A HAMILTON-JACOBI TREATMENT OF DISSIPATIVE SYSTEMS

Transcription

1 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

2 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

3 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN ( 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

4 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

5 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

6 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

7 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

8 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

9 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

10 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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

11 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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) Baur, PS,(1931), Dissipaiv Dynamical Sysms Procdings of h Naional Acadmy of Scincs of h Unid Sa of Amrica, 17(5) Das, U, Ghosh, S, Sarkar, P, and Talukdar, B,(5),Quanizaion of Dissipaiv Sysms wih Fricion Linar in Vlociy Physica Scripa, 71 (3)

12 Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: (Prin) - ISSN 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) Rabi, E and Gulr, Y(199), amilon-jacobi Tramn of Scond-Class Consrains Physical Rviw A, 46(6) 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