Fractional Euler-Lagrange Equations of Order ( α, β ) for Lie Algebroids

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1 Studes n Mthemtcl Scences Vol. 1 No pp ISSN [Prnt] ISSN [Onlne] Frctonl Euler-Lgrnge Equtons of Order ( α β ) for Le Algerods El-Nuls Ahmd Rm 1* Astrct: The mn purpose of ths pper s to derve the frctonl Euler-Lgrnge equtons whch depend on the Remnn-Louvlle dervtves of order ( α β) α > 0 β > 0 for Le lgerods. The frctonl Hmltonn formlsm ws lso dscussed. Two exmples n prtculr the frctonl geodescs for Le lgerods nd the Wong's frctonl equtons whch rse n the dynmcs of colored prtcle n Yng-Mlls feld nd on the fllng ct theorem were lso derved. Key Words: Frctonl Acton-lke Vrtonl Approch; Frctonl Lgrngn nd Hmltonn Formlsms; Le Algerods 1. INTRODUCTION The Frctonl Clculus of Vrtons (FCV) sed on frctonl clculus [1-18] ws proved recently to e useful tool for descrpton of wek dssptve nd nonconservtve dynmcl systems wth holonomc nd nonholonomc constrnts. The respectve Euler-Lgrnge type equtons re suject of current strong reserch nd nvestgtons [19-25]. An extenson of Noether's symmetry theorem to the FCV hs recently ntroduced y the uthor s the frctonl cton-lke vrtonl pproch (FALVA) wth mny nterestng pplctons nd fetures [26-32]. In ths work our mn m s to derve the frctonl Euler-Lgrnge equton on Le lgerods. However the frctonl Euler-Lgrnge equtons sed on the FALVA wth one prmeter α defned on Le lgerods were nvestgted more recently [33]. In ths pper we wll enlrge the prolem nd descre the frctonl Euler-Lgrnge equtons whch depend on the Remnn-Louvlle dervtves of order ( α β) α > 0 β > 0 for Le lgerods. We wll n ddton explore the correspondng frctonl Hmlton equtons. In fct snce Le lgerod s concept whch smply unfes tngent undles nd Le lgers nturlly one cn expects ther relton to clsscl mechncs. Further t hs proved to e powerful tool n the nvestgtng of mny fundmentl prolems n ppled mthemtcs n generl nd dfferentl geometry n prtculr [34-38]. In the context of clsscl mechncs theory of Lgrngn nd Hmltonn systems on Le lgerods hs een explored n some detls usng the lner Posson structure nd the dul of the Le lgerod nd the Legendre trnsformton ssocted wth the regulr Lgrngn [34]. Thus powerful mthemtcl structure hs emerged. Wthn FALVA frmework t ws proved tht the set of dmssle curve on Le lgerod wth fxed endponts cn e endowed wth structure of Bnch mnfold tht the frctonl cton ntegrl n FALVA s contnuously dfferentle nd tht the 1 Deprtment of Nucler nd Energy Engneerng Cheju Ntonl Unversty Ar-dong 1 Jeju South Kore. * Correspondng uthor. E-ml ddress: nulshmdrm@yhoo.fr. Receved 5 Novemer 2010; ccepted 28 Novemer

