Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 A AMILTON-JACOBI TREATMENT OF DISSIPATIVE SYSTEMS Ola A Jarab'ah Tafila Tchnical Univrsiy, Tafila, Jordan Khald I Nawaflh Mu'ah Univrsiy, Al-Karak, Jordan umam B Ghassib Th Univrsiy of Jordan, Amman, Jordan Absrac Dissipaiv sysms ar invsigad wihin h framwork of h amilon-jacobi quaion Th principal funcion is drmind using h mhod of sparaion of variabls Th quaion of moion can hn b radily obaind Thr xampls ar givn o illusra our formalism: h dampd harmonic oscillaor, a sysm wih a variabl mass, and a chargd paricl in a magnic fild Kywords: amilon-jacobi Equaion, Dissipaiv Sysms 1 Inroducion I is wll known ha h nrgy concp is almos indispnsabl in h analysis of physical sysms Such sysms can b sudid in rms of hir sourcs and sinks of nrgy A dissipaiv sysm is ha which canno sor all h nrgy impard o i by an xrnal sourc, losing nrgy hrough som sink (Grinr, 1953) In his work, dissipaiv sysms ar invsigad using h amilon- Jacobi quaion (JE) This quaion is simplifid using h sparaion-ofvariabls chniqu Th corrsponding principal funcion S is found Th quaion of moion can hn b drivd from his funcion, which rprsns h nrgy of h sysm in rms of h gnralizd coordinas and momna This, in urn, is usd as a basis for hso-calld canonical quanizaion using h WKB approximaion, hrby obaining h corrsponding amilonian and Schrödingr's quaion (Das, 5) In h amilonian formulaion of non-consrvaiv sysms, svral mhods hav bn dvisd o includ dissipaiv ffcs Th arlis mhod invoks h so-calld Rayligh dissipaion funcion, which is valid whn h fricional forcs ar proporional o h vlociy owvr, in his 7
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 mhod, anohr scalar funcion is ndd, in addiion o h Lagrangian, o spcify h quaions of moion This funcion dos no appar in h amilonian; so i is of no us whn amping o quaniz fricional sysms Anohr mhod, dvlopd by Baman (Baman, 1931), inroducs auxiliary coordinas in h Lagrangian ha dscrib a rvrsim sysm wih ngaiv fricion This mhod lads o xranous soluions, and h physical maning of h momna is no clar Furhr, Baur (Baur, 1931) provd ha i is impossibl o us a variaional principl o driv a singl linar dissipaiv quaion of moion wih consan cofficins Rcnly, a complly diffrn approach -h canonical- has bn dvlopd for invsigaing singular sysms(rabi, 199)A gnral mhod for solving h amilon-jacobi parial diffrnial quaion(jpde) for consraind sysms has bn proposd (Nawaflh, 4) and (Rabi, ) Th prsn work xnds his framwork, for h firs im, o dissipaiv Lagrangian sysms Th firs sp hr, hn, is o consruc JPDE for dissipaiv sysms (Scion) Thr sysms ar xamind wihin his framwork (Scion3): h dampd harmonic oscillaor(oghr wih h RLC circui and a viscous liquid); a sysm wih a variabl mass; and a chargd paricl in a magnic fild amilon-jacobi Formalism W sar wih h Lagrangian L = L ( q, q ) (1) L ( q, r q ) sands for h Lagrangian of h corrsponding consrvaiv sysm; i rprsns h sysm's physical Lagrangian, which mans h kinic nrgy minus h ponial nrgy Th dissipaion is incorporad hrough, which is a damping facor ( ) As usual, h gnralizd momnum is dfind by (Fowls, 1993) p = L q Th corrsponding amilonian is p i q = i L () JE for dissipaiv sysms is a firs-ordr, non-linar parial diffrnial quaion, h kinic nrgy bing, in gnral, a quadraic funcion of momnum of h form 71
