Hamiltonian Representation of Higher Order Partial Differential Equations with Boundary Energy Flows

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1 ounl of Aled Mhemcs nd Physcs Publshed Onlne Novembe 05 n ScRes. h:// h://dx.do.og/0.436/jm Hmlonn Reesenon of Hghe Ode Pl Dffeenl Euons wh Boundy Enegy Flows Gou Nshd Demen of Mechncl nd Envonmenl nfomcs Gdue School of nfomon Scence nd Engneeng Tokyo nsue of Technology Tokyo n Receved 7 Seembe 05; cceed 4 Novembe 05; ublshed 7 Novembe 05 Coygh 05 by uho nd Scenfc Resech Publshng nc. Ths wok s lcensed unde he Ceve Commons Abuon nenonl Lcense (CC BY. h://cevecommons.og/lcenses/by/4.0/ Absc Ths e esens sysem eesenon h cn be led o he descon of he necon beween sysems conneced hough common boundes. The sysems conss of l dffeenl euons h e fs ode wh esec o me bu slly hghe ode. The eesenon s deved fom he nsnneous mulsymlecc Hmlonn fomlsm; heefoe ossesses he hyscl conssency wh esec o enegy. n he neconnecon cul s of conol nus nd obsevng ouus clled o vbles defned on he boundes e used. The o vbles e sysemclly noduced fom he eesenon. eywods Symlecc Sucue Dc Sucue Hmlonn Sysems Pssvy Pl Dffeenl Euons Nonlne Sysems. noducon Enegy s one of he mos mon conces fo descbng hyscl sysems. n nlycl mechncs n enegy n he sysems cn be neeed s Hmlonn. Hmlonn sysems cn be chcezed by symlecc sucues [] deved fom he skew symmey h congen bundles ossess. Hmlonn sysems nd he symlecc sucues hve been wdely led no only n hyscs bu n engneeng cully conol heoy [] [3]. Secfclly n n eleccl ccu n enegy s defned s he me negl of he oduc of cuens nd volges. Enegy flows beween ech ccu lso blnce f hee s no dssve elemen. Fuhemoe he sum of cuens blnces beween nflows nd ouflows ny node nd he deced sum of vol- How o ce hs e: Nshd G. (05 Hmlonn Reesenon of Hghe Ode Pl Dffeenl Euons wh Boundy Enegy Flows. ounl of Aled Mhemcs nd Physcs h://dx.do.og/0.436/jm

2 G. Nshd ges ound ny closed loo s zeo ccodng o chhoff lw. ndeed such oees hve been genelzed o vous hyscl sysems n ems of o-hmlonn sysems [] ogned n bondgh heoy [4]. The bove cul s wh he hyscl dmenson of owe e clled o vbles nd he enegy blnces cn be egded s ssvy []. A sysem s ssve f nd only f fne moun of enegy cn be exced fom he sysem. n ohe wods enegy chnges n necons cn be obseved by he o vbles nd suled enegy s less hn soed enegy f sysem s ssve. Pssvy-bsed conols v o-hmlonn sysem eesenons hve been feuenly used n conol desgns [3]. Ths e ooses he o-hmlonn eesenon of sysems of hghe ode l dffeenl euons defned on domn wh boundy. The eesenon cn be fomlly fomuled fom he vewon of he mulsymlecc fomlsm [5] [6] unde he ssumon of fs ode wh esec o me bu ossbly hghe ode wh esec o sl vbles. The o eesenon fo Hmlonn sysems wh boundy enegy flows ws ned by he dsbued o-hmlonn sysem n [7]. The sysems ssfy owe blnce defned on he boundy; heefoe cn descbe he necon beween he sysems conneced hough common boundes. Thus ssvy-bsed conols n hs fomulon cn be enhnced s boundy enegy conols. Vous secs of he dsbued o-hmlonn sysems hve been suded e.g. he mlc eesenon of dsbued o-hmlonn sysems [8] nd he elonsh beween feld euons nd dsbued o-hmlonn sysems [9]-[]. The hghe ode eesenon of he dsbued o-hmlonn sysems hs been oosed n e.g. [] [3]; howeve hey e no eled wh he mulsymlecc fomlsm. Thus we fs ele hghe ode l dffeenl euons wh he mlc Hmlonn sysems [4]. Nex we descbe he mlc eesenon s Dc sucue defned ove he mulsymlecc mnfold n nlogy wh he fs ode fomlzons [5]-[7]. Dc sucues [8] [9] e unfed conce of symlecc nd Posson sucues. Then we deve he Sokes vonl dffeenl fom he fc h hghe ode devves yeld vons of boundy o vbles hough negon by s nd Sokes heoem. Fnlly we shows h he boundy enegy blnce nd he Sokes-Dc sucue [7] [0] h s n exended Dc sucue fo dsbued o-hmlonn sysems cn be defned n he oosed hghe ode feld o Hmlonn sysems wh boundy enegy flows. Ths e s ognzed s follows: n Secon we mke bef summy of o-hmlonn sysems nd exln he movon of hs sudy. Secon 3 noduces mhemcl elmnes fom some efeences. Secon 4 esens he followng hee conces unde he ssumon of me-sl slng: n mlc Hmlonn eesenon usng he dul sucue deved fom he mulsymlecc nsnneous fomlsm Sokes vonl dffeenl deved fom he negon by s fomul nd 3 he mlc hghe ode feld o Hmlonn eesenon wh boundy. Secon 5 noduces he foml o eesenon fo hghe ode l dffeenl euons fom he mlc Hmlonn eesenon. We cll hghe ode feld o Hmlonn sysems wh boundy enegy flows. Fnlly Secon 6 lluses wo modelng exmles.. Summy of Po-Hmlonn Reesenons Ths secon exlns he conce of o-hmlonn sysems by mens of smle exmle of couled mulhyscl models nd he movon of hs wok... Po Reesenon fo Lumed Pmee Enegy Consevng Physcl Sysems Le us consde he followng model of he dec cuen moo conssng of n eleccl ccu nd n mue: d dω L + R = u ω + Bω = N ( d d whee = = d d s he cuen h s he me devve of he elecc chge = ω = ω = dθ d s he velocy of he ngle θ = θ nd u s he nu volge. n ( he followng consns e defned: he nducnce L he essnce R he bck elecomove foce consn he ne momen he vscous fcon consn B nd he oue consn N. When he dssve elemens nd he nu e null.e. R = B = 0 nd u = 0 he sysem ( s enegy consevng nd cn be fomuled s he followng sndd Hmlonn sysem: 473