2 El-Nuls Ahmd Rm /Studes n Mthemtcl Scences Vol.1 No equtons for the crtcl ponts re precsely of the frctonl Euler-Lgrnge equtons otned n FALVA for the gven Lgrnge system [33]. The fundmentl prolem of the FCV wth Remnn-Louvlle frctonl ntegrl of order ( α β) α > 0 β > 0 s ntroduced y El-Nuls-Torres n [39] s the followng: consder smooth mnfold M nd let L e n dmssle smooth Lgrngn functon d d L : d 1. For ny pecewse smooth pth q:[ t0 t1] M stsfyng the oundry condtons q = q nd q = q the frctonl functonl ssocted to L s defned y [ ] 1 ( α ) ( ) S q = L D q τ q τ τ t τ dτ Γ (1) the frctonl dervtve opertor of order ( α β ) s defned y The crtcl ponts of [ q ] 1 α β α β = D D D D D. (2) 2 2 S were proved to stsfy the followng Euler-Lgrnge equtons: D β α τ L q ( βα ) L D q q D ; τ D q q τ τ τ τ τ τ q ( ( τ ) ( τ) τ) represents the frctonl tme dervtve wth respect to τ. 1 α L = D q q. (3) t τ q 2. FRACTIONAL EULER-LAGRANGE EQUATIONS ( α β ) FOR LIE ALGEBROIDS A Le lgerod structure on vector undle π : E M s gven y vector undle morphsm ρ : E TM over the dentty n M clled the nchor mp together wth Le lger structure on the spce C ( M) -module of sectons of E determned y the Le rcket whch nduces Le lger homomorphsm ρ from Sec( E) χ( M) y the nchor mp ρ : E TM gven y σ Sec( E) ρσ ( x) χ( M) ρ( σ)( x) = ρ( sx ) x M stsfyng the Lenz comptlty dentty: [ σ f] ( ρ( σ) f) f [ σ ] E = + f C ( M) σ Sec( E). (4) A vector undle ( E ξ M) endowed wth Le lgerod structure ([ ] E ρ) s clled Le lgerod over M nd s denoted y ( E[] ρ). E A locl coordnte system ( x ) n the se mnfold M nd locl ss { e } of secton of E determne the locl coordnte system ( x y ) on E. The nchor nd the rcket re loclly determned y the locl structure functons ρ k nd C jk structure functons ρ Cj C ( M) of ( E[] E ρ) whch stsfes the followng reltons whch results from the Lenz dentty nd the Jco dentty: E 14

3 El-Nuls Ahmd Rm /Studes n Mthemtcl Scences Vol.1 No σ σ ρ j ρ σ k j = k C j ρ( e ) = ρ nd e ej = Cjek. ρ ρ ρ = 1 n = 1 m (5) l C jk h l ρ + CjkCh = 0 (6) cyclc( j k) k We consder now the spce of E -pths on the Le lgerod denoted y P ( J E) J = [ ] dfferentle Bnch mnfold [33]. whch s Theorem 2.1 Let LD ( q( τ ) q( τ) τ) e Lgrngn on the Le lgerod E wth dmssle curve q n E nd wth two fxed endponts B A { P π π } A B M P( J E) = ( J E) ( q) = A nd ( q) = B. The crtcl ponts of the frctonl cton ntegrl S ( ) : P ( J E) defned y: [ ] 1 ( α ) α ( ) 1 τ τ τ τ τ( α β) ( 01] S q = L D q q t d Γ on the Bnch mnfold P ( J E) B A re exctly those elements of tht spce whch stsfy the followng frctonl Euler-Lgrnge equtons of order ( α β ) : Here τ L = ( L y ) dy n locl coordntes. r r LD q q δ ( ( τ) ( τ) τ ) = 0 < δ LD ( q( τ) q( τ) τ) >=< dld ( q( τ) q( τ) τ) ( σ) > βα τ ; τ LD ( q( τ) q( τ) τ) D < q σ > 1 α < τ q σ >. (7) LD ( q( τ) q( τ) τ) Proof. The tngent spce on P ( J E) B A t q P ( J E) B A s nothng thn the set of vector felds long q re of the form Ξq( σ ): σ Sec( E)/ σ = σ = 0. The frctonl cton s smooth then: 1 ( ) q Γ( α ) q 0 =< ds Ξ ( f σ ) >= < dl( D q( τ) q( τ) τ) Ξ ( f σ) > dτ 1 = f( τ) < dl( D q( τ) q( τ) τ) Ξ q ( σ) > Γ( α ) LD ( q( τ) q( τ) τ) D < τ q σ > t τ q α τ q σ t τ d τ α 2 + ( 1) < > LD ( q( τ) q( τ) τ) 15

