NLELE ŞTIINŢIICE LE UNIERSITĂŢII L.I.CUZ IŞI Toul XLII, s.i, Mtetă, 2001, f.2. STRENGTH IELDS ND LGRNGINS ON GOs 2 M BY DRIN SNDOICI strt. In ths pper we stud the strength felds of the seond order on the geoetrl odel gven GOs 2 M nd Lgrngns nvolvng guge felds, defned through strength felds. full Lgrngn should e the su of the Lgrngn of guge felds nd lol guge nvrnt Lgrngn of tter felds. We shll onsder full Lgrngn L 0u nd, n Lgrnge nner, rewrte the equtons of oton. 1. Introduton. The for of the ntertons etween soe well known felds n e deterned postultng nvrne under ertn group of trnsfortons.the sstetl stud of nvrnt theoretl nterpretton of nterton n etween tter felds ws ntted Ut n [12]. Benu solved the prole of nvrne n the se of tngent undle TM, where M s rel, sooth, n-densonl nfold [3]. In onogrph [5] Mron gves n orgnl onstruton of the geoetr of the hgher order Lgrnge spes. These spes onsttute n dequte geoetrl frework for the developent of n ntegrted guge theor of the tter felds. Moreover, n ths work there re forulted the fundentls of the extenson of the lssl guge theor to the guge trnsfortons on the k osultor undle, endowed wth the Lgrngns of order k. The sgnfnt results onernng the guge theor of hgher order were otned Muntenu [2], [7], [8], [9]. In the pper we shll stud the strength felds of the seond order on the geoetrl odel gven GOs 2 M. These felds shll e ntrodued n nner, dfferent fro tht n [9]. lso, we shll stud Lgrngns nvolvng guge felds, defned through strength felds. s n lssl guge
214 DRIN SNDOICI 2 theor, the full Lgrngn should e the su of the Lgrngn of guge felds nd lol guge nvrnt Lgrngn of tter felds. We shll onsder full Lgrngn L 0 u nd look for the equtons of oton. The nottons nd the geoetrl oets tken nto onsderton re those used n [5], [7] nd [9]. 2. Strength felds nd Lgrngns for guge felds on GOs 2 M. Let us onsder M rel, sooth, n densonl nfold, G opt sugroup n GL, R, nd the nfold GOs 2 M orrespondng to M nd G see [5]. lso, we onsder L 0 u, u = x, 1, 2 Lgrngn defned on the don Ω R 3n, N = N, N nonlner 1 2 guge onneton on GOs 2 M, nd g = g etr struture on GOs 2 M. Suppose tht there exst p dfferentle slr felds phsl felds, = 1, p so tht the Lgrngn L 0 depends onl on the vrle x, 1, 2 through nd ther dervtves x,, α = 1, 2. More α presel, L 0 s slr feld on GOs 2 M gven : L 0 x, 1, 2 = L, x, α The tter felds re supposed to e guge slrs nd sne the nonlner onneton s guge, ther dervtves re guge vetors. s n lssl guge theores, new Lgrngn s onsdered: L 0 u = L, x, α, H u, α u, n whh H α u nd u, α = 1, 2 re the oponents of three guge d ovetors, lled lol guge felds, stsfng the followng nonhoogeneous ondtons of vrtons : H u = ε u f H ε x α u = ε u f α ε α Let us defne the followng guge dervtve opertors:
3 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 215 h D = x H [X ] B B v α D = α α [X ] B B, α = 1, 2 Now we n look t the Lgrngn L 0 s eng: L 0 u = L, h D, vα D The lol guge nvrne of the 2 order Lgrngn L s dedued fro the glol guge nvrne of the Lgrngn L see [9]. Next, we shll onstrut soe Lgrngns for the guge felds H u nd α u. rst, we note tht the lol ton of the Le group G outes wth the opertors nd x. We n wrte : 2.1. 2.2. 2.3. 2.4. α H x = ε x f H ε f H x x H α = ε α f H ε f H α ɛ α x ε α x = ε x f α ε f α x ε x α α = β ε β f α ε f α β β x ε α where we denote f the struture onstnts of the Le group G. ordng to the ft tht we hve to work wth two tpes of guge felds, we look for Lgrngns of the followng for: 2.5. L 0 u = L H, α, H x, H α, α x, α β
216 DRIN SNDOICI 4 where L s dfferentle funton on don n R 3n6n2. The ondton for the lol guge nvrne of L 0 s gven : L 0 = L H H L α α 2.6. L H x H x L α x α x L H α H α β=1 L α β α = 0 β Note tht ondton 2.6 expresses the lol guge nvrne of L 0 on the don of oordntes neghourhood of GOs 2 M. In ft, we n prove tht 2.6 hs n nvrnt hrter on GOs 2 M. More presel, the vnshng of L 0 does not depend on the oordntes on GOs 2 M. Usng the lnguge of Lgrnge geoetr we n stte: Proposton 2.1. L 0 s nvrnt wth respet to the lol guge ton of G f nd onl f the followng reltons re stsfed: 2.7. L H x L H = 0 x 2.8. 2.9. L H α L α β L α x β α = 0 L = 0
