Hamiltonian multivector fields and Poisson forms in multisymplectic field theory

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS 46, Hamltonan multvecto felds and Posson foms n multsymplectc feld theoy Mchael Foge a Depatamento de Matemátca Aplcada, Insttuto de Matemátca e Estatístca, Unvesdade de São Paulo, Caxa Postal 66281, BR São Paulo, S.P., Bazl Conelus Paufle b Matematsk Fysk/Alba Nova, Kunglga Teknska Högskolan, SE Stockholm, Sweden Hatmann Röme c Fakultät fü Mathematk und Physk, Physkalsches Insttut, Albet-Ludwgs-Unvestät Febug m Besgau, Hemann-Hede-Stasse 3, D Febug.B., Gemany Receved 4 August 2005; accepted 16 Septembe 2005 We pesent a geneal classfcaton of Hamltonan multvecto felds and of Posson foms on the extended multphase space appeang n the geometc fomulaton of fst ode classcal feld theoes. Ths s a peequste fo computng explct expessons fo the Posson backet between two Posson foms Amecan Insttute of Physcs. DOI: / I. INTRODUCTION AND GENERAL SETUP The pesent pape s a contnuaton of pevous wok on Posson backets of dffeental foms n the multphase space appoach to classcal feld theoy. 1,2 Ou am s to specalze the geneal constuctons of Ref. 2 fom abstact exact multsymplectc manfolds to the extended multphase spaces of feld theoy, whch at pesent seem to be the only known examples of multsymplectc manfolds, to clafy the stuctue of Hamltonan multvecto felds, of Hamltonan foms and of Posson foms on these spaces and to gve explct fomulas fo the Posson backet between the latte ntoduced n Refs. 1 and 2. The stuctue of ths pape s as follows. In the emande of ths ntoducton, we befly evew the geometc constuctons needed n the pape. We put patcula emphass on the consequences that ase fom the exstence of a cetan vecto feld, the scalng o Eule vecto feld. Also, we fx the notaton to be used n what follows. In Sec. II, we pesent an explct classfcaton of locally Hamltonan multvecto felds on extended multphase space n tems of adapted local coodnates and, followng the logcal ncluson fom locally Hamltonan to globally Hamltonan to exact Hamltonan multvecto felds, show how the last two ae stuated wthn the fst. Sec. III s devoted to the study of Hamltonan foms and Posson foms that ae assocated wth globally Hamltonan multvecto felds. In Sec. IV, we use the outcome of ou pevous analyss to deve expessons fo the Posson backet between two Posson foms. In Sec. V, we summaze ou man conclusons and comment on the elaton of ou esults to othe appoaches, as well as on pespectves fo futue eseach. Fnally, n ode to make the pape self-contaned, we nclude n an appendx a poposton that s not new but s needed n some of the poofs. We begn wth a few comments on the constucton of the extended multphase space of feld theoy, 3 7 whch stats out fom a gven geneal fbe bundle ove space-tme, wth base space M dm M =n, total space E, bundle pojecton :E M and typcal fbe Q dm Q=N. Its a Electonc mal: foge@me.usp.b b Electonc mal: pcp@theophys.kth.se c Electonc mal: hatmann.oeme@physk.un-febug.de /2005/4611/1/0/$ , Amecan Insttute of Physcs

2 1-2 Foge, Paufle, and Röme J. Math. Phys. 46, usually efeed to as the confguaton bundle snce ts sectons consttute the possble feld confguatons of the system. Of couse, the manfold M epesents space-tme, wheeas the manfold Q plays the ole of a confguaton space. The extended multphase space, whch we shall smply denote by P, s then the total space of a lage fbe bundle ove M and n fact the total space of a vecto bundle ove E whch can be defned n seveal equvalent ways, e.g., by takng the twsted affne dual J n E of the fst ode jet bundle JE of E o by takng the bundle n 1 T * E of n 1-hozontal n-foms on E; see Refs. 2, 5, and 7 fo detals. Theefoe, thee s a natual class of local coodnate systems on P, namely those that ase fom combnng fbe bundle chats of E ove M wth vecto bundle chats of P ove E: these so-called adapted local coodnates x,q, p, p ae completely fxed by specfyng local