3 G. Nshd = = θ = θ = θ whee we hve defned he Hmlonn nd he momen s follows: = + θθ = L + θ κθ + κθ (3 = = L + κθ θ = = θκ θ wh : = κ κ nd N : = κ+ κ fo cen κ nd κ. We shll ugmen he Hmlonn sysem ( s he followng o-hmlonn sysem wh he dssons R nd Bθ nd he nu u: f R e e [ 0 ] = u y = f e e f B e θ e y [ 0 ] = f 0 = e e whee we hve defned he followng vbles clled o vbles: d d f = f = e = = e = = κθ d d d θ dθ f = f = e = = θ e = = κ d d θ θ nd n cul f j nd j j = e clled flows nd effos esecvely. Hee we cn see h he eleccl nd mechncl subsysems e couled by he neconnecon of he effo vbles: nd θ. Then he summon conssng of he oducs of he s of he o vbles s euvlen o he me devve of he Hmlonn.e. he ol enegy chnge of he sysem. ndeed we cn decly clcule he followng owe blnce: e fo = { } nd { } d = f e + f e + f e + f e = R B θ + uy (7 d whee we hve used he elon y = = e. The elon (7 mens h he dssons R nd Bθ sblze he sysem by decesng he enegy nd he oduc uy of he nu nd he ouu my ffecs he dsson e. Fuhemoe he Hmlonn cn be conolled f we cn fnd suble nu ssfyng ( 0 = uy d fo he desed Hmlonn ( + n he me nevl [ ] 0. A fne moun of 0 enegy chnges n necons cn be ecsely obseved by he oduc of n nu-ouu f he sysem s ssve. Hence hese conols e clled ssvy-bsed conols... Dsbued Po-Hmlonn Sysems n he cse of dsbued mee sysems he eesenon s clled dsbued o-hmlonn sysem nd hs he followng foml sucue: b e f 0 d e f = b = f d 0 e e e clled he Sokes-Dc sucue defned on he sysem domn wh he boundy (see [7] fo dels whee d s he exeo dffeenl oeo we hve defned θ ( (4 (5 (6 (8 474

4 G. Nshd α α b n f = Ω f = Ω f Ω ( δ n δ n b n e = Ω e = Ω e Ω ( δα δα (9 fo Hmlonn funconl k H = Ω s he sce of dffeenl k-foms δ δα s he vonl devve wh esec o he dffeenl fom α = + nd + = n+. Then he owe blnce (7 s exended o he followng elon descbed by dffeenl foms: dh d = f e + f e = e e. The boundy negl em n (0 s geneed fom he domn negl em by Sokes heoem: dω = ω fo n n-fom dω n he n-dmensonl domn. Hence he ssvy-bsed conols cn be enhnced s boundy enegy conols by egdng e s nu-ouu.e. boundy o vbles..3. Movon e nd (0 As we hve seen bove he o-hmlonn eesenons e mon fo he conol of e.g. nonlne sysems dsbued mee sysems hghe ode sysems nd mul-hyscl sysems fom whch s dffcul o obn nlycl soluons n closed fom. Ths e deves foml o-hmlonn eesenon of gven l dffeenl euon ncludng hghe ode devves n ems of he mulsymlecc fomlsm. n hs e we ssume h gven sysem of l dffeenl euons s deemned by vonl oblems. Such sysem mus be egded s n enegy consevng hyscl sysem [5] hough Legende nsfomons h m Lgngn sysems o Hmlonn sysems. Ths ssumon comes fom he fc h ny sysem cn be decomosed no vonl subsysem h cn be deemned by vonl clculus nd non-vonl subsysem h cnno be noduced fom ny Lgngn on concble mnfold []. Fo exmle s we seen n ( he dssve ems R nd Bθ cnno be deved fom ny Lgngn o Hmlonn. On he ohe hnd Lgngns of he fs subsysem cn be exlcly clculed by homooy oeos n ems of he excness of vecl dffeenl foms n vonl b-comlex [] [3]. Hence we consde only he vonl subsysems n hs sudy. 3. Mhemcl Pelmny Mhemcl noons used n hs e bsclly confom o hose of he efeences [] [4] [5]. 3.. Mul ndex fo Hghe Ode Devves Le X be n ( m + -dmensonl mnfold. Le be fbe mnfold on X nd consde he -h ode je bundle ove. We denoe he locl coodnes of X nd esecvely by x ( x nd ( x whee 0 m nd l. The mul ndex descbes ll vbles of he eeed combnon j of x h men hghe ode devves wh esec o he vble e.g. j= x x. We he ode xx 0 of by nd s used s 0 h mens ll devves u o he -h ode. Le x be he me s coodne fo nd le x be he sl coodnes fo s m. n some cse we use he bbevon such s = 0 = x 0. Exmle. Le m = l = nd =. We defne he locl coodnes of X by ( x 0 x = ( y nd hose of by ( 0 0 x x; = ( y ;. Then he locl coodnes of he je bundle e ( x x ; = ( y ; y y yy fo ll 0. Noe h s descbed s funcon of y; howeve ech elemen of s egded s ndeenden vbles on he bundle. By usng he summon convenon fo exm- d = d + d fo =. le we cn nee such s y y 475

