LAGRANGIAN AND HAMILTONIAN MECHANICS WITH FRACTIONAL DERIVATIVES

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1 LAGRANGIAN AND HAMILTONIAN MEHANIS WITH FRATIONAL DERIVATIVES EMIL POPESU 2,1 1 Asronomicl Insiue of Romnin Acdemy Sr uiul de Argin 5, Buchres, Romni 2 Technicl Universiy of ivil Engineering, Bd Lcul Tei 124, Buchres, Romni Emil: epopescu@ucbro Absrc In his pper we discuss he frcionl exenion of clssicl Lgrngin nd Hmilonin mechnics We give view of he mhemicl ools ssocied wih frcionl clculus s well s descripion of some pplicions Key words: frcionl derivive - Lgrngin - Hmilonin - Euler-Lgrnge equions 1 INTRODUTION Lgrngin mechnics nd Hmilonin mechnics re lernive formulions of clssicl Newonin mechnics Their impornce is represened by he fc h ny of hem could be used o solve problem in clssicl mechnics We emphsize h he Newonin mechnics requires he concep of force, while Lgrngin nd Hmilonin sysems re expressed in erms of energy For sysem of n inercing pricles m i > 0, i = 1,2,,n, le q i = (x i,y i,z i, q i R 3 be heir posiion vecors wih respec o n rbirry origin nd le q = (q 1,q 2,,q n R 3n be he configurion of he sysem, q i = q i ( We hve verified he Newon s second lw F i = m i q i, for ech pricle i, i = 1,2,,n, where F i is he force on he pricle i The liner momenum of he sysem is n p = m i qi nd he kineic energy is K = 1 2 i=1 n m i qi 2, i=1 Romnin Asron J, Vol 23, No 2, p 85 97, Buchres, 2013

2 86 Emil POPESU 2 where q i 2 = qi q i is he Euclidin norm The dynmics of his n body sysem in he field deriving from he funcion poenil V (q is m i q i = V q i, i = 1,2,,n In his cse, he ol energy E := K + V is conserved nd he sysem is clled conservive This sysem of equions is equivlen o he Euler-Lgrnge equions ( d d q = 0, i = 1,2,,n i q i for he Lgrngin L (,q, q := 1 2 n m i qi 2 V (q i=1 We remrk h L = K V The following vriionl principle, nmed Hmilon s principle of les cion, is vlid: for ny differenible Lgrngin L, he Euler-Lgrnge equions re equivlen o δs = 0, where S [q] := nd δs is he firs vriion of S, δs = s b b L (,q(, q( d L (,q(,s, q(,s d, s=0 for deformion q(,s of q( leving he endpoins fixed The bove sysem of equions is equivlen o Hmilon s equions, for he Hmilonin q = H p, H (q,p := 1 2 n i=1 ṗ = H q 1 m i p i 2 + V (q, where p i = m i qi, i = 1,2,,n nd p = (p 1,p 2,,p n For such p, we hve H (q,p = K + V We cn lso define Hmilon s equions on R 3n for ny H, no necessrily

3 3 Lgrngin nd Hmilonin Mechnics wih frcionl derivives 87 derived from Lgrngin: q = H p, ṗ = H q, for some H : R 3n R 3n R, clled Hmilonin Of course, he Hmilonin is conserved The Euler-Lgrnge equions nd Hmilon s principle form he bsis of Lgrngin (or Hmilonin mechnics The power of Lgrngin (Hmilonin mechnics is h he given equions re chrcerized wih only one sclr funcionl, he Lgrngin L, or he Hmilonin H Generly, hese funcionls only describe conservive sysems There hve been some pproches describing non-conservive sysems in such formlism The mehod presened by Ryleigh inroduce funcionl R (clled Ryleigh s dissipion funcion nd rewrie he Euler-Lgrnge equions in he following form: ( d d q i q i + R q i = 0, i = 1,2,,n, where L is he sysem s Lgrngin These equions provide wy o re dissipive sysems, bu by mens of wo sclr funcionls, no one This is n objecion o his mehod Riewe proposed n pproch o incorpore non-conservive sysems o he field of vriionl mechnics using frcionl clculus (Riewe, 1996; Riewe, 1997 He hs shown h Lgrngin wih frcionl derivive led o equions of moion wih non-conservive clssicl forces such s fricion nd his formlism could be pplied o frcionl force proporionl o he velociy On he oher hnd, Riewe (1997 presened mehod o obin poenils for non-conservive forces in order o inroduce dissipive effecs o he Lgrngin nd Hmilonin mechnics The Hmilonin nd Lgrngin involving frcionl derivive is lso used o derive he equion dmped hrmonic oscillor (see Trwneh e l, 2010 Therefore, he dynmicl sysems wih frcionl order cn be dissipive For his reson, he heory nd mehods of frcionl clculus re exensively used for describing criicl phenomen in non-equilibrium sysems of physics nd mechnics, especilly in he complex sysems 2 FRATIONAL DERIVATIVES Frcionl clculus, which generlize he clssicl clculus, is he heory of derivives nd inegrls of rbirry non-ineger order In he ls yers ineres in frcionl clculus hs been simuled by he pplicions in differen res of science nd engineering In he mhemicl modelling of mny sysems nd processes,