coodnates x fo M the space-tme coodnates, local coodnates q fo Q the poston vaables and a local tvalzaton of E ove M, and ae such that the nduced local coodnates p the multmomentum vaables and p the enegy vaable ae lnea along the fbes of P ove E. Fo detals, we efe to Ref. 2, whee one can also fnd the explct tansfomaton law fo the multmomentum vaables and the enegy vaable nduced by a change of the space-tme coodnates, of the poston vaables and of the local tvalzaton. A fst mpotant featue of the extended multphase space P s that t caes a natually defned multcanoncal fom whose exteo devatve s, up to a sgn, the multsymplectc fom, = d. The global constucton can be found n Refs. 2, 5, and 7, so we shall just state the explct fom n adapted local coodnates, 1 = p dq d n x + pd n x. 2 =dq dp d n x dp d n x. Hee, we have aleady employed pat of the followng conventons concenng local dffeental foms defned by a system of adapted local coodnates, whch wll be used systematcally thoughout ths pape, d n x =dx 1 dx n, d n x 1... = 1 d n x. Fo late use, we also ecall the defnton of the Le devatve of a dffeental fom along an -multvecto feld X, L X =d X 1 X d, whch leads to the followng elatons, vald fo any dffeental fom and any two multvecto felds X and Y of tenso degees and s, espectvely, 3 4 dl X = 1 1 L X d, X,Y = 1 1s L X Y Y L X, 5 6 L X,Y = 1 1s 1 L X L Y L Y L X, 7 L X Y = 1 s Y L X + L Y X, whee X,Y denotes the Schouten backet of X and Y. Fo decomposable multvecto felds X=X 1 X and Y =Y 1 Y s, t can be defned n tems of the Le backet of vecto felds accodng to the fomula 8

3 1-3 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, X,Y = =1 s j=1 1 +j X,X j X 1 Xˆ X Y 1 Yˆ j Ys, whee as usual the hat ove a symbol denotes ts omsson. We shall also wte L X Y = X,Y, fo any two multvecto felds X and Y. Fo popetes of the Schouten backet, we efe to Ref. 8. A poof of the above denttes elatng the Schouten backet and the Le devatve of foms along multvecto felds can be found n the appendx of Ref. 2. A second popety of the extended multphase space P whch povdes addtonal stuctues fo tenso calculus on ths manfold s that t s the total space of a fbe bundle, whch mples that we may speak of vetcal vectos and hozontal covectos. In fact, t s so n no less than thee dffeent ways. Namely, P s the total space of a fbe bundle ove M wth espect to the so-called souce pojecton, the total space of a vecto bundle ove E wth espect to the so-called taget pojecton and the total space of an affne lne bundle ove the odnay multphase space P 0. 2 Theefoe, the notons of vetcalty fo multvecto felds and of hozontalty fo dffeental foms on P admt dffeent ntepetatons, dependng on whch pojecton s used. In any case, one stats by defnng tangent vectos to the total space of a fbe bundle to be vetcal f they ae annhlated by the tangent map to the bundle pojecton, o what amounts to the same thng, f they ae tangent to the fbes. Dually, a k-fom on the total space of a fbe bundle s sad to be l-hozontal f t vanshes wheneve one nsets at least k l+1 vetcal tangent vectos; the standad hozontal foms ae obtaned by takng l=k. Fnally, an -multvecto on the total space of a fbe bundle s sad to be s-vetcal f ts contacton wth any s+1-hozontal fom vanshes. It s not dffcult to show that these defntons ae equvalent to equng that, locally, an l-hozontal k-fom should be a sum of exteo poducts of k one-foms, among whch thee ae at least l hozontal ones, and that an s-vetcal -multvecto feld should be a sum of exteo poducts of tangent vectos, among whch thee ae at least s vetcal ones. Usng ths ule, popetes of vetcalty fo multvectos o hozontalty fo foms ae easly deved fom the coespondng popetes fo vectos o one-foms, espectvely. In what follows, the tems vetcal and hozontal wll usually efe to the souce pojecton, except when explctly stated othewse. A thd mpotant featue of the extended multphase space P s that t caes a natually defned vecto feld, the scalng vecto feld o Eule vecto feld, whch exsts on any manfold that s the total space of a vecto bundle. In adapted local coodnates, = p p + p p. It s then easy to vefy the followng elatons see Poposton 2.1 of Ref. 2: =0, =, L =, L =. 