5 G. Nshd 3.. Tme-Sl Slng of Se Sce Vonl oblems cn be fomlzed s follows. m 0 : = dx Ω + s Lgngn densy of funconl Defnon. Consde he ( 0 m + -fom dx. We defne he vonl devve d of he Lgngn densy s he ( m + -fom m dv dx= d dx Ω + ( ( h deemnes he sony condon dv 0 of he vonl oblem whee 0 nd d v s he vecl dffeenl oeo (see Secons B nd C. n he followng dscussons he vonl devve d s lfed on. Le us consde he followng conol sysem s mn objecve. Assumon. A gven sysem s defned on concble domn wh boundy nd cn be deved fom Lgngn densy n funconl foms ncludng devves h e fs ode wh esec o he me coodne nd u o ( -h ode wh esec o sl coodnes. Unde hs ssumon he mul ndex u o -h ode used fo descbng -h ode Lgngns cn be = whee he mul ndex s of sl coodnes fo 0. defned s { } Exmle. n he cse of m = l = nd = le ( x 0 x x ; ( yz ; of on X. Then by defnng = { yz } nd { } { y z y z} 0 cn be descbed s ( x x x; ( yz ; = be locl coodnes = = he locl coodnes of =. Whou Assumon my nclude yy zz. Fuhemoe he followng second ssumon s mon when we use he mulsymlecc nsnneous fomlsm (see Secon E. Defnon. Le us consde he me-sl sl domn = conssng of me nevl nd sl domn. Then sysem defned on n nsnneous me s defned on. m 0 Ω + esced o cn be descbed s Assumon. A Lgngn densy 0 dx s Ω m ( me n = whee we denoe he sl (hozonl volume fom dx dx m by dx s. Noe h we e vbles ncludng me devves e.g. n hs seng. n bundle esced o he sl domn 4. mlc Hmlonn Reesenon nduced fom Dsbuons n hs secon we esen symlecc sucue fo dsbuons deemned by l dffeenl euons n ems of he mlc Hmlonn eesenon [4]. A dsbuon s subbundle of ngen bundle h s defned by sysem dynmcs exenl consns nd nenl consns geneed by degenee Lgngns. On he ohe hnd feld Hmlonn sysem s defned by he covn Hmlonn n he mulsymlecc fomlsm [6]. Howeve he covn Hmlonn does no coesond o he ycl Hmlonn h e consn wh esec o me evoluon e.g. fo cle sysems bu he nsnneous Hmlonn deved fom he me-sl slng. 4.. Dsbuons on n n-dmensonl mnfold M whee ω fo ech s dffeenl -fom. Then he submnfold N of M s clled n negl mnfold of f ξω = 0 fo ny veco ξ on he ngen sce TN ech on N whee l. Defnon 4. Le M be n n-dmensonl mnfold M. A mohsm ssocng n -dmensonl subsce of he ngen sce TM wh ech on M s clled n -h ode dsbuon. The dsbuon s clled egul f he dmenson s nvn. Defnon 3. Consde sysem of Pfff euons = { ω = 0 l} 476

6 G. Nshd The negbly of cn be ehsed by dsbuons. Th s he -dmensonl dsbuon s de- = ξ TM ω = 0 l fo ech on M whee = n l. fned by { ξ } 4.. Symlecc Sucue Resced o Dsbuons The elonsh beween Lgngn nd Hmlonn sysems s gven by Legende nsfomons (see (65 n Secon D. Fo clsscl feld euons he Legende nsfomons (o Lgngns e no egul n genel nd hus hey e no one-o-one. Howeve he Legende nsfomon cn esonbly wok unde he followng weke condon. Pooson. [5] Le P be n embedded closed subbundle of. Then he Legende nsfomon σ : s clled lmos egul f σ s submeson (.e. Tσ s sujecve wh esec o he mge P. n hs cse hee exss veco feld ξ on M such h Tσ ξ = ν fo ny veco feld ν on P. Pooson. [5] The followng condons e euvlen: φ Γ ssfes he Eule-Lgnge euons j φ Γ( ssfes he Cn euons 3 f σ s lmos egul δ = σ j φ Γ( P ssfes he Hmlon-De Donde euons whee Γ s he sce of ll secons nd j φ s he ( -h je of he secon φ. Remk. Noe h φ = π ρ s no lwys he exemum of he ognl vonl oblem.e. Eu- 0 le-lgnge euons even f cen ( ρ Γ s Cn euon whee π 0 : s he nul ojecon. Ths coesondence s vld only f σ s egul [5]. Hence we s fom Eule-Lgnge euons n hs e. Unde he ssumon of fs ode Lgngns wh esec o me he covn Legende nsfomon σ : n (65 s esced he followng nsnneous Legende nsfomon: = f = ( s = Ds f < m whee 0 s he mul ndex wh esec o sl vbles nd Ds = D = s he sl ol s dvegence. The em D s n he second euon of ( s noduced fom = 0 n D of (65 becuse Lgngn s fs ode wh esec o me. Fom he bove elmnes we shll ele dsbuon of Eule-Lgnge euons wh dsbuon of Hmlonn sysems on he mulsymlecc mnfold hough he elons n Pooson. ndeed he followng Hmlonn eesenon fo cn be defned. Defnon 5. Consde he ( -h mulsymlecc mnfold wh he locl coodnes ; x unde Assumon. Then he locl coodnes of V cn be wen s ( ( x ; coodnes ( he elons: δ δ becuse hs esuls fom he ddon of he locl coodnes of nd he δ δ geneed by he dffeenl oeo δ wh esec o nd we hve used δ = D = ; δ δ V h s defned by ξ ( dθ ( x ξ δ = D. On he ohe hnd he locl coodnes of V * becuse s gven by he ng of e * dθ V nd by veco beween he vecl ngen nd congen bundles []. Noe h hee s no djon vble of defnng n ffne sucue [5] (. 4. Now we cn consde he followng nduced symlecc sucue nduced fom dsbuons. Pooson 3. Le σ : be he lmos egul Legende nsfomon nd le V be egul dsbuon on h s esced o. We esc he mulsymlecc ( m + -fom Ω ove (see Secon D o.e. we defne skew symmec blne fom by Ω =Ω. Then hee 477

