well as (57). If oe treats ths ew system just lke the orgal oe (54) ad the proceeds, the oe ultmately obtas a represetato of (54) the form: (58) X f = 0,, X l+h f = 0, P f = 0,, P l f = 0 (2l + h = 2 m), where the covarat ( ρσ σ ρ ) vashes for ay two equatos X f = 0 ad X k f = 0 ad ay two equatos P f = 0 ad P k f = 0, whle for ay two equatos X f = 0 ad P k f = 0 they have the value ε k. Wth S. Kator, we call (58) a caocal bass for the system (54), ad we call X +l f = 0,, X +h f = 0 the dstgushed equatos of the system. The system (53 ), whch s recprocal to (54), cossts of all equatos that are cojugate to all equatos of (54). As a cosequece, the dstgushed equatos X +l f =