9 The man utlty of s that takng the Le devatve L along povdes a devce fo contollng the dependence of functons and, moe geneally, of tenso felds on P on the multmomentum vaables and the enegy vaable, that s, along the fbes of P ove E: L has only ntege egenvalues, and egenfunctons of L wth egenvalue k ae homogeneous polynomals of degee k n these vaables. As we shall see soon, homogenety unde L plays a cental ole n the analyss of vaous classes of multvecto felds and dffeental foms on P. Let us ecall a few defntons. An -multvecto feld X on P s called locally Hamltonan f X s closed, o equvalently, f L X =0. 10

4 1-4 Foge, Paufle, and Röme J. Math. Phys. 46, It s called globally Hamltonan f X s exact, that s, f thee exsts an n--fom f on P such that X =df. In ths case, f s sad to be a Hamltonan fom assocated wth X. Fnally, t s called exact Hamltonan f L X =0. Of couse, exact Hamltonan multvecto felds ae globally Hamltonan to show ths, set f = 1 1 X and apply Eqs. 4 and 1, and globally Hamltonan multvecto felds ae obvously locally Hamltonan. Convesely, an n--fom f on P s called a Hamltonan fom f thee exsts an -multvecto feld X on P such that Eq. 11 holds; n ths case, X s sad to be a Hamltonan multvecto feld assocated wth f. Moeove, f s called a Posson fom f n addton, t vanshes on the kenel of, that s, f fo any multvecto feld Z, we have Z =0 Z f =0. A tval example of a Posson fom s the multsymplectc fom tself. Anothe example s povded by the multcanoncal fom, snce t can be wtten as =. Concenng stablty unde the Le devatve along the scalng vecto feld, we have the followng. Poposton 1.1: The space X LH P of locally Hamltonan multvecto felds, the space X H P of globally Hamltonan multvecto felds, the space X EH P of exact Hamltonan multvecto felds and the space X 0 P of multvecto felds takng values n the kenel of ae all nvaant unde the Le devatve along the scalng vecto feld, L X =0 L,X =0, X =df,x =dl f f, L X =0 L,X =0, =0, = Poof: All these elatons can be shown by dect calculaton. Dually, we have the followng. Poposton 1.2: The space H P of Hamltonan foms, the space 0 P of foms that vansh on the kenel of and the space P P of Posson foms ae all nvaant unde the Le devatve along the scalng vecto feld, df = X dl f = X+,X. 18 Poof: The fst statement s a consequence of Eq. 18, whch follows dectly fom combnng Eqs. 5 and 6 wth Eq. 9. Fo the second statement, assume that f vanshes on the kenel of. Then f s any multvecto feld takng values n the kenel of, the multvecto feld, takes values n the kenel of as well cf. Eq. 17, so that accodng to Eq. 6, L f = L f, f =0. But ths means that L f vanshes on the kenel of. Fnally, the thd statement follows by combnng the fst two. A specal class of multvecto felds and of dffeental foms on P whch wll be of patcula mpotance n what follows s that of fbewse polynomal multvecto felds and of fbewse polynomal dffeental foms on P: the coeffcents ae polynomals along the fbes of P ove E,

5 1-5 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, o n othe wods, polynomals n the multmomentum vaables and the enegy vaable. The man advantage of wokng wth tenso felds on the total space of a vecto bundle whch ae fbewse polynomal s that they allow a unque and globally defned o n othe wods, coodnate ndependent decomposton nto homogeneous components, accodng to the dffeent egenspaces of the Le devatve L along ; the coespondng egenvalue wll n what follows be called the scalng degee to dstngush t fom the odnay tenso degee. In dong so, t must be bone n mnd that, n an expanson wth espect to an adapted local coodnate system, the scalng degee eceves contbutons not only fom the coeffcent functons but also fom some of the coodnate vecto felds and