7 G. Nshd exss he followng subbundle ( α * V V fo ech z nd fxed : * { z z z z z αz z ( z z z } z = v V V v z ( w =Ω z v w w z (3 whee we hve defned σ ( π0 Vπ 0 V = V V V s he dsbuon lfed long he vecl ngen m Vσ s he vecl ngen m of σ nd Vπ V V of 0 : he cnoncl ojecon π 0 :. Poof. Accodng o Pooson hee exss veco feld ξ T fo ny veco feld ν. Thus hee exss lso ζ = ϖ ξ whee : ϖ T V s he nul ojecon. Hence he elon n (3 s esuled fom he nue of he symlecc sucue. ndeed fo gven vz ( z V z hee * lwys exss he coesondng ( m + -fom α z V z h deemnes Hmlonn veco feld. Le us noduce Dc sucues on veco sces fom he efeences [] [8] [9]. Defnon 6. A Dc sucue on veco sce A s subsce A A such h = whee A s he dul sce of A s he ohogonl sce of wh esec o he symmec ng on A A such h ( v α ( v α = α v + α v fo (( v α ( v α A A nd s he nul ng beween A nd A. Coolly. n (3 s he (lmos Dc sucue. Poof. f we fx he coodne of.e. he covn Hmlonn hen Euon (3 s he ycl fom of nduced Dc sucues [5]. 5. Dsubed Po-Hmlonn Sysems wh Hghe Ode Boundy Enegy Flows n hs secon we deve foml sucue of dsbued o-hmlonn sysems wh boundy enegy flows ncludng hghe ode devves fom he evously dscussed mlc Hmlonn eesenon. The enegy flows ssng hough boundes of sysem domns e used fo boundy neconnecons o ssvybsed boundy conols. 5.. Boundy Tems Geneed by negon by Ps n hghe ode vonl oblems he zeo boundy condon s usully ssumed fo smlfcon o some ohe eson. Then boundy ems geneed by Sokes heoem fe lyng negon by s e elmned. Acully hese boundy ems e eled wh he boundy enegy flows. Le us ecll such clculon h yelds he boundy em n vonl clculus. We fs defne he followng noon fo smlfcon. Defnon 7. Fom he Legende nsfomon σ n (65 we cn deve he followng vble: : = = + D (4 whee 0 0 m D s he ol dffeenl oeo (6 nd we hve se = 0 fo =. Now we consde he Lgngn densy funconl dx. The vonl devve of he Lgngn densy funconl cn be nsfomed by he negon by s fomul s follows: d dx = d dx = D ( d dx D d d x (5 whee 0. By Sokes heoem [] fo ny 0 m he fs em n he gh sde of (5 cn be nsfomed no whee m D s he ol dvegence h cs s = D 0 D d dx = d d x x nd dx s he volume ( m x (6 -fom on. Fo he 478

8 G. Nshd negnds n (5 nd (6 consde he oeon of dffeenl foms h see coeffcen fom vecl bss. Fo exmle d dx cn be decomosed no nd d. Ths oeon s defned s follows. m Defnon 8. Fo n ( m + -fom dv dx = ddx Ω + ( we defne he negon by s oeo s he followng locl exesson: + + Ω Ω k j m m d : ; x ; k j+ k j+ x ; D x ; whee we denoed Ω m+ ( = Ω m+ ( Ω m (.e. he syle {} (7 A B mens he se of he nsfomed locl coodnes (A on nd (B on esecvely. Hee unnsfomed coodnes unde -h ode e omed n (7 nd he numbes cn be exlcly clculed by k = 0 k nd j = k. Remk. Fo Lgngn densy d s m ( x Ω esced o he sl domn n he mesl sl sce he oeon (7 cn be lso well defned. n hs cse he boundy ems geneed by D n he ol dvegence e elmned becuse hee Sokes heoem wh esec o he me devve s no boundy of on n he me xs. The eeed lcon of he negon by s oeo cn yeld ll vons of boundy ems eed n vonl clculus. Pooson 4. Fo some whee v he v-h degee negon by s oeos s defned by v ( ( d = ( v+ ( ( ( d v (8 h cn be exessed s he coodne nsfomon = 0 f = (9 = D f 0. < v Poof. Fom he dec clculon of ( ( d ( m Ω + m h s deved fom h on ( we obn he followng eesenon on Ω + ; v 0 0 ; x D x D ( D D 0. (0 = 0 = = The fs bcke of (0 ncludes he nonzeo of he Eule-Lgnge euon; heefoe hs oeon s euvlen o he vonl dffeenl. The ls elemens n he second bcke coesond o n (9 fo Sokes Vonl Dffeenl The symlecc sucue nduced fom dsbuons does no hve ny nfomon on boundy enegy flows. n hs secon we defne vonl dffeenl oeo wh boundy ems geneed by negon by s nd Sokes heoem clled he Sokes vonl dffeenl. The Sokes vonl dffeenl cn be used n he nduced symlecc sucue fo elng gven Lgngn wh o-hmlonn eesenons. m 0 Pooson 5. Fo n -h ode Lgngn densy Ω + ( h s fs ode wh esec o me nd ( -h ode wh esec o sl coodnes he followng vonl dffeenl oeon cn be defned: ( m+ 0 * : d = χ d ι : Ω V ( 479

9 G. Nshd χ whee * * : V V (gven s Ω κ. We cll d he Sokes vonl dffeenl on wh. Befoe he oof of Pooson 5 we should ee he followng bundle ms. We fs consde he m beween V * nd V ove. Lemm. We cn defne he followng bundle m Ω : Ω V V * : ; ( x δ δ ( x δ δ ; ;. Poof. Ths cn be oven by he dec clculon wh esec o he symlecc fom (see Defnon 5. Lemm. By usng he Legende nsfomon σ : we cn defne he followng bundle m unde he fs ode ssumon: κ : V V ; * ( x δ δ ( x δ δ ; ; whee nd s defned by σ. Poof. Thee exss he followng bundle m κ unde he ssumon: κ * * : VV V V ; ( x δ δ ( x δ δ ; ;. * Ths cn be oven n nlogy wh he dffeomohsm n he lumed mee cse.e. κ : TT * T T ; ( x ; δ δ ( x ; δ δ ([6]. 40 [6]. The nvese m of (4 cn be exended s * m on he ffne bundle (3. ndeed we hve he bundle m beween he bundle V nd he ognl bundle V of he ngen ffne bundle V V. Becuse he locl coodnes of * * * x ; n he cse of fs ode wh esec o me he locl coodnes of e ( e ( ; V V x nd hose of *. nd V * e ( ; s esecvely ( (3 (4 x. Thus unde he ssumon we cn egd VV * nd V V * Nex we egd he devve of Lgngn densy d s dffeenl ( m + he bove dscusson by usng. Lemm 3. The sndd vonl devve ( ( m+ 0 m+ d v ι : Ω Ω ; dx d dx by denfyng -fom on cn be egded s he followng devve unde he ssumon of fs ode Lgngns wh esec o me: ( m 0 d v ι : Ω + V ; dx d + d d x whee we hve defned he nclusons : ι ; ( ; ( ; x x nd : ; ( ; ( ; ι x x. Poof. By : ; ι ( x ; ( x ; nd ι ( x ; ( ; x we cn ewe (5 s m + 0 ( ( v ι : ; d : Ω V ; dx d + d d x (7 nd n (5 (6 480

