The theory of integral invariants is a corollary to the theory of differential invariants

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1 De Theore der Itegralvarate st e Korollar der Theore der Dfferetalvarate, Lepzger. Berchte (897), Heft III, submtted o , pp Preseted at the sesso o ; Gesammelte Abhadluge, v. IV, art. XXVII, pp The theory of tegral varats s a corollary to the theory of dfferetal varats By SOPHUS LIE Traslated by D. H. Delphech. I ths Note, I shall frst show that my geeral theory of dfferetal varats mples the etre theory of tegral varats wth o further assumptos. That wll allow me to clarfy ther depedecy relatoshp, whch ot oly the relevat vestgatos of Zorawsky, Carta, ad Hurwtz ( ), but also some papers of Pocaré, ad especally a Note of Kögs, relate to my older work. As far as I ca see, Carta s the oly oe whose has preseted the matter correctly. At the same tme, however, I would lke to express my thaks to Zorawsky ad Hurwtz for the fact that they have, ay evet, referred to the relatoshp wth my vestgatos ther ow work. As far as the work of Kögs s cocered, I must assume that my paper that goes back to the begg of the year 877 the Norwega Archv, Bd. II: De Störugstheore ud de Berührugstrasformatoe [ths collecto, v. III, art. XX] has escaped hs atteto. I. Geeral defto of the cocept of tegral varat. 2. If we have bee gve a fte or fte cotuous trasformato group the varables: ( q ) x,, x, x,, x,, x,, x, z,., z, m z,., z ( q ) m,, z,, z ( q ) the we wll say that a tegral: ( ) Zorawsky, Akademe zu Krakau, Aprl 895. Kögs, Comptes redus, Pars 9 Dec Carta, Bullet de la Socété math., Pars 896. A. Hurwtz, Gesellsch. d. Wss. zu Göttge, March 897. Pocaré, Acta math., t. 3, 890. Sophus Le, Norweg. Archv, Bd. II, 877, ad Theore der Tras., Lepzg , Bd. I ad III; Ges. d. Wss. zu Chrstaa 883 [ths collecto, v. III, art. XX, ad art. XLI]

2 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 2 2 z z Ω x,, x,, x, z,, z,,,, dω m 2 (,,,, ( q dω = dx ) dx dx ) ( q ) represets a tegral varat of the group questo whe the varato of the tegral: δ Ω( ) dx dx vashes for all ftesmal trasformatos of the group other words, whe the form of the tegral s preserved for all fte trasformatos of the group. ( ) I ths, we have assumed that each z ( ) ca be cosdered to be a fucto of x, ( ) 2 k ( ) x,, x ( ), ad furthermore that Ω cludes ot just the x ad z, but also hgher-order dervatves of the z wth respect to x, ad fally that the quattes x (), z () trasform amogst themselves. The fact that both of the deftos of the cocept of a tegral varat that wll be formulated here are equvalet s a specal case of my geeral theorem that every fte trasformato of a arbtrary (fte or fte), cotuous group of trasformatos ca be replaced by the composto of a sequece of ftely-may ftesmal trasformatos of the group. 3. I wll ow show that my geeral theory of dfferetal varats acheves the determato of all tegral varats that are preset every dvdual case mmedately because t wll covert the problem to the tegrato of a complete system. I order to make the valdty of ths oteworthy geeral theorem clear to the reader, I fd t coveet to frst cosder a example that I have log sce treated thoroughly. Namely, I wll frst recall how I approached the questo at the tme of whether a gve group of pot trasformatos of space x,, x leaves a dfferetal expresso: H (x,, x, dx,, dx ) varat that s homogeeous of frst order the dfferetals dx,, dx, ad thus represets a elemet of arc legth, the Rema sese of the word. If ay such expresso H s gve the: dx2 dx H x,, x,,,, dx dx dx wll obvously be a tegral varat of the group questo.

3 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 3 II. Trasformato groups that leave a arc legth varat. 4. Already the year 886 ( ), I cocered myself detal wth all groups of trasformatos: x k = f k (x,, x, a,, a r ) (k =,, ) for whch a fucto: H (x,, x, dx,, dx ) that s homogeeous of frst degree the dfferetals dx,, dx remas varat. I chap. 25 of the frst volume of my Theore der Trasformatosgruppe ad chaps. 22, 23 of the thrd volume (888 ad 893, resp.), I gave the most elemetary presetato of ths specal theory that s possble, whch s cotaed mplctly as a specal case my geeral theory of dfferetal varats. I wll ow permt myself to summarze the developmet that I gave the aforemetoed place. 5. If X f,, X r f, where: X k f = ξk ( x,, x ), are r depedet ftesmal trasformatos of a r-parameter group, ad f: ξk X k f = ξk( x,, x) + x ν (k =,, r), ν are the ftesmal trasformatos of the oce-exteded group the the r + ftesmal trasformatos X f,, X r f, ad: ν X 0 f = = x wll also geerate a group ther ow rght, sce the r expressos: X 0 X f X X 0 f wll all vash detcally. However, two cases ca come about here: It s, at frst, cocevable that X 0 f ca be represeted as a sum of the X f, whe they are multpled by sutable quattes: ( ) Cf., Lepzger Berchte 886, pps. 34, 342; 890, pps. 292, 293. Grudlage der Geometre [ths collecto, v. II, art. V, at the ed; art. VI, 2].

4 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 4 () X 0 f = ϕ X f + + ϕr X r f. I that case, all of the varat dfferetal expressos: Ω (x,, x, dx,, dx ) that rema varat uder the r-parameter group X f,, X r f wll be of order zero the dfferetals dx,, dx. There wll the exst o varat dfferetal expressos of frst order the dx, ad therefore o varat arc legth, ether. The curves space x,, x wll the have o arc legth that s varat uder the group. By cotrast, f o relato of the form () exsts the the trasformatos X f,, X r f, ad X 0 f wll geerate a (r + )-parameter group the varables x,, x, x,, x, ad t possble, moreover, to fd solutos of the equatos: (2) X f = 0,, X f = 0 0 ϕ x 0 = X f + ϕ x + ϕ Yf ; ϕ r the X f,, X r f, ad Yf wll obvously geerate a (r + )-parameter group x,, x, x,, x, ad ϕ. If Φ(x,, x, x,, x, ϕ) s a arbtrary soluto of the complete system (2), ad f the equato: yelds: Φ(x,, x, x,, x, ϕ) = cost. ϕ = ϕ(x,, x, x,, x ) upo solvg t the ϕ wll be a dfferetal varat of the group X f,, X r f, that s homogeeous of frst degree the x k. Moreover: ds = ϕ (x,, x, dx,, dx ) s the a arc legth that s varat uder the group X f,, X r f. Therefore: dx ϕ ( x,, x, x,, x ) x s a tegral varat of our group. Obvously, all tegral varats of our group that relate to a curve space x,, x wll be foud to have degree oe. I preseted the matter a essetally more geeral way the cted place volume three of my Trasformatosgruppe. O the oe had, I preseted the theory a form that was geeral eough that the fte groups were dealt wth, but o the other had, t was suggested that I had carred out the determato of all tegral varats.

5 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 5 III. A geeral theorem o tegral varats. 6. As we dd I, we would lke to assume that we are gve a fte or fte cotuous trasformato group the varables: ( ) x,, x ( ), ( ) ( j) z,, z ( j ) (, j =, 2,, q), ( j) m ad that we are seekg all of the assocated tegral varats: J = z Ω x,, x, z,, z,,, dx dx m ; t s therefore our assumpto ths that Ω cludes ot just the depedet varables x ad ther fuctos z, but also the assocated dervatves of order oe, two,, up to v. If we ow set: dx dx dx dx dω, for brevty, ad correspodgly: the the varato: 2 J = Ω dω, δ Ω dω = ( δ Ω dω + Ω dδω) must vash for all ftesmal trasformatos of our group. From the rules of the calculus of varatos, order for ths to be true, t s ecessary ad suffcet that the equato: δ (Ω dω) = δ Ω dω + Ω δ dω = 0, or the equvalet oe: X (Ω dω) = X Ω dω + Ω X (dω) = 0, must always be true, o matter whch ftesmal trasformato Xf of the group we also choose to be the varato operator. Now, f: Xf = ξ ( x, z) + ζ ( x, z) z s the geeral symbol of a ftesmal trasformato of our group the, from a kow theorem of hydrodyamcs (whch we shall derve later o a group-theoretc way), moreover, oe wll have: X (dω) = dω ξ,