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 ( q1, q,, qn ;,,, ; ) = q1 q qn Th gnralizd momna do no appar in his quaion, xcp as drivaivs of amilon's principal funcion S, which is a funcion of h N gnralizd coordinas q 1, q,, qn and h im Now, if S ( q1, q,, q N ; α1, α,, α N ) is a compl ingral of JE, h ingrals of amilon's quaions of moion will b givn by β j, α = j β j bing som consans This quaion can b invrd o find h N gnralizd coordinas q as funcions of α j, β j and Th gnralizd momna ar i = q j p j Thus, h amilon-jacobi funcion is givn by ( q, ) ( q, p) (5) L = T V Sinc is h physical Lagrangian of h sysm, i follows ha is h physical amilonian rprsning h sysm's oal nrgy: TV (Goldsin, 198) Th rsuling acion S is S = Ld = ( p q i i ) d To build JPDE, w mus wri S in h sparabl form S ( q, α, ) = W ( q, α) f ( ), (6) whr h im-indpndn funcion W(q,α) is h so-calld amilon's characrisic funcion Diffrniaing Eq (6) wih rspc o, w find ha f = From Eq(5), i follows ha (3) (4) (7) 7
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 f = (8) Th lf-hand sid of his quaion dpnds on alon; whras h righ-hand sid dpnds on q alon Each sid mus hn b qual o a consan indpndn of boh q and Thrfor, h im drivaiv S in JE mus b a consan, usually dnod by (-α) Thus, S( q, α, ) = W ( q, α) α (9) I follows ha W(q) q, = α q 3 Exampls 31 Dampd armonic Oscillaor Th following Lagrangian is suiabl for his sysm in on dimnsion (Baman, 1931): 1 1 L( q, q, ) = mq mω q, (1) m bing h mass of h oscillaor andω h frquncy of oscillaion Th linar momnum is givn by L p = mq q = (11) This quaion can radily b solvd o giv p q = m Th canonical amilonian has h sandard form (1) = pq L (13) Subsiuing Eqs(1),(11) and (1) ino (13), w g h amilonian = p m 1 mω q (14) W shall now us a chang of variabls o solv his quaion Suppos ha 73
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 y = q (15) Using h chain rul, w find ha y p = = = (16) q y q y Thn w hav p = = q y = (17) Subsiuing Eqs(17)and (15) ino (14), w find 1 y m y 1 mω = (18) Thn JE aks h form 1 m y 1 mω y p y = (19) This diffrnial quaion is h wll-known amilon- Jacobi quaion for h simpl harmonic oscillaor Is soluion is α y sin mω (( β ) ω) = () In rms of q, using Eq(15), w g α q = sin( ( β ) ω) (1) mω In h limi, Eq (1) is in agrmn wih h wll-known rsul for h fr harmonic oscillaor, as i should On can follow h sam sps oulind in his xampl o sudy ohr dissipaiv sysms, such as h RLC circui and a viscous liquid, as follows: For h RLC circui, an appropria Lagrangian is 1 Q L( Q, Q, ) = LQ C 74
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 I follows ha 1 y = L y C Th J funcion can b obaind as y L S = Lα dy α C Th rsuling quaion of moion is 1 Q = Asin ( β ) CL For a viscous liquid in a ub, w hav h following Lagrangian: 1 L( q, q, ) = lq gq, whr l is h lngh of h liquid column, g is h graviaional acclraion akn hr as consan, and q rprsns h variaions in h liquid high Is amilonian is givn by 1 = gy l y Th J funcion can b obaind as S = lα gly dy α Finally, h quaion of moion is α g q = sin( β ) g l 3 Sysm wih a Variabl Mass A suiabl Lagrangian for his sysm is (Razavy,5): 1 L( q, q, ) = mq mgq Suppos ha h mass changs wih im according o m = m () 75
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 Thn 1 ( q, q, ) = mq m gq L (3) Clarly, h damping facor hr ariss from h variaion of h mass wih im Th linar momnum is givn by (4) Th usual ramn givs p = m q p = m = ; (5) gq m Furhr, h principal funcion aks h form S ( q, ) = qn( ) D( ) (6) So on gs and = qn ( ) D ( ), = N() q p q Wih =, w hav p = = q ( N( ) ) Th corrsponding JE aks h form 1 m ( N( ) ) gq m Maching powrs of q, w g 1 m and ( N( ) ) = (), (7) (8) (9) q N ( ) D ( ) = (3) D (31) 76