10 G. Nshd m whee we hve used Ω + ( Y VY. Moeove he ncluson ι : V ; ι V V yelds he dul m ( x δ δ ( x δ δ ; ;. Unde he ssumon we cn secfy by = n he bove euons. Poof of Pooson 5: Fom Lemms -3 we cn see h he m d s well-defned. ( Locl Exesson of nduced Symlecc Sucues Fom he evous eons we cn deve he elonsh beween he dsbuon nd n nsnneous Hmlonn sysem on h s descbed by he nduced symlecc sucue usng he Sokes vonl dffeenl d. m 0 = dx Ω + be Lgngn densy h s fs ode wh esec o me Defnon 9. Le nd le V be egul dsbuon on. Consde he symlecc sucue nduced on n (3 nd he Sokes vonl dffeenl : ( ( P = V V σ be he mge of d on wh n (. Le Ω m + 0 * V whee we hve defned ( Vπ 0 defne he mlc hghe ode feld Hmlonn eesenon on wh s whee X P. ( =. Then we X d (9 Theoem. The locl exesson of he mlc hghe ode feld o Hmlonn eesenon on X d s gven s follows: wh deemned by ( ( D = D D D (30 = = whee s he covn momenum nd we hve defned he null sce 0 = α V α v = v P V { } (. On he boundy hee s he followng ( m j j s s D D d w -fom: whee j s he mul ndex wh esec o sl coodnes s he mul ndex geneed by eeed emuons (see Secon B nd we hve defned he sl ol dvegence Ds = D =. Poof. Le X = ( x; D D be he veco feld on V. Fo veco feld X we consde he followng locl exesson of he symlecc fom Ω whee (( ( Ω x u α x u α = α u α u (3 α u : V V s he nul ng beween u X V nd ohe hnd fom he locl exesson of he Sokes vonl dffeenl on we denoe he -fom felds by elon n (3 nd (3 we obn he condon ( x m α V (3. On he d = ; δ δ (33 * u : δ = nd * α : δ * * u u α α α u α u =. Fom he euvlen condon beween he ls + = (34 48

11 G. Nshd δ u + α = α D D u (35 fo ny u nd α. Then Euon (35 yelds (30 becuse = nd δ = h cn be deved fom he Eule-Lgnge euon. ndeed he ng wh esec o u gves he hd elon n (30 s follows: D δ ( D D δ = +Φ whee Φ coesonds o he boundy em n (3. The em (3 s obned fom he clculon n (0. Hee he ol dvegence D ( wh esec o me hs been elmned. Conseuenly n (30 he fs elon mens gven veco feld he second elon s he defnon of je vbles he hd elon s he Eule-Lgnge euon befoe lyng negon by s nd he fouh elon s he defnon of he momenum. The bove eesenon cn be conveed o he followng foml fom of o eesenons [7]. Coolly. The mlc hghe ode feld o Hmlonn sysem defned on wh cn be ewen s he followng o Hmlonn sysem: whee l nd we hve defned he vbles f 0 D e = f D Τ 0 e ( f e ( D ( f e ( D j j ( f e = Ds e ( Ds e j j = = (38 = = fo j. We cll f nd e boundy o vbles. j j Poof. Fom he hd nd second elons n (30 we cn obn he fs nd second ows n (37 esecvely. The oduc of he of he hd elon n (38 h s euvlen o he boundy em n (3 whee n (3 hs been neeed s n nfnesml von of. d 5.4. Powe Blnce nd Pssvy Ths secon deves he owe blnce of he Hmlonn eesenon dscussed n he evous secon nd defne he foml eesenon of hghe ode feld o Hmlonn sysems wh boundy enegy flows. n he me-sl sl sce he nsnneous Hmlonn (70 on s gven by s H = d x (39 = 0 whee j. The followng elon coesonds o he owe blnce of dsbued o-hmlonn sysems. Pooson 6. The sysem (37 ssfes he owe blnce s s ( d d 0. j j s e f + e f x + e f x = (40 x j= Poof. The owe blnce cn be deved fom he neo oduc beween he devve of he nsnneous x = x be he genelzed Hmlonn (39 nd by veco feld s zeo. Le ( ( 0 enegy densy of (39 defned on ( Ω v M whee 0 nd (36 (37 s he Whney 48

12 . Nex consde he - bundle wh he locl coodnes ( x ; fo 0 nd 0 ng beween d v nd he veco feld ( ( ( µ D D V M dh d whee we hve used = µ D d + d d = µ DD d + d d G. Nshd = + s follows: = µ dv = µ dv ( = µ d + d d s s s s s = DD s + D = D = + D = s s nd he me ol dvegence sme wy of he oof of Theoem. Fom he enegy consevon d 0 we obn On he ohe hnd we hve d = d + d µ v µ s j j j= µ v = (4 D s elmned n he D =. j j = µ D d + D d D s ( Ds D s (4 j= w = D D D e f = 0. By subsung (4 no (4 he negnd of (40 s gven s follows: d = + + = 0. (43 D D e f µ v s j j j= By lyng Sokes heoem o he negl of he hd em of he second euon n (43 (40 s gven. Pooson 7. The sysem (37 s ssve. Poof. The Hmlonn (39 nd he owe blnce (40 coesond o he fne consn nd he duly oduc befoe he me negon esecvely n he defnon of he ssvy (see Secon A. Conseuenly we ls ech he fnl esul h mens he sysem (37 s jus hghe ode eesenon of dsbued o-hmlonn sysems. Theoem. The sysem (37 s he Sokes-Dc sucue. Poof. We hve ledy oven h he sysem (37 s Dc sucue n Pooson. On he ohe hnd he owe blnce (40 coesonds wh he mn oey of dsbued o-hmlonn sysems descbed by he Sokes-Dc sucue nd cn be egded s he hghe ode veson of he sucue []. 6. Exmles Ths secon esens wo modelng exmles. 6.. Tmoshenko Bem Euon The -dmensonl Tmoshenko bem euon ( θ 0 ( Aρ GADy y = ρθ Eθyy GA y θ = 0 (44 483