dffeentals snce the vecto felds /x, /q, /p, and /p cay scalng degee 0, 0, 1, and 1, espectvely, whle the dffeentals dx,dq,dp, and dp cay scalng degee 0, 0, +1, and +1, espectvely; moeove, the scalng degee s addtve unde the exteo poduct, snce L s a devaton. Theefoe, a fbewse polynomal -multvecto feld on P admts a globally defned decomposton nto a fnte sum X = X s, s whee X s s ts homogeneous component of scalng degee s, L X s = sx s. Each X s can be obtaned fom X by applyng a pojecto whch s tself a polynomal n L, X s = s ss 1 s s L sx. Smlaly, a fbewse polynomal n--fom f on P admts a globally defned decomposton nto a fnte sum f = f s, s0 whee f s s ts homogeneous component of scalng degee s, Agan, the f s can be obtaned fom f, f s = s0 ss L f s = sf s. 1 s s L sf. The elevance of these decompostons fo locally Hamltonan multvecto felds and fo Hamltonan foms on the extended multphase space P stems fom the followng theoem, whose poof wll follow fom statements to be deved n the couse of the next two sectons, by means of explct calculatons n adapted local coodnates. Theoem 1.3: Fo 0n and up to tval contbutons (-multvecto felds takng values n the kenel of and closed (n )-foms, espectvely), locally Hamltonan -multvecto felds and Hamltonan (n )-foms on P ae fbewse polynomal and have non-tval homogeneous components of scalng degee s only fo s= 1,0,..., 1 and fo s=0,1,...,, espectvely. Moe pecsely, we have Evey fbewse polynomal locally Hamltonan (Hamltonan, exact Hamltonan) -multvecto feld X on P, admts a unque, globally defned decomposton nto homogeneous components wth espect to scalng degee, whch can be wtten n the fom (we abbevate X 1 as X

6 1-6 Foge, Paufle, and Röme J. Math. Phys. 46, X = X + X wth X + = X s, whee each X s s locally Hamltonan (Hamltonan, exact Hamltonan) and = s + s 20 s 2 s s a fbewse polynomal -multvecto feld on P takng values n the kenel of. Evey fbewse polynomal Hamltonan fom (Posson fom) f of degee n onp, admts a unque, globally defned decomposton nto homogeneous components wth espect to scalng degee, whch can be wtten n the fom s=0 19 f = f 0 + f + + f c whee each f s s Hamltonan (Posson) and f c = f c s s+1 s a fbewse polynomal closed n -fom on P. wth f + = f s, The cases =0 and =n ae exceptonal and must be dealt wth sepaately; see Popostons 2.2 and 3.2 fo =0 and Popostons 2.3 and 3.1 fo =n. In vew of ths theoem, t s suffcent to study locally Hamltonan multvecto felds and Hamltonan foms whch ae homogeneous unde the Le devatve along the scalng vecto feld. Ths condton of homogenety s also compatble wth the coespondence between globally Hamltonan multvecto felds X and Hamltonan foms f establshed by the fundamental elaton 11, because tself s homogeneous: accodng to Eq. 9, has scalng degee 1. Indeed, except fo the ambguty nheent n ths coespondence f detemnes X only up to a multvecto feld takng values n the kenel of and X detemnes f only up to a closed fom, Eq. 11 peseves the scalng degee, up to a shft by 1: X s homogeneous wth scalng degee s 1 f and only f f s homogeneous wth scalng degee s, L X = s 1X modulo multvecto felds takng values n the kenel of L f = sf modulo closed foms. Fo a poof, note that the condton on the left-hand sde lhs amounts to equng that,x =s 1 X, whle the condton on the ght-hand sde hs amounts to equng that dl f =s df, so the equvalence stated n Eq. 23 s an mmedate consequence of Eq. 18. A patcula case occus when, snce the locally Hamltonan multvecto felds whch ae homogeneous of scalng degee 0 ae pecsely the exact Hamltonan multvecto felds: fo L X =0, 23 L X =0 modulo multvecto felds takng values n the kenel of Indeed, the popetes of and gve L X =0. 24 L X = L X = 1 X, L X = 1 1,X. 25 Moe geneally, the fundamental elaton 11 peseves the popety of beng fbewse polynomal, n the followng sense: If X s a fbewse polynomal Hamltonan -multvecto feld and f s a Hamltonan n -fom assocated wth X, then modfyng f by addton of an appopate