5 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 217 2.10. L H L f H x H L α x f α = 0 L α 1 2 L R H 0α x n n 2.11. L H α f H β=1 L β x n B βα n β=1 L β α f β 1 2 β=1γ=1 L β γ n α C γβ n = 0 L H H L α α 2.12. L H H x x L H α H α L α x α x α,β=1 L α β α β f = 0 felds where R x, h 0α, B 1, h αβ 2 nd α C h γβ re gven the Le rkets of the vetor. ro 2.7 to 2.9 t follows tht H α x, x,
218 DRIN SNDOICI 6 H α, α enter the Lgrngn tht we seek onl ens of funtons β gven : 2.13. h = H x v α,v β = α β H x, β α h,v α = H α α x Now, usng the ove funtons, the struture onstnts of the Le group G, the guge felds H, α nd the nonlner onneton on GOs 2 M, we defne the followng dfferentle funtons: 2.14. h = h f H H 2 R. α 0α 2.15. h,v α = h,vα f H α 2 B. β β=1 αβ 2.16. v α,v β = vα,v β f α β 2 γ C γ γ=1αβ urther, we re nterested n studng the ehvour of h, h,vα wth respet to oth the lol ton of G nd the trnsforton of oordntes on GOs 2 M. We hve the followng two results: Proposton 2.2. The lol ton of G on the funtons h, vα,v β, h,vα,
7 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 219 v α,v β s hoogeneous: h = ε f h 2.17. h,vα = ε f h,vα vα,v β = ε f vα,v β Proposton 2.3. The dfferentle funtons h, h,vα nd vα,v β re the oponents of soe d tensor felds of tpe 0, 2 for eh =1,...,. Moreover, h nd vα,v β re nt setr d tensor felds. We ll h, h,vα nd vα,v β the horzontl strength felds, the xed strength felds nd v α vertl strength felds respetvel. Next, we shll see tht strength felds ke gret ontruton to the onstruton of Lgrngns for guge felds. We frst onsder the Lgrngn L 0 fro 2.5. s dfferentle funton whh depends onl on h h,v α v α,v β nd g u,.e. we hve: 2.18. L 0 = L h, h,vα, vα,v β, g u where L s dfferentle funton. Then fro 2.13 to 2.16 we esl otn:,, 2.19. L H x = 2 L α x = L h L, h,vα, L H α L α β = L h,vα L = 2 vα,v β
220 DRIN SNDOICI 8 2.20. L H = 2 L f h H L h,v α f α L α = L h R 0α 2.21. L h,v α f H β=1 L h,vβ B βα 2 β=1 L vα,v β f β γ=1β=1 L vβ,v γ α C γβ Usng 2.18. 2.21. t s es to hek tht L stsfes equtons 2.7. 2.11.. Thus, the onl ondton for the lol guge nvrne of the Lgrngn whh rens for further nvestgtons s 2.12.. fter lultons, we otn tht: Proposton 2.4. The ondton of lol guge nvrne of L s gven : fd L h h h,vα h,v α L 2.22. β=1 L vα,v β vα,v β = 0 We now rell tht the Kllng for of the Le lger G s lner for K on G G whose oponents re gven K = f d e f e d. The
9 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 221 Le group G s sd to e se sple f the Kllng for K of G s non degenerte. t ths pont we suppose tht the Le group G nvolved n our guge theor s se sple nd opt. In ft the lss of Le groups tht we shll work wth s of spel nterest n phss. In order to onstrut Lgrngns for guge felds we onsder generlzed Lgrnge etr of order 2. Then, nspred the Yng Mlls felds fro lssl guge theor, we onsder the followng three Lgrngns: 2.23. L H = 1 4 K g u g n u h u h n u 2.24. L = 1 H, α 4 K g u g n u h,vα u h,vα n u 2.25. L = 1 α, β 4 K g u g n u vα,v β u vα,v β n u Esl, we n verf tht L, L nd L re Lgrngns for H H, α α, β guge felds. s n lssl guge theor, the full Lgrngn should e the su of the Lgrngn of guge felds nd the lol guge nvrnt Lgrngn of tter felds. 3. Equtons of otons for the full Lgrngns. We egn wth generlzed Lgrnge etr of order 2 on GOs 2 M, gven the oponents g u, wth full Lgrngn L 0 u nd guge nonlner onneton N = N 1, N 2. Then we otn the Lgrngn denst: 3.1. 3 L u = L 0 u G, where G = detg u Next, we suppose tht the equtons of oton re ust the Euler Lgrnge equtons wth respet to L u nd the ndependent tter felds u