13 G. Nshd 0 3 s deved fom he Lgngn densy funconl on Ω ( ρ ρ ( { ε } θ y θ θy =. Fom ( L = dx = A + GA E d x (45 whee [ 0 ] s he me coodne y s he sl coodne long he longudnl xs s he sheng θ s he oon ech on n y A ρ s he un mss ρ s he momen of ne E s he elsc sffness nd GA s he sheng sffness. 0 Le m = l = nd 3 x x ; = y; y θ y nd he mxmum of hghe ode degees mx = n (44 we deve = D fo 0 3. By defnng 3 wh he locl coodne = Dθ we se dx d dy = 0. n (38 fom k = 0 j = nd = we obn = nd [ ] ( f e = ( D = ( D Aρ ( f e = (( D θ = ( D ( ρθ θ ( f e = ( Dθ = ( Dθ GA( y θ y fy ey = Dy = ( D y GA( y θ w( y y ( fy ey = Dθy ( D y E y w( y = θ θ ( f e = ( Ds ( Ds ey = ( ey ( f e = ( Ds ( Ds ey = ( θ ey whee Ds = Dy nd we hve defned = = ε θ = =. θ ε θ Hence we hve y y y f 0 0 Dy 0 0 e D f y e f = D y e f y ey f 0 Dy y e y f f e e = = f e f e whee wo lnes fom he fs euon n (48 s euvlen o (44 nd hee lnes fom he boom e eules. Moeove he sysem (48 ssfes whee noe h Sokes heoem cnno be led o ( no been defned. s s e f + e f + e f + e f + e f dx + e f + e f s dx = 0 (49 y y y y x y (46 (47 (48 e f ; heefoe he coesondng boundy em hs 484

14 G. Nshd 6.. Poenl Boussnes Euon The -dmensonl oenl Boussnes euon ([7]. 37 h exesses shllow we wves yy + Dy ( yy + yyyyyy = 0 ( s deved fom he Lgngn densy funconl defned on Ω ( whee [ ] L = + d x (5 = nd [ ] 3 y yy yyy 6 0 s he me coodne y s he sl coodne long he we sufce nd s he hegh of he wve. Le m = l = nd = 7. We hve defned 7 wh he locl coodne = D fo dx d dy = by ( x x ; = y; ( y nd he mxmum ode 0 mx = 6 n (50. By subsung k = j 3 nd = o (38 we ge y ( f e = (( Dy = ( Dyy yy ( fyy eyy = D yy = D yy yy w( yy yyy ( fyyy eyyy = D yyy ( D yyy yyy w( yyy = ( f e = ( Ds ( Ds eyy = ( D y eyy ( f e = ( Ds ( Ds eyy = ( De y yy 3 ( f 3 e 3 = ( Ds ( Ds eyyy = ( D y eyyy 3 ( f 3 e 3 = ( Ds ( Ds eyyy = ( D y De y yyy 33 3 ( f 33 e 33 = ( Ds ( Ds eyyy = ( Dyeyyy whee Ds = Dy nd we hve defned = = = y yy yyy y yy yyy (5 (53 Hence he followng sysem eesenon s gven: 3 f 0 Dy D y e fyy = Dy 0 0 eyy 3 f yyy Dy 0 0 e yyy f e f e f = f 3 e = e 3 f 3 e 3 f 33 e 33 (54 485

15 G. Nshd whee he fs lne of he fs euon n (54 s euvlen o (50 nd wo lnes fom he boom e eules. Moeove he sysem (54 ssfes 7. Conclusons Τ { } e f + e f + e f x + e f x = (55 s s yy yy yyy yyy d s d 0. x Ths e deved he hghe ode feld o Hmlonn sysem wh boundy enegy flows fom sysems of hghe ode l dffeenl euons h e deemned by vonl oblems n ems of he mulsymlecc nsnneous fomlsm. By defnng he symlecc sucue nduced fom dsbuons nd he Sokes vonl dffeenl ncludng he negon by s oeos we clfed he mlc Hmlonn eesenon of he sysems of hghe ode l dffeenl euons nd s locl exesson coesonds o he dsbued o-hmlonn sysem. n hs e we ssumed h Lgngns e fs ode wh esec o me bu ossbly hghe ode wh esec o sl vbles fo smlfcon. Ths ssumon cn be genelzed. On he ohe hnd he foml eesenon ncludng me devves u o fs ode coesonds o he dsbued o-hmlonn sysems. Acknowledgemens The uho hnks Pofesso Benhd Mschke fo fuful dscussons on hs sudy. Ths wok ws suoed by SPS Gns-n-Ad fo Scenfc Resech (C No nd SPS Gns-n-Ad fo Chllengng Exlooy Resech No Refeences [] Abhm R. nd Msden. (008 Foundons of Mechncs. nd Edon AMS Chelse Pub. Chelse. [] vn de Schf A.. (000 L -Gn nd Pssvy Technues n Nonlne Conol. nd Revsed nd Enlged Edon Snge-Velg London. h://dx.do.og/0.007/ [3] Dundm V. Mcchell A. Smgol S. nd Buynnckx H. Eds. (009 Modelng nd Conol of Comlex Physcl Sysems The Po-Hmlonn Aoch. Snge Beln. h://dx.do.og/0.007/ [4] no D.C. Mgols D.L. nd Rosenbeg R.C. (006 Sysem Dynmcs Modelng nd Smulon of Mechonc Sysems. 4h Edon Wley Hoboken. [5] Goy M.. (99 A Mulsymlecc Fmewok fo Clsscl Feld Theoy nd he Clculus of Vons:. Covn Hmlonn Fomlsm. n: Fncvgl M. Ed. Mechncs Anlyss nd Geomey: 00 Yes fe Lgnge Elseve Scence Pub. B.V. Amsedm h://dx.do.og/0.06/b [6] Goy M.. (99 A Mulsymlecc Fmewok fo Clsscl Feld Theoy nd he Clculus of Vons : Sce + Tme Decomoson. Dffeenl Geomey nd s Alcons h://dx.do.og/0.06/096-45( [7] vn de Schf A.. nd Mschke B.M. (00 Hmlonn Fomulon of Dsbued-Pmee Sysems wh Boundy Enegy Flow. ounl of Geomey nd Physcs h://dx.do.og/0.06/s ( [8] Mcchell A. (04 Pssvy-Bsed Conol of mlc Po-Hmlonn Sysems. SAM ounl on Conol nd Omzon h://dx.do.og/0.37/30988 [9] Nshd G. nd Ymk M. (005 Foml Dsbued Po-Hmlonn Reesenon of Feld Euons. Poceedngs of he 44h EEE Confeence on Decson nd Conol Sevlle -5 Decembe h://dx.do.og/0.09/cdc [0] Nshd G. nd Mschke B. (0 mlc Reesenon fo Pssvy-Bsed Boundy Conols. Poceedngs of he 4h FAC Woksho on Lgngn nd Hmlonn Mehods fo Non-Lne Conol Benoo 9-3 Augus [] Schöbel M. nd Suk A. (04 e Bundle Fomulon of nfne-dmensonl Po-Hmlonn Sysems Usng Dffeenl Oeos. Auomc h://dx.do.og/0.06/j.uomc [] Nshd G. nd Ymk M. (004 A Hghe Ode Sokes-Dc Sucue fo Dsbued-Pmee Po-Hmlonn Sysems. Poceedngs of he 004 Amecn Conol Confeence Boson 30 une- uly

16 G. Nshd [3] Le Goec Y. w H. nd Mschke B. (005 Dc Sucues nd Boundy Conol Sysems Assoced wh Skew-Symmec Dffeenl Oeos. SAM ounl on Conol nd Omzon h://dx.do.og/0.37/ [4] vn de Schf A.. (998 mlc Hmlonn Sysems wh Symmey. Reos on Mhemcl Physcs h://dx.do.og/0.06/s ( [5] Dlsmo M. nd vn de Schf A. (998 On Reesenons nd negbly of Mhemcl Sucues n Enegy- Consevng Physcl Sysems. SAM ounl on Conol nd Omzon h://dx.do.og/0.37/s [6] Yoshmu H. nd Msden.E. (006 Dc Sucues n Lgngn Mechncs P : mlc Lgngn Sysems. ounl of Geomey nd Physcs h://dx.do.og/0.06/j.geomhys [7] Vnkeschve. Yoshmu H. nd Leok M. (0 The Hmlon-Ponygn Pncle nd Mul-Dc Sucues fo Clsscl Feld Theoes. ounl of Mhemcl Physcs 53 Acle D: h://dx.do.og/0.063/ [8] Coun T. (990 Dc Mnfolds. Tnscons of he Amecn Mhemcl Socey h://dx.do.og/0.090/s [9] Dofmn. (993 Dc Sucues nd negbly of Nonlne Evoluon Euons. ohn Wley Chchese. [0] Nshd G. Mschke B. nd keu R. (05 Boundy negbly of Mulle Sokes-Dc Sucues. SAM ounl Conol nd Omzon h://dx.do.og/0.37/ [] Nshd G. Ymk M. nd Luo. (007 Vul Lgngn Consucon Mehod fo nfne Dmensonl Sysems wh Homooy Oeos. n: Allgüwe F. Flemng P. okoovc P. uzhnsk A.B. wkenk H. Rnze A. e l. Eds. Lgngn nd Hmlonn Mehods fo Nonlne Conol 006 Lecue Noes n Conol nd nfomon Scences Vol. 366 Snge Beln h://dx.do.og/0.007/ _5 [] Olve P.. (993 Alcons of Le Gous o Dffeenl Euons. nd Edon Snge-Velg New Yok. h://dx.do.og/0.007/ [3] Andeson.M. (99 noducon o he Vonl Bcomlex. Conemoy Mhemcs h://dx.do.og/0.090/conm/3/88434 [4] Sundes D.. (989 The Geomey of e Bundles. Cmbdge Unvesy Pess Cmbdge. h://dx.do.og/0.07/cbo [5] Gche G. Mngo L. nd Sdnshvly G. (997 New Lgngn nd Hmlonn Mehods n Feld Theoy. Wold Scenfc Sngoe. h://dx.do.og/0.4/99 [6] Tulczyjew W.M. (977 The Legende Tnsfomon. Annles de l nsu Hen Poncé (A Physue Théoue [7] Olve P.. (995 Euvlence nvns nd Symmey. Cmbdge Unvesy Pess Cmbdge. h://dx.do.og/0.07/cbo Aendx A. Pssvy Consde he followng ng beween fo u L ( U e nd y L e ( U Le U nd e 0 L U : : T yu = y u d (56 T whee s he duly oduc U s fne dmensonl lne sce U s s dul sce nd we hve defned he exended L e sce h s he se of ll mesuble funcons n L unced o fne me nevl. Noe h he duly oduc y u( coesonds o owe. G: L U L U. Then G s ssve f hee exss consn H such h Defnon 0. Le e e 487