6 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 6 whch oe must observe that every dervatve ( ξ : ) the calculato, lke the z ad ther dervatves wth respect to the x, should be regarded as fuctos of x. Now, f: Ω dω ad Ω dω 2 are two such tegral varats, ad correspodgly, the two relatos: ξ X Ω + Ω dω = 0, ξ X Ω + Ω 2 2 dω = 0 are verfed for each ftesmal trasformato Xf of the group questo (or the equvalet oes: ξ X Ω + Ω = 0, ξ X Ω + Ω 2 2 = 0, for that matter), the oe wll have: 2 X Ω Ω = 0. That s, the rato of the two quattes Ω ad Ω 2 s a dfferetal varat of the group that s subject to o other restrcto o ts form whatsoever besde that t should deped upo oly the x, the z, ad the dervatves of the z wth respect to the x. 7. Oe the has the followg theorem: Theorem. If oe s gve a fte or fte cotuous trasformato group the varables: ( ) ( ) ( j) ( j) x,, x ( ), z,, z ( j ) (, j =, 2,, q), ad f: s a kow tegral varat, whle: m Ω dx dx dx 2 z z U x,, x,, x, z,, z ( q ),,,, m 2 deotes ay dfferetal varat of the group questo that depeds upo oly the x, the z, ad the dervatves of the z wth respect to the x, the the geeral formula: U Ω dx dx dx

7 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 7 wll yeld all of the assocated tegral varats of the group. 8. Several sets of tegral varats belog to ay cotuous group, each of whch wll ecompass ftely may members. If: Ω dx dx ( q ) s a tegral varat the Ω tself wll ot be a dfferetal varat, geeral; from the dscusso above, the expresso X Ω wll the have the value: Ω ξ, ad therefore, t s oly whe all of the expressos: ξ vash detcally that Ω wll represet a dfferetal varat of our group. By cotrast, the expresso: Ω dω Ω dx dx dx ( q ) s, tur, a dfferetal varat of the group; that dfferetal varat s specfed by the fact that t cotas the quatty dω as a factor. Later, we shall fd a more precse expresso for ths state of affars. IV. Ay cotuous group has ftely may tegral varats. 9. Before we go further, we ow set: (q) =, m + m + + m (q) = m, ad correspodgly deote the depedet varables x by: ( ) k x, x 2,, x, ( ) ad, o the other had, deote the fuctos z j by: v

8 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 8 z, z 2,, z m. We deote the assocated frst-order dervatves by p, p 2,, the secod-order dervatves by r, r 2,, ad so o. Now, f: Uf = ξk ( x,, x, z,, zm) + ζ ( x, z) z s the geeral symbol of the ftesmal trasformatos of a cotuous (fte or fte) group, ad o the other had f: Ω (x,, x, z,, z m, p,, r, ) dx dx deotes ay assocated tegral varat the, as we saw, the equato: U(Ω) + Ω ξ = 0 k Ω dω wll be verfed, ad coversely, ay soluto Ω of ths equato wll gve a tegral varat. I ths, oe must observe that Ω must frst take the form of a fucto of the x, the z, the frst-order dervatves p, the secod-order dervatves r, ad so o; however, for our purposes, the z wll be fuctos of the x, ad therefore we shall also regard Ω as a fucto of the x. If we would therefore lke to calculate the expresso U(Ω) the we would frst have to exhbt the v-fold exteded ftesmal trasformato: ad the set: U (v) f ξ + ζ + π + ρ z p r +, U(Ω) = U (v) Ω. O the other had, we must uderstad the deote more specfcally by: ξ ( j) + p. j z j ξ to mea the quattes that we If Uf represets the geeral symbol of a ftesmal trasformato of our group the the desred fucto Ω of the x, z, p, r, wll be defed by the fact that every equato of the form: 0 = U (v) ξ ξ ( j) Ω + Ω + p j z j ξ

9 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 9 should be fulflled. We have to exame whether these equatos possess a commo soluto. That s the questo that shall be aswered. 0. From prcples that have bee kow for some tme, ths questo cocdes wth the questo of whether the homogeeous, lear, partal dfferetal equatos: U (v) Ω Ω ξ ξ ( j) + p = 0 j z j Ω possess commo solutos, ad the latter questo wll be dealt wth a geeral way by my theory. Namely, sce Uf mples the geeral symbol of the ftesmal trasformatos of a group, ad U (v) f represets the assocated v-fold exteded trasformato, the ftesmal trasformatos: U (v) Ω Ω ξ ξ ( j) p z j Ω + x j wll also geerate a fte group ( ) the varables: x, z, p, r,, Ω. Oe the has oly to decde whether ths fte group possesses varats by usg my usual rules. Ay such varat that cludes Ω wll yeld a tegral varat: Ω dx dx, ad oe wll fd all tegral varats that way. I partcular, f the gve group s fte the t ca be proved that v ca be chose to be large eough that oe has tegral varats of order v.. By cotrast, f the group of Uf s fte the a specal aalyss wll be requred. For example, f we take the group of all pot trasformatos: ξ ( x, y) + η( x, y) y, ad f we apply these trasformato to the par of curves: ( ) Cf., Lepzger Berchte, Aprl 895, pp [here, art. XX, pp. 579, et seq.]

10 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 0 y = ϕ (x ), y 2 = ϕ 2 (x 2 ) the we wll have to exame whether our group possesses tegral varats of the form: We ow have to set: Ω( x, x2, y, y, y,, y2, y 2, y 2) dx dx 2. Uf = ξ ( x, y) + η( x, y ) + ξ ( x2, y2) + η( x2, y2) y y 2 2 Now, f U (v) f s the v-fold exteded ftesmal trasformato the the ftesmal trasformatos: U (v) 2 ξ ( xk, yk ) ξ ( xk, yk ) f Ω + y k k = k yk Ω wll also geerate a fte cotuous group that caot be defed by dfferetal equatos, geeral. If oe would lke to decde whether that group possesses varats for a certa v the oe would have to observe that there s o coecto betwee the varous frst, secod,, up to v th -order dervatves of ξ(x, y ) ad the correspodg dervatves of ξ(x 2, y 2 ). The desred varats must therefore be, at the same tme, varats of a fte group that oe fds whe oe sets:. V f = ξ( x, y) + η( x, y ) + ξ2( x2, y2) + η2( x2, y2) y y 2 2, ad the costructs the v-fold exteded trasformato V (v) f, ad fally, the equato: (3) V (v) 2 f Ω ξk ( xk, yk ) ξk ( xk, yk ) + y k k = k yk Ω = 0. Here, the left-had sde represets the ftesmal trasformato of a fte group that s defed by dfferetal equatos. Sce the quattes ξ, η, ξ 2, η 2, ad the assocated dervatves of order oe up to v are ot lked by ay (lear) relatos, equato (3) wll decompose to: 4 ( (v )) = 2 (v + ) (v + 2) dfferet lear partal dfferetal equatos. The umber of depedet varables x, y, ( v) ( v) y,, y, x 2, y 2,, y s, however, equal to: 2 2 (v + 2),

11 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. ad sce v must be greater tha zero, the umber of depedet varables wll be small tha the umber of equatos for ay v. Oe ca therefore expect that the group: V (v) Ω ( ) Ω possesses o varats, ad the correspodgly, the fte group Uf wll possess o tegral varats that relate to a par of curves the x, y-plae the desred way. 2. I order to gve the smplest-possble form to the fact that the theory of dfferetal varats orgates the theory of tegral varats ( as a specal case, fact) t wll be better to express that dea the followg way: If we keep the prevous otatos, should the tegral: 2 z z Ω x, x2,, z, z2,,,., 2 dx dx 2 dx, whch ca be brought to the form: Ω ±,. dτ dτ 2 dτ τ τ by the troducto of the parameters τ,, τ, keep ts form for all trasformatos of a certa fte or fte, cotuous, dscotuous, or mxed group the x ad z, ad thus represet a so-called tegral varat, the t would be ecessary ad suffcet that the product: z 2 Ω x, x2,, z, z2,,, ± τ τ 2 τ should represet a dfferetal varat of the group questo. I geeral, the prevously-chose form s ofte preferable for practcal calculatos. 3. As oe sees, the coecto betwee the theory of tegral varats ad the theory of dfferetal varats s mmedately obvous for ay mathematca that has avaled hmself of the frst elemets of my theory of trasformato groups. At o pot have I ever doubted that so dstgushed a mathematca as Pocaré has clearly recogzed that state of affars o the frst glace. If he dd ot ecessarly dscover what was proved here the that would probably be due to the fact that he addressed oly oe-parameter groups. However, o matter how self-explaatory my cocepto of the expresso:

12 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 2 Xf = ξk ( x,, x) as the symbol of a ftesmal trasformato ad the geerator of a oe-parameter group mght seem, t stll remas for oe to cosder that t s precsely that cocepto of me that defes the startg pot for vestgatos that have already exerted a powerful fluece o the developmet of pure, as well as appled, mathematcs. As far as the theory of bodes that Pocaré has treated s cocered, allow me to show that my older vestgatos to fucto groups have led to that theory the same way that Lagrage s vestgatos to algebrac equatos of degree three ad four led to the geeral theory of algebrac equatos. k V. O caocal systems ad dfferetal (tegral, resp.) varats. 4. I my older studes (cf., e.g., De Störugstheore ud de Berührugstrasformatoe, Norweg. Archv, Bd. II, Jauary 877 [ths coll., v. III, art. XX, pp ]), I showed (amog other thgs) that the reducto of a smultaeous system: dx k = α k (x,, x 2 ) (k =,, 2) dt to the so-called caocal form ca be accomplshed f ad oly f a certa Pfaffa expresso: X (x) dx + + X 2 dx 2 = X dx exsts that fulflls a equato of the form: ad furthermore, that 2 quattes: d ( X kdxk ) = dω (x,, x 2 ), dt y,, y, q,, q must be gve that fulfll a relato of the form: qkdyk = X kdxk. I partcular, that would mply that the characterstc fucto of the caocal system questo wll be homogeeous of frst degree the q whe the two dfferetals dω ad dv are equal to zero. I the latter case, t s obvous that the tegral:

13 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 3 ( Xdx + + X 2dx2 ), whe exteded alog a arbtrary curve the space of x, preserves ts value uder the ftesmal trasformato: δx k = α k (x,, x 2 ) δt, ad, at the same tme, for all fte trasformato of the assocated oe-parameter group. I dd ot fd t ecessary to make ths last remark explctly; f my work, especally my older work, s ot, fact, kow to the reader the t would ot be clear mmedately that ay trasformato that leaves a dfferetal dω varat wll smultaeously reproduce the tegral dω. I the treatse questo, I also gave (amog other thgs) the geeral determato of all trasformato x,, x, p,, p that take a gve caocal system to a caocal system. I thus foud trasformatos that geerally dd ot represet ay cotact trasformato the x, p. 5. Recetly (Comptes redus, Pars, Dec. 895), Kögs has addressed precsely some of these problems that I treated geeral hs ote: Applcato des varats tégraux à la reducto au type caoque Hs ote mght have some formal value. As far as realty s cocered, he has ot added aythg to my ow developmets, my opo. Now, I realze that Kögs was probably stll uaware of my aforemetoed work the edtg of hs ote. I my opo, however, he must, ay evet, establsh the relatoshps that exst betwee hs ote ad stll-older work that lkewse goes back to myself ad s hardly kow to hm. I order to llustrate how much further that my studes at the tme wet beyod the ote of Kögs, permt me to reproduce the followg fal theorem of my paper [ths coll., v. III, art. XX, pp. 37, les 3-2]. Coversely, let a complete system: A f = 0,, A q f = 0 be gve. I assume that I kow a expresso: that fulflls q relatos of the form: A ( X kdxk ) X dx + + X m dx m = dω (or = 0). I pose the problem of evaluatg ths stuato to the greatest extet possble. If q =, m = 2, ad therefore the ormal form for X dx cotas 2 depedet fuctos, the the tegrato of A f = 0 wll requre oly:

14 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 4 operatos. 2 2, 2 4,, 6, 4, 2 VI. Hstorcal remarks. 6. The cocept of tegral varat had already occurred to the older mathematcas, who, fact, completely uderstood the cocepts of arc legth, surface area, volume, ad so o, seemgly greater geeralty, ad obvously ther relatoshps wth the famly of all motos, as well. By ad by, the cocept of tegral varat appeared ts geeral form. Thus, for example, Lobatscheffsky, Bolya, Gauss, ad Rema also troduced the cocepts of arc legth, surface, area, ad so o to o-euclda geometry. From the ature of thgs, extedg the cocept of tegral varat to arbtrary trasformato groups was frst possble oly after I troduced the geeral cocept of a cotuous group. 7. I all of my studes of cotact trasformatos ad Pfaffa expressos, whch bega the year 87 (ad especally the aforemetoed ote the year 877), I have exclusvely addressed lear, homogeeous frst-order dfferetal equatos: X k (x,, x ) dx k that rema varat uder oe-parameter groups. I partcular, I have developed theores the year 877 that Kögs recetly reproduced, ad whch were, fact, derved from Pocaré s etrely specalzed cosderatos o tegral varats that were publshed the year Moreover, the year 886, these Berchte [ths coll, v. II, art. V], I hted at vestgatos that relate to trasformato groups for whch the arc legth tegral remas varat. I gave a rgorous presetato of these vestgatos varous places, ad the most thorough of them was volume three of my Theore der Trasformatosgruppe ( partcular, see pp. 499, et seq.) I rgorously dscussed the coecto betwee that theory ad my geeral studes to the varats of several ftely-separated pots uder all trasformatos of a cotuous group. I partcular, I drect your atteto to the followg theorem [Th. d. Tras., Bd. III, pp. 506, le 7-23], whch clearly shows that I was possesso of the geeral theory of tegral varats at the tme: The foregog cosderatos regardg the relatos betwee the varats of ftely-separated pots ad those of ftely-close pots ca be completed substatally; oe ca also geeralze them whe oe troduces three ad more pots,

15 Le Theory of dfferetal varats s a corollary to the theory of tegral varats. 5 whch brg wth them the dfferetals of the x v of order two ad hgher, ad so o. We reserve the thorough treatmet of these questos to aother occaso. 6. I regularly touched upo the cocept of tegral varat my older lectures; but the, I also oly touched upo t. I the year 895 ( Aprl), I preset a treatse to the local socety (whose cotet I had already submtted to the socety the year 894, moreover) that exhbted a formula o pp that yelded the foudato for the actual calculato of the tegral varats [here, art. XX, pp. 579, et seq.]. At precsely the same tme (Aprl 895), Zorawsky, who had studed my theores wth me the years 889 ad 89, publshed a udoubtedly worthwhle paper o tegral varats. Ufortuately, I was able to read oly the formulas, but ot the text, of hs Polsh paper. Later, Carta publshed a very terestg paper that theory. Carta, whose excellet papers o the composto of groups assume a especally pre-emet place amog the recet group-theoretc vestgatos, sad, ad wth complete justfcato, my opo: The questo thus-posed reduces to purely a problem the calculus, thaks to the theores of Le that permt oe to form all of these quattes (tegral varats). O aother occaso, I would lke to be permtted to show that Carta s beautful remarks o the optcs of o-euclda space ca be coected wth my cocepto of Huyges s wave theory as a chapter the theory of cotact trasformatos, ad o the other had, wth by studes of surfaces whose prcpal taget curves belog to le complexes. I a followg ote, I wll (amog other thgs) deftvely address the questo of the exstece of tegral varat for fte cotuous groups, as well.