222 DRIN SNDOICI 10 nd guge felds H, α, α {1, 2}. Hene we hve the followng equtons of oton: L 3.2. L x L x α = 0 α 3.3. L H L x H x α L H = 0 α 3.4. L α x L α x β=1 β L α β = 0 Tkng nto ount the ft tht L 0 depends on x, on ens of x 1 x nd, nd on 1 2 x, nd, we n wrte: 1 2 ens of ens of 3.5. L = x L x 3.6. L 1 = L 1 L 1 N 1 x
11 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 223 3.7. L = 2 L 1, 2 N 1 2 L 1 1 N 2 L x We ke the followng nottons: 3.8. h = L x v 1 v 2 L = 1 L = 1, 2 1 2, v 1 nd v 2 The dfferentle funtons h re the oponents of d tensor felds of tpe 1,0. In ths w, 3.5. 3.7. n e wrtten n the followng for: 3.9. 3.10. 3.11. 1 L h = G x L 1 = G L 1, 2 = G 2 v1 N h 1 v2 N v 1 N h 1 2 Usng 3.9. 3.11. n 3.2., we otn: h 3.12 G L x N 1 2 v N 1 2 h 2 v 1 1 v 2 N 1 h 2 = G x h G 1 1 v 1 G 2 v 2
224 DRIN SNDOICI 12 3.12. ertnl looks ore oplted thn 3.2., ut t s the Lgrnge geoetr whh wll help us to otn sple for of 3.12.. rst of ll, we onsder guge N lner onneton on GOs 2 M wth the oeffents L k, C, C. Usng the guge h nd v α ovrnt dervtves, 1 2 we hve: 3.13. k k h = h L h x vα = vα α α C α vα The dentt 3.12. n e wrtten n the followng for: 3.14. 3.15. where: E = 1 G L h v α α Γ L h Γ = G x h G L = L N 1 1 1 N 2 v 1 2 = E C vα α G 2 v 2 C = C 1 1 N 1 2 ndc = C 2 2 Usng 3.14. we onlude tht the E re p guge slr felds on
13 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 225 GOs 2 M. We ke the followng nottons: 3.16. h H = L H, x v 1 H = L H 1 1, v 2 H = L H 1, 2 2 The dfferentle funtons h H, v 1 H, v 2 H re the oponents of d tensor felds of tpe 2,0. Moreover, the h H re nt setr. B dret lultons nd usng the Lgrnge geoetr t follows tht 3.3. n e wrtten n the followng for: 3.17. L H h H v α α H =H E where: 3.18. L H = L H L h H E = 1 G H Γ L h H C v 1 H C v 2 H 1 2 H C vα H α H H Γ = G x h G 1 v 1 H G 2 v 2 H On the other hnd, we n prove tht the dfferentle funtons L H, = 1,..., n re the lol oponents of soe d vetor felds. Usng 3.17. nd the ove rerk we onlude tht H E, {1,..., n} re d vetor
226 DRIN SNDOICI 14 felds. Next, we defne: 3.19. h,v α = 1,v α = L α x L α 1 2,v α 1 = L α 2 1, 2 The dfferentle funtons h,vα, 1,vα nd 2,vα h,vα re the oponents of d tensor felds of tpe 2,0. Moreover, the re nt setr. Usng gn the lnguge of Lgrnge geoetr we n wrte 3.4. n the followng for: 3.20. L α h,vα β=1 β,v α β = α E where: 3.21. L α = L α E α Γ = 1 G α L h,vα C 1,vα C 2,vα 1 2 α Γ L h,vα = G x h,vα G 1 C β,vα β β=1 1,vα G 2 2,vα lso, we n prove tht the dfferentle funtons L α, = 1,..., n re d vetor felds. Usng 3.20. nd the ove rerk, we onlude tht α E, {1,..., n} re d vetor felds. Therefore, we proved the followng result:
15 STRENGTH IELDS ND LGRNGINS ON GOs 2 M 227 Theore. The equtons of oton generted full Lgrngn re gven the reltons 3.14., 3.17. nd 3.20.. 4. Conlusons. In the lst prt of the pper we rewrte the Euler Lgrnge equtons of full Lgrngn of order 2, nl usng the notons of the nonlner onneton of order 2 nd of Lgrnge etr of order 2. The n otvton of ths rewrtng onssts n the ft tht, n the equvlent equtons tht we hve got, we hve underlned the dstngushed H nd α geoetr oets onsderng the defnton n [6] E, E E, ent s generlzed guge energes. These hve een used to deterne the lws of onservton nd the generlzed Mxwell equtons of order 2 orrespondng to guge full Lgrngn see [10] nd [11]. knowledgeent. The uthor s grteful to the revewer for hs rerks whh prove the prevous verson of the pper. REERENCES 1. snov, g.s. nsler Geoetr, Reltvt nd Guge Theores, D. Redel, Dordreht, 1985. 2. Bln, v., Muntenu, gh. nd Stvrnos, p.. Generlzed Guge snov Equtons on Os 2 M Bundle., Proeedngs of the Workshop on Glol nlss, Dfferentl Geoetr nd Le lgers, 1995, 21 32. 3. Benu,. nsler Geoetr nd ppltons, Ells Horwood Lted, 1990. 4. Chhn,. nd Nelp, n.f. Introduton to Guge eld Theores, Sprnger erlg, 1984. 5. Mron, r. The Geoetr of Hgher Order Lgrnge Spes. ppltons to Mehns nd Phss, Kluwer de Pulshers, 1997. 6. Mron, r. nd nstse,. The Geoetr of Lgrnge Spes: Theor nd ppltons, Kluwer de Pulshers,1994.
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