17 G. Nshd Gu u H u L U T 0 (57 T whee he lef-sde of (57 s ssumed o be well-defned. Hence G s ssve f nd only f fne moun of enegy cn be exced fom he sysem defned by G. Coolly 3. Fo on n he me xs B. Dffeenl Foms on Bundles A dffeenl ( dh Gu u u L ( U. (58 d j -fom gη ψ defned on he -h ode je bundle e defned by hozonl -fom d d ( ψ = x x Ωh nd vecl j-fom d d j j η = Ω j v j ( Ω s he sce of dffeenl j-foms on ( g g x = s smooh funcon defned on whee nd k nd k e dffeen combnon seleced fom nd fo k j. Le j Ω ( be he sce of dffeenl ( j -foms defned on. ( j -foms such h n= + j n e clled n-foms nd he sce s denoed by Ω (. The exeo dffeenl oeo d d d j -foms s defned by he vecl dffeenl oeo nd he hozonl dffeenl oeo = + fo v h j j+ g d v : ( ( ; g d Ω Ω η ψ η ψ (59 + m j j k d h : Ω Ω ; gη ψ Dg k η d x ψ k = whee he ol dffeenl oeo wh esec o D (60 x s defned by = + (6 x l = = 0 Dd = d D = d. Noe h = x nd h s euvlen o l dffeenl nd ( = ( w( w( whee w( s he wegh of he ndex [] [4] nd ( geneed by he eeed emuon of he combnon n. C. Eule-Lgnge Euons = w fo he ndex An Eule-Lgnge euon s deemned by he sony condon d dx = 0 of he vonl devve m0 d of Lgnge densy = dx Ω (. f vbles on boundes e zeo he locl exesson of Eule-Lgnge euons s gven by he sony condon of dv dx = d dx = D d d x (6 0 whee dx= dx dx m s he ( m + -volume fom D s he ol dffeenl oeo m o ll ndex n negon by s s used n he second euly nd he em Sokes heoem unde he ssumon of he zeo boundy condon. D. Mulsymlecc Covn Fomlsm D wh esec s elmned by = D 0 The Hmlonn eesenon of lumed mee sysems e deemned by he symlecc -fom * ω = dθ on congen bundle T whee θ = d s he cnoncl -fom. Then fo gven Hmlonn * H: T Hmlonn veco feld µ s defned by ([]. 87 whee * µω = dh T s 488

18 G. Nshd he neo oduc. The (covn Hmlonn eesenon of feld euons e deemned by he mulsymlecc Ω=dΘ on he mulsymlecc mnfold ([5]. whee Θ s he cnoncl ( m + m + ove defnng ( m + -foms fom Then s defned s he subbundle of m+ { ( = z z V ξ ν = νξ } m whee he sce Ω + ( of ( m + -foms ove ( m + -h degee exeo owe congen ( been defned s veco subbundle m + - -fom. 0 (63 s defned by he sce of ll secons of -h je bundle [5] nd he vecl ngen bundle V hs T V = etπ deemned by he ngen m Tπ : T TX fo x x of he ngen bundle TX when hose of he m- bundle π : X. Noe h he locl coodnes ( nfold X e ( x nd he locl coodnes of TY e ( x y x y e ( x y whee hose of VY e ( x y y. Le ( ; x fo 0 be he locl coo- dnes of. Any z n (63 cn be loclly wen s when hose of he bundle π :Y X z = d x + d d x (64 x = 0 whee we ssume h he mul ndex ssfes = +. Then Θ= z s defned by usng (64. On he ohe hnd he covn Lgngn sysem cn be defned s dθ on by he Cn fom m Θ Ω + ( whee σ Θ=Θ nd σ : s he covn Legende nsfomon on = + c f = = D + c f < (65 c = 0 f = = 0 x e he locl coodnes of fo 0. The funcons c nd c n (65 gve bness n he globl exesson of Θ ; heefoe hs s no used n he locl exesson.e. c = c = 0. n he bove he covn Lgngn sysem deemned by he vonl oblem of he -h ode Lgngn densy dx Ω ( on s defned by he Cn fom Θ Ω + ( m0 m on ([5]. 0. Noe h he covn Hmlonn deemnes he ffne sucue of h s he essenl of he symlecc sucue. whee σ s he ull-bck nd ( ; E. Mulsymlecc nsnneous Fomlsm The nsnneous fomlsm s he covn eesenon wh me-sl slng. The me-sl slng s euvlen o choosng n nfnesml suesufce mezed by me n he confguon sce. Bundles wh me-sl slng conss of he Cuchy sufce X nd me-sl veco felds ζ on. Then ζ mens he decon of he me evoluon of he sysem nd ζ on nsveslly nesecs o eveywhee whee mens he esced o. The nsnneous eesenon s defned on he sce Γ( h consss of ll secons γ of 489

19 G. Nshd esced o. Fo gven globl secon ( γ Γ gven s γ = γ ([6]. 379 whee he subbundle j of Γ ( he locl coodnes of Γ( s he locl coodnes of s fo 0. Then h cn be denfed wh he ngen bundle T he decon of he me evoluon of ζ whee he locl coodnes of j e obned fom usng he mul ndex U wh esec o me fo 0 U. Hence by escng he sysem o he sl devves n γ e elmned. T nd * T ( T esecvely ( γ L fo 0 nd (.e. * ( T T e he veco bundle ove T by choosng j γ U by wh numbes of locl coodnes γ π fo 0. On he dul bundle of T he nsnneous Hmlonn sysems e deved fom he cnoncl fom whee he nsnneous momenum π L s ϑ = πdl d x (66 L = 0 L L L = ( wl ( D = 0 (67 s clculed by negon by s nd he wegh ϑ = σ ϑ [6. 38] whee he n- bsed on he combnon hve been defned. nsnneous Lgnge sysems e deemned on T snneous Legende nsfomon s gven by γ ( L U γl π The mge n * ( T T. The nsnneous Hmlonn Becuse sysems on P (! (!! wl = L+ L (68 by * σ (69 * : T T T fo 0 U nd 0 L h e mul ndexes wh esec o me. of he bundle m σ s he nsnneous ml consn se H on : P P s defned s follows ([6]. 384: P L s H = π L d x. (70 L = 0 ossesses he (e-symlecc sucue ω of * T ( T P e gven by evoluonl veco felds ν such h νω = dh. nsnneous Hmlonn 490