4 88 Emil POPESU 4 he new frcionl-order models re more deque hn he ineger-order models In he recen yers, frcionl derivives hve plyed imporn role in very diverse opics such s clssicl mechnics, scling phenomen, frcls nd muli-frcls dynmics, dispersion nd urbulence, srophysics, poenil heory, viscoelsicy, elecrodynmic, opics, nd hermodynmic In he sequel, we give descripion of he bsic conceps of frcionl clculus Severl definiions of frcionl derivive hve been proposed, s Riemnn Liouville, Grunwld Lenikov, Weyl, puo, Mrchud, nd Riesz frcionl derivives (see Smko e l, 1993; Podlubny, 1999; Hilfer, 2000; Kilbs e l, 2006; Hilfer, 2008 We presen wo definiions: Riemnn-Liouville frcionl derivive nd puo frcionl derivive Le f : [,b] R be funcion, where nd b cn even be infinie The lef Riemnn-Liouville frcionl derivive (wih fixed lower erminl nd moving upper erminl is defined by D α f ( = ( 1 d n Γ(n α d f (u ( u α+1 n du nd he righ Riemnn-Liouville frcionl derivive (wih moving lower erminl nd fixed upper erminl b is defined by ( Db α f ( = 1 d n b Γ(n α d f (u (u where n 1 α < n, Γ represens he Euler gmm funcion Γ(z = 0 z 1 e d α+1 n du, nd ( d d n snds for ordinry derivives of ineger order n Alernive definiions of Riemnn-Liouville frcionl derivives re puo derivives The lef puo frcionl derivive defined s D α 1 ( f ( = ( u n α 1 d n f (udu Γ(n α du nd he righ puo frcionl derivive Db α f ( = 1 Γ(n α b ( (u n α 1 d n f (udu, du

5 5 Lgrngin nd Hmilonin Mechnics wih frcionl derivives 89 where he order α sisfies n 1 α < n The Riemnn-Liouville derivive of consn is no zero, lhough puo derivive of consn is zero Following Podlubny (1999, he lef derivive nd he righ derivive re operions performed on he ps ses, respecively on he fuure ses, of he process f If u <, where is he presen momen, hen he se f (u belongs o he ps of he process f; if u >, hen f (u belongs o he fuure of he process f Thus, he presen se of he process f (, which sred u =, depends on ll is ps ses f (u, u < If α is n ineger, he Riemnn-Liouville derivives re defined in he usul sense, ie ( d α ( D α f ( = f (, Db α d f ( = d d α f (, where α = 1,2,3, For α R +, we cn lso define he operors J α nd J α b on L1 ([,b] : nd J α f ( = 1 Γ(α Jb α f ( = 1 b Γ(α f (u ( u f (u (u 1 α du 1 α du These operors re clled he lef nd he righ frcionl Riemnn-Liouville inegrls of order α R +, respecively I is esy o see h he Riemnn-Liouville frcionl inegrls converge for ny inegrble funcion f For ineger α, α = n, he frcionl Riemnn-Liouville inegrls coincide wih he usul ineger order n-fold inegrion (uchy formul for n-fold inegrion: n n f ( 1 d 1 d 2 d n 1 d n = 1 Γ(n f (u ( u 1 n du The inegrion operors J α nd Jb α ply role in he definiion of frcionl clculus The lef nd he righ Riemnn-Liouville frcionl derivive of order α > 0 re D α f ( = D n J n α f (, Db α f ( = ( 1n D n J n α f (, wih n = [α] + 1 nd D n is he ordinry derivive of ineger order n The lef nd he righ puo frcionl derivives of order α R + re D α f ( = J n α D n f (, Db α f ( = ( 1n J n α b D n f ( b

6 90 Emil POPESU 6 We observe h he formul of frcionl derivive involves n inegrion which is non-locl operor, so frcionl derivive is non-locl operor Thus, clculing ime-frcionl derivive of funcion f( some ime, i is required ll he previous hisory from 0 o Time-frcionl derivives re nurlly reled o sysems wih memory These sysems re closely reled o frcls, which re presen in mos physicl sysems A propery of he frcionl operor is D p ( D q f ( = D p q f (, where f is coninuous nd 0 q p For p > 0 we obin he fundmenl propery of Riemnn-Liouville frcionl derivive D p ( D p f ( = f ( This formul mens h he Riemnn-Liouville frcionl differeniion operor is lef inverse o he Riemnn-Liouville frcionl inegrion operor of some order We lso remrk he following properies: D α f ( = dn d n D α n f (, Db α f ( = d n ( 1n d n D α n b f (, D α f ( = D α f ( α Γ(1 α f (, Db α f ( = Db α (1 α f ( Γ(1 α f (b The Mig-Leffler funcions E α nd E α,β nurlly occur in soluions of frcionl order differenil equions Mig-Leffler (1903 defined he funcion E α s he power series E α (z = k=0 z k Γ(αk + 1, α > 0 nd Wimn (1905 obined he generlision of E α, denoed E α,β (E α = E α,1 ; E 1 = expz: E α,β (z = k=0 z k, α > 0, β > 0 Γ(αk + β

7 7 Lgrngin nd Hmilonin Mechnics wih frcionl derivives 91 3 FRATIONAL EULER-LAGRANGE EQUATIONS Riewe hs used he frcionl clculus o develop formlism which cn be used for boh conservive nd non conservive sysems (see Riewe, 1996; Riewe, 1997 Using he frcionl pproch, one cn obin he Euler-Lgrnge nd he Hmilonin equions of moion for he non-conservive sysems The clssicl clculus of vriions ws exended by Agrwl (2002 for sysems conining Riemnn- Liouville frcionl derivives Mhemicl ools nlogous o clculus of vriions will be needed o minimize cerin funcionls Mny of he conceps nd resuls of clssicl clculus of vriions cn be exended wih minor modificions o frcionl clculus of vriions The frcionl Euler-Lgrnge equions re se of differenil equions involving boh he lef nd he righ frcionl derivives A frcionl clculus of vriions problem conins les one frcionl derivive erm We denoe by F 1 he se of ll funcions q( which hve coninuous lef Riemnn-Liouville frcionl derivive of order α nd righ Riemnn-Liouville frcionl derivive of order β for x [,b] nd sisfy he condiions q( = q, q(b = q b The problem cn be defined s follows: find he funcion q F 1 for which he funcionl b S[q] = L (,q, D α q (, D β b q ( d hs n exremum, where L(,q,u,v be funcion wih coninuous firs nd second pril derivives wih respec o ll is rgumens A necessry condiion for S[q] o hve n exremum for given funcion q( is Euler Lgrnge equion (Agrwl, 2002: q + D α b D α q + D β The generlized momen re inroduced s p α = D β b q = 0 D α q, p β = D β b q nd he Hmilonin depending on he frcionl ime derivives is H = p α D α q + p β D β b q L (,q, D α q (, D β b q ( The Hmilon s equions of moion re obined in similr mnner o he usul mechnics (Rbei e l, 2007: H =, H p α = D α q,

8 92 Emil POPESU 8 H p β = D β b q, H q = D α b p α + D β qp β We remrk h he frcionl Hmilonin is no consn of moion even if he Lgrngin does no explicily depend on he ime When α = β = 1, D α q = dq d, he bove funcionl reduces o he simples form nd Euler Lgrnge equion is S[q] = d d b D β b q = dq d L (,q, q d ( q q = 0 We cn generlize in srigh forwrd mnner o problems conining severl unknown funcions We denoe by F n he se of ll funcions q 1 (, q 2 (,, q n ( which hve coninuous lef Riemnn-Liouville frcionl derivive of order α nd righ Riemnn-Liouville frcionl derivive of order β for x [,b] nd sisfy he condiions q i ( = q i, q i (b = q i b, i = 1,2,,n The problem cn be defined s follows: find he funcions q 1,q 2,,q n from F n, for which he funcionl = b S[q 1,q 2,,q n ] = ( L,q 1 (,q 2 (,,q n (, D α q 1 (,, D α q n (, D β b q 1 (,, D β b q n ( d hs n exremum, where L(,q 1,,q n,u 1,,u n,v 1,,v n is funcion wih coninuous firs nd second pril derivives wih respec o ll is rgumens A necessry condiion for S[q 1,q 2,,q n ] o dmi n exremum is h q 1 (, q 2 (,, q n ( sisfy Euler-Lgrnge equions: q i + D α b D αq + D β i D β b q = 0 i = 1,2,,n i

9 9 Lgrngin nd Hmilonin Mechnics wih frcionl derivives 93 In vecor noion, he bove condiion cn be wrien s where q R n q + D α b D α q + D β D β b q = 0, 4 SOME APPLIATIONS ON ONSERVATIVE AND NON-ONSERVATIVE SYSTEMS A simple hrmonic oscillor is conservive sysem This sysem consiss of force F which pulls he mss m in he direcion of he poin x = 0 nd depends only on he mss s posiion x nd consn k The Newon s second lw for his sysem is F = m d2 x d 2 = kx The moion is described by he funcion x( = Acos(ω + ϕ, where k ω = m = 2π T The force F is conservive wih he poenil energy funcion V (x = 1 2 kx2 The Lgrngin of he pricle cn be wrien L (,x,ẋ = 1 2 mẋ2 V (x nd he equions of moion re rerieved by pplying he Euler Lgrnge equion ( d d ẋ x = 0 We hve Thus x = dv dx, ẋ dv m + dx = mẋ ẋ = 0, mẋ + kx = 0 We observe h if he Lgrngin of sysem is known, hen he equions of moion my be obined by he Euler Lgrnge equions The Lgrngin of sysem is no unique Lgrngins which describe he sme sysem cn differ by he ol derivive wih respec o ime of some funcion, bu hey will give he sme equions of moion

10 94 Emil POPESU 10 We consider now frcionl Lgrngin of he bove oscillory sysem L(,x, D α x = 1 2 m ( D α x kx2 Then he frcionl Euler Lgrnge equion is m D α b ( D α x kx = 0, This equion reduces o he equion of moion of he hrmonic oscillor when α 1 If we consider he sysem of wo plnr pendul, boh of lengh l nd mss m, suspended sme disnce pr on horizonl line so h hey moving in he sme plne, he kineic energy is K = 1 ( q 2 m q 2 2 nd he poenil energy is V = 1 mg ( q 2 2 l 1 + q2 2, where q 1 nd q 2 denoe he corresponding coordines nd g is he grviy consn (see Blenu e l (2012, for wo-elecric pendulum The Lgrngin hs he following form: L (,q 1, q 1,q 2, q 2 = 1 2 m ( q q mg ( q 2 2 l 1 + q2 2 The frcionl form of his Lgrngin is given by L(,q 1,q 2, D α q 1, D α q 2 = 1 [ 2 m ( D α q ( D α q 2 2] 1 mg ( q 2 2 l 1 + q2 2 To obin Euler-Lgrnge equions, we use I follows q i + D α b D αq + D β i D β b q = 0 i = 1,2 i D α b D α q 1 g l q 1 = 0, D α b D α q 2 g l q 2 = 0 The clssicl Euler-Lgrnge equions re obined if α 1 : q 1 + g l q 1 = 0, q 2 + g l q 2 = 0 As non-conservive sysem, we consider he dmped hrmonic oscillor In his cse, he fricionl force F f cn be modeled s being proporionl o he velociy v of he objec, F f = cv, where c is he viscous dmping coefficien From

11 11 Lgrngin nd Hmilonin Mechnics wih frcionl derivives 95 Newon s second lw, i follows h ie F = kx c dx d = md2 x d 2, ẋ m + cẋ + kx = 0 The frcionl Lgrngin nd he frcionl Ryleigh s dissipion funcion which describe his moion re L(,x, D α x = 1 2 m ( D α x kx2 nd R = 1 2 ( D α x 2 In his cse, F f = k D α x nd is derivble from frcionl Ryleigh s dissipive funcion R = 1 2 ( D α x 2 We modify he sndrd frcionl Euler-Lgrnge equions by including he frcionl Ryleigh s dissipion funcion wih ime frcionl derivive of he displcemen The frcionl Euler Lgrnge equion kes he form x + D α b D α x + D β Subsiuing L nd R in his equion we obin D β b x R D α x = 0 m D α b x ( D α x c (D α x kx = 0 For α 1 we ge he equion of moion of he dmped hrmonic oscillor m ẋ + cẋ + kx = 0 We cn ry o consruc he clssicl mechnics reled o he frcionl clculus We cn inroduce he frcionl velociy v( nd frcionl ccelerion ( s follows : v ( = 0 D α x(, ( = 0 D α v (, where 0 Dα f ( is he lef puo frcionl derivive (wih fixed lower erminl 0 nd moving upper erminl 0 D α 1 f ( = ( u α d f (udu, 0 < α < 1 Γ(1 α du 0 In he frcionl mechnics, we cn define Newon s equion by F = m = m 0 D α v (,

12 96 Emil POPESU 12 where m is mss of he body For body in resising medium in which here exiss rerding force proporionl o he frcionl velociy in uniform grviionl field, he equion of vericl moion is given by m 0 D α v = mg kv, v (0 = 0, which is frcionl uchy problem If we inegre his equion, we obin v ( = gd α (1 k m D α (v ( In order o find he formul of v (, we use resul from Sxen e l (2010, which yields v ( = mg [1 E α ( km ] k α In he clssicl mechnics, hs he soluion m v = mg kv, v (0 = 0, v ( = mg k [ 1 e k m ] We noe h he erminl velociy is he sme in boh cses mg lim v ( = k An exension of frcionl uchy problem for generl operors is given in Popescu (2010 Using spce-ime frcionl equion, in Popescu nd Popescu (2010, he scling nd inermien behvior of probbiliy densiy funcions of solr wind plsm prmeers flucuions is nlized 5 ONLUSIONS This pper inended o repor some of he imporn resuls in he re of frcionl clculus wih pplicions o mechnics Frcionl differenil models ply significn role in he descripion of he dynmics of mny complex sysems I is presened n exension of vriionl clculus wihin he frmework of frcionl clculus Frcionl vriionl principles conin clssicl ones s priculr cse when frcionl operors converge o ordinry differenil operors Frcionl mechnics, which is non-locl heory, describes boh conservive nd non-conservive sysems Using frcionl derivives nd iniil vlues of clssicl ineger-order derivive wih known physicl inerpreions, some illusrive pplicions on conservive nd non-conservive sysems were given

13 13 Lgrngin nd Hmilonin Mechnics wih frcionl derivives 97 REFERENES Agrwl, OP: 2002, J Mh Anl Appl 272, 368 Agrwl, OP: 2006, Journl of Physics A, 39(33, Blenu, D, Asd, JH, nd Pers, I: 2012, Romnin Repors in Physics, 64, 907 Hilfer, R: 2000, Applicions of Frcionl lculus in Physics, World Scienific Publishing o, New York, 2000 Hilfer, R: 2008, in Anomlous Trnspor: Foundions nd Applicions, R Klges, G Rdons nd I Sokolov (eds, Wiley-VH, 17 Kilbs, AA, Srivsv, HM, nd Trujillo, JJ: 2006, Theory nd Applicions of Frcionl Differenil Equions, Norh-Hollnd Mhemics Sudies 204, Elsevier, Amserdm Mig-Leffler, GM: 1903, R Acd Sci Pris, 137, 554 Podlubny, I: 1999, Frcionl Differenil Equions, Acdemic Press, New York Popescu, E: 2010, in Progress in Indusril Mhemics EMI 2008, A D Fi, J Norbury, H Ockendon, E Wilson (eds, Springer-Verlg Berlin Heidelberg, 983 Popescu, E, nd Popescu, N A: 2010, in Inernionl Associion of Geomgneism nd Aeronomy (IAGA 2nd Symposium: Solr Wind Spce Environmen Inercion, L Dm, A Hdy (eds, iro Universiy Press, 103 Rbei, EM, Nwfleh, KI, Hijjwi, RS, Muslih, SI, nd Blenu, D: 2007, J Mh Anl Appl, 327, 891 Riewe, F: 1996, Phys Rev E, 53, 1890 Riewe, F: 1997, Phys Rev E, 55, 3581 Smko, SG, Kilbs, AA, nd Mrichev,O I: 1993, Frcionl Inegrls nd Derivives: Theory nd Applicions, Gordon nd Brech, Yverdon Sxen, R, Mhi, A, nd Hubold, H: 2010, Proceedings of he Third UN/ESA/NASA Workshop on he Inernionl Heliophysicl Yer 2007 nd Bsic Spce Science, HJ Hubold nd AM Mhi (eds, Springer Berlin Heidelberg, 35 Trwneh, KM, Rbei, EM, nd Ghssib, HB: 2010, J Dyn Sys Geom Theories, 8, 59 Wimn, A: 1905, Ac Mhemic, 29, 191 Received on 25 November 2013