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 gq m qn ( ) = (3) N( ) = N mg and Afr ingraion:, gn D( ) = mg N D 3 m Puing Eqs(33)and (34) ino (6), w hav S = m gq N q m g N 3 m Thus, D gn β = = q g N m N Tha is, q g β N =, m and p = = N q m g (33) (34) (35) (36) (37) (38) 33 A Chargd Paricl in a Magnic Fild As a final xampl, l us considr h moion in wo dimnsions of a chargd paricl undr h influnc of a cnral forc ponial, V=kr /, as wll as an xrnal consan magnic fild prpndicular o h plan of B = B kˆ moion: Th vcor ponial is 77
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 1 1 A = B r = B ( yiˆ Th Lagrangian is [9] xˆ) j 1 q k L = m( x y ) ( v A) ( x y ) c In h prsnc of damping ffcs, h Lagrangian bcoms L = 1 m( x y ) v = xi ˆ y ˆj Wih, L = 1 m( x y ) q c ( v A) qb ( xy yx ) c To simplify, plan polar coordinas ar usd: x = r cosθ; y = r sinθ Thn Eq(4) bcoms 1 L = m( r r θ ) Th conjuga momna ar L = mr ; p r r = L p θ = = mr θ θ qb θ c r qb c r Th final form of h Lagrangian is, hn, L = pr m p θ Th amilonian is pr m = wih mr 1 mr k ( x y ) (39) q B r p θ 8mc k ( x y ) k r qb r c k r (4) (41) k r ; (4) 78
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 p r p θ = ; r = θ Th corrsponding JE is 1 r m 1 mr θ qb r c k r = (43) Sincθ is a cyclic coordina, h conjuga momnum mus b P θ = S = To simplify, w choos γ = θ consan: γ As a rsul, JE rducs o 1 m or r q B r 8mc k r = ; (44) 1 m whr r q B 8mc C = 16mc Cr k =, Now, using a chang of variabls y = r : p = = r y (45) From Eq(45), w find JE: 79
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 1 m (46) S y Cy = This quaion can asily b solvd o giv α C y = sin( ) β C m In rms of r w g, r = α C sin( ) β, C m (47) 4 Conclusion In his work, dissipaiv sysms hav bn invsigad wihin h framwork of h amilon-jacobi mhod Th amilon-jacobi parial diffrnial quaion for hs sysms has bn obaind wihin h canonical mhod Th principal funcion S has bn drmind by invoking sparaion of variabls and h chain rul, in h sam mannr as for rgular im-indpndn Lagrangians Th quaion of moion can hn b radily obaind, hrby finding familiar rsuls bu wih unfamiliar chniqus In ordr o s our proposd mhod, w hav xamind hr xampls: h dampd harmonic oscillaor (oghr wih wo "varians": h RLC circui and a viscous liquid); a sysm wih a variabl mass; and a chargd paricl in a magnic fild Our formalism may shd furhr ligh on such sysms as wo inracing paricls moving in a viscous mdium, and h classical radiaing lcron, among ohrs Rfrncs: Baman,,(1931),On Dissipaiv Sysms and Rlad Variaional Principls Physical Rviw, 38 (4) 815-819 Baur, PS,(1931), Dissipaiv Dynamical Sysms Procdings of h Naional Acadmy of Scincs of h Unid Sa of Amrica, 17(5) 311-314 Das, U, Ghosh, S, Sarkar, P, and Talukdar, B,(5),Quanizaion of Dissipaiv Sysms wih Fricion Linar in Vlociy Physica Scripa, 71 (3) 35 37 8
Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 Fowls, G,(1993), Analyical Mchanics, (5 h d) Florida: arcour Brac and Company Goldsin,,(198),Classical Mchanics, ( nd d) Rading-Massachuss: Addison-Wsly Grinr,W,(1953),Classical Mchanics: Sysms of Paricls and amilonian Dynamics, Nw York:Springr-Vrlag Nawaflh, K, Rabi, E, and Ghassib, (4),amilon-Jacobi Tramn of Consraind Sysms Inrnaional Journal of Modrn Physics A, 19 (3) 347-354 Rabi, E and Gulr, Y(199), amilon-jacobi Tramn of Scond-Class Consrains Physical Rviw A, 46(6) 3513-3515 Rabi, E, Nawaflh, K, and Ghassib, (),Quanizaion of Consraind Sysms Using h WKB Approximaion Physical Rviw A, 66 () 411 Razavy, M (5), Clasical and Quanum Dissipaiv Sysms, London:Imprial Collg Prss 81