4 El-Nuls Ahmd Rm /Studes n Mthemtcl Scences Vol.1 No f( τ ) τ q σ ( t τ α + < > ) LD ( q( τ) q( τ) τ) 1 r = f ( τ ) < δ LD ( q( τ) q( τ) τ) σ( τ) > dτ Γ ( α ) v we hve used the fct tht Ξ q( fσ ) = f ( σ) + D q σq. v ρ v = ( q) s the ctul velocty nd D σ q s the Remnn-Louvlle frctonl dervtve of the cnoncl vertcl lft of σ. Ths equton s stsfed for every functon f C ( ) nd every secton σ Sec( E). Thus the crtcl ponts re stsfed y r LD q q δ ( ( τ) ( τ) τ ) = 0. Defnton 2.2 In locl coordntes τ L = ( L y ) dy the frctonl Euler-Lgrnge equtons of order ( α β ) s: ρ D x = y. Defnton 2.3 The term ( ) L D x x D ; τ ( D x x ) L βα ρ τ τ y ( τ ) L k j 1 α L Cj y + D x x = 0 (8) k t y τ y 1 α < Fτ q σ >= < τ q σ > (9) LD ( q( τ) q( τ) τ) s clled the frctonl decyng frcton force for Le lgerods. Remrk 2.1 When β = 1 equton (7) s the sme s the one otned n [33]. 3. THE FRACTIONAL HAMILTONIAN FORMALISM FOR LIE ALGEBROIDS We now consder the followng generl frctonl vrtonl prolem n locl coordntes: 1 S x u = L( y( τ) x( τ) τ)( t τ) dτ mn Γ (10) [ ] ( α ) ( ) D x τ = ϕ u τ x τ τ. (11) A necessry optmlty condtons to the followng prolem for Le lgerods my e otned f we ntroduce the ugmented frctonl cton ntegrl [39]: ( τ τ τ τ) 1 S x u p u x p = Γ( α ) H p ( τ ) D x( τ) dτ (12) p α β s the frctonl Lgrnge multpler nd 16

5 El-Nuls Ahmd Rm /Studes n Mthemtcl Scences Vol.1 No H ( u( τ) x( τ) p ( τ) τ) = L( u( τ) x( τ) τ)( t τ) ( ) τ ϕ τ τ τ + p u x (13) s the frctonl Hmltonn. Theorem 3.1: Let Lu ( ( τ ) x( τ) τ) e hyperregulr Lgrngn on the Le lgerod E wth dmssle curve q n E nd wth two fxed endponts B A { π π } A B M P( J E) = P ( J E) ( q) = A nd ( q) = B. If ( x u ) s mnmzer of prolem (9) then there exsts co-vector functon p α β such tht the followng condtons hold: The frctonl Hmltonn system for Le lgerods: ( ) βα α H τ ; = p D x ( τ ) ρ u τ x τ p τ τ (14) ( ) βα H τ = D ; p ( τ) ρ u τ x τ p τ τ k H Cj pk ( τ ) ( u( τ) x( τ) p ( τ) τ ) (15) p The frctonl sttonry condton for Le lgerods: H u j ( u( τ) x( τ) p ( τ) τ) Remrk 3.1 From equton (15) equtons (13) nd (14) yeld: = 0. (16) ( ) βα L D τ ; D x x τ t τ u ( ) L L k j ρ D x x τ t τ + Cj y = 0 k y (17) u( τ ) = D x( τ ). Equton (17) s equvlent to equton (9). Thus theorem 3.1 s generlzton of theorem 2.1 to the frctonl optml control prolem for Le lgerods. Remrk 3.2 Equtons (14) (15) nd (16) re otned esly f we pply the frctonl Euler-Lgrnge equtons (9) of order ( α β ) to the ugmented frctonl cton ntegrl wth respect to p α β x nd u respectvely. Remrk 3.3 The frctonl Hmlton dynmcs for equton (9) on the dul undle E s represented y the frctonl vector feld: D r ( x ξ) ρ H = α β p 17

6 El-Nuls Ahmd Rm /Studes n Mthemtcl Scences Vol.1 No H k α β H ρ + Cj pk ( τ). (18) α β p ξ j Remrk 3.4 It s well-know tht Noether's theorem s consequence of the exstence of vrtonl descrpton of the dynmcl prolem. In the stndrd cse when α = β = 1 the Noether energy s conserved. Ths sttement does not hold for the cse ( ) ( 01] over Le lgerods. To solve the prolem βα one my use the followng new noton of frctonl constnt of noton C : D τ ; C = 0. Exmple 3.1 As smple exmple we wll dscuss the geodescs for Le lgerods. In locl coordntes the Lgrngn s L = (1 2) gj ( x) y y j the metrc g = gj ( x) e e j s expected to nduce n somorphsm of the vector undles g : E E. The frctonl Euler-Lgrnge equtons red: βα D ; gk y τ ( D x x τ ) = 1 α gk y ( D x x τ ) j 1 gj α β j α β + Ck gsj + ρk y ( D x x τ) y ( D x x τ) 2 whch cn e rewrtten n the form βα l j 1 α l τ ; j (19) D y +Γ y y y = 0 k = 1 n (20) l 1 kl gk g jk g Γ j = g ρj + ρ ρk 2 j s s k sj jk s re the Chrstoffel symols for Le lgerods. Equton (20) wth r geodescs equtons of order ( α β ) for ρ : E TM. C g C g (21) ρ D x = y re the frctonl Exmple 3.2 Another exmple concerns the Wong's equtons whch rse n the dynmcs of colored prtcle n Yng-Mlls feld nd on the fllng ct theorem [36-38]. The Lgrngn nd the Hmltonn of the theory on the Le lgerod E re gven y: α β α β σ 1 j ( j σ ) 2 1 j σ = ( j + ) Lx ( D x v ) = hv v + g uu (22) H ( x p p ) h p p g ppσ (23) 2 u( τ ) = D x( τ ) (24) ( x D x v ) s the correspondng dul fered coordntes on TQ / G nd ( x p p ) s the dul coordntes on TQ/ G G eng compct Le group E = TM A wth dmtm = m nd dma = n A eng n rtrry Le -lger of dmenson n. h s metrc on A nd g s ssumed to e Remnnn metrc on M. The correspondng frctonl Euler-Lgrnge equtons of order ( α β ) re gven y 18

7 El-Nuls Ahmd Rm /Studes n Mthemtcl Scences Vol.1 No βα k l k j s 1 α k τ ; j ls D v = C h h v v + v k = 1 n (25) β α β α σ σ β α λ β α ε 1 α β α σ D τ ; ( D τ ; x ) =Γ λεd τ ; x D τ ; x + D τ ; x (26) t τ whch re the frctonl Wong's equtons of order ( α β ) nd they re exctly the frctonl Euler-Lgrnge equtons of order ( α β ). Remrk 3.5 Beng tht symmetry plys cptl role on clsscl nd modern physcs one cn wrte the frctonl Euler-Lgrnge equtons of order ( α β ) for systems wth symmetry sed on the Atyh lgerod. Ths wll e explored n future work. See [40-43] for other nterestng pplctons. 4. CONCLUSIONS Usng the frctonl cton-lke vrtonl pproch together wth Remnn-Louvlle frctonl dervtves of order ( α β ) we generlze prevous results of the FCV for Le lgerods. The generlzed frctonl Euler-Lgrnge equtons the generlzed Hmlton equtons nd the generlzed geodescs equton for Le lgerods re derved. It ws lso proved tht the frctonl Wong's equtons of order ( α β ) re exctly the frctonl Euler-Lgrnge equtons of sme order. It would e of nterest n the future to explore the frctonl Lgrnge-d'Alemert-Poncré equtons on Atyh lgerods nd prolems wth symmetry. Another nterestng prolem concerns the guge nvrnt Lgrngns n terms of groupods frctonl cton of order ( α β ) on complex plne [44]. REFERENCES [1] El-Nuls R. A. (2005). FzA [2] El-Nuls R. A. (2010). Centrl Europ. J. Phys [3] El-Nuls R. A. (2009). Int. J. Mod. Phys. B [4] Almed R. Mlnowsk A. B. & Torres D. F. M. (2010). J. Mth. Phys [5] Mlnowsk A. B. Amm M. R. S. & Torres D. F. M. (2010). Comm. Frc. Clc [6] Almed R. & Torres D. F. M. Commun. (2011). Nonl. Sc. Numer. Smult [7] Bstos N. R. O. Ferrer R. A. C. & Torres D. F. M. (2011). Dscrete Contn. Dyn. Syst [8] Frederco G. S. F. & Torres D. F. M. (2007). Int. J. Ecol. Econ. Stt [9] Frederco G. S. F. & Torres D. F. M. (2005). Int. J. Appl. Mth [10] Frederco G. S. F. & Torres D. F. M. (2007). Appl. Anl [11] Frederco G. S. F. & Torres D. F. M. (2007). J. Mth. Anl. Appl [12] Frederco G. S. F. & Torres D. F. M. (2008). Int. Mth. Forum [13] Frederco G. S. F. & Torres D. F. M. (2008). Nonlner Dynm [14] Agrwl O. P. (2002). J. Mth. Anl. Appl [15] Klmek M. (2001). Czech. J. Phys [16] Blenu D. & Mustf O. G. (2010). Comp. Mth. Appl [17] Blenu D. & Trujllo J. I. (2010). Comm. Nonln. Sc. Num. Smul [18] Blenu D. Defterl O. & Agrwl O. P. (2009). J. Vrton nd Contr

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