7 1-7 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, closed n -fom f necessay, we may always assume, wthout loss of genealty, that f s fbewse polynomal as well. Convesely, f f s a fbewse polynomal Hamltonan n -fom and X s a Hamltonan -multvecto feld assocated wth f, then modfyng X by addton of an appopate -multvecto feld takng values n the kenel of f necessay, we may always assume, wthout loss of genealty, that X s fbewse polynomal as well. II. HAMILTONIAN MULTIVECTOR FIELDS Ou am n ths secton s to detemne the explct fom, n adapted local coodnates, of locally Hamltonan -multvecto felds on the extended multphase space P, whee 0n+1. Multvecto felds of tenso degee n+1 ae unnteestng snce they always take the values n the kenel of. As a fst step towads ths goal, we shall detemne the explct fom, n adapted local coodnates, of the multvecto felds on P takng values n the kenel of ; ths wll also seve to dentfy, n the next secton, the content of the kenel condton 13 that chaactezes Posson foms. To ths end, note fst that beng a homogeneous dffeental fom of degee n+1, ts kenel s gaded, that s, f an nhomogeneous multvecto feld takes values n the kenel of, so do all ts homogeneous components. Poposton 2.1: Evey -multvecto feld X on P admts, n adapted local coodnates, a unque decomposton of the fom X = 1! X 1 x 1 x + 1 1! X, 2 + 1! X 1 p 1 x 2 q x 2 x + 1 X 2 1! p x x 2 x +, whee all coeffcents ae totally antsymmetc n the space-tme ndces and takes values n the kenel of. Poof: Ths s an mmedate consequence of the patcula fom of n adapted local coodnates, Eq. 3. Fo moe detals, see Ref. 9. Wth ths local coodnate epesentaton at hand, we ae n a poston to analyze the estctons mposed on the coeffcents X 1, X, 2, X 1, and X 2 by equng X to be locally Hamltonan. Of couse, t makes no sense to dscuss the queston whch locally Hamltonan multvecto felds ae also globally Hamltonan when wokng n local coodnates. As a wam-up execse, we shall settle the exteme cases of tenso degee 0 and n+1. Poposton 2.2: A functon on P, egaded as a 0-multvecto feld, s locally Hamltonan f and only f t s constant; t s then also exact Hamltonan. Smlaly, an n+1-multvecto feld on P, wth standad local coodnate epesentaton X = X p x 1 x n +, 27 whee takes values n the kenel of, s locally Hamltonan f and only f the coeffcent functon X s constant and s exact Hamltonan f and only f t vanshes. Poof: Fo functons, we use the fact that the opeato 1 coespondng to the constant functon 1 on a manfold s defned to be the dentty, so that the opeato f coespondng to an abtay functon f on a manfold s smply multplcaton by f. Theefoe, we have fo any dffeental fom L f =d f f d =df f d =df, mplyng that f f s constant, L f =0 no matte what one chooses. On the othe hand, an explct calculaton n adapted local coodnates shows that the condton L f =0 foces all patal deva- 26

8 1-8 Foge, Paufle, and Röme J. Math. Phys. 46, tves of f to vansh; see Ref. 9. Smlaly, fo multvecto felds of degee n+1, t s clea that, as equals n+1, the fst thee tems n Eq. 26 also take values n the kenel of and can thus be ncopoated nto. Theefoe, by settng X 1 n = 1 n X, we see that X = X. But L X =d X 1 n X d=d X and L X =d X 1 n+1 X d= 1 n+1 X, so the poposton follows. The ntemedate cases 0n ae much moe nteestng. Howeve, the stuaton fo tenso degee n s substantally dffeent fom that fo tenso degee n and hence wll be dealt wth fst. To smplfy the notaton, we wte X 1 n = 1 nx, X, 2 n = 2 n X, so that we obtan X 1 n = 1 nx, X 2 n = 2 n X, and X = 1 n 1 X dp + X dp 1 n 1 X dq X dx, 28 X = px + 1 n 1 p X, espectvely. Poposton 2.3: An n-multvecto feld X on P s locally Hamltonan f and only f, locally and modulo tems takng values n the kenel of, t can be wtten n tems of a sngle functon f, as follows: 1 X = n 1! 2 n f x p 1 f n p x 1 + n 1! 2 f n p q 1 f n q p x 2 x 2 x. 29 n Moeove, X s exact Hamltonan f and only f f s a lnea functon of the multmomentum vaables p and the enegy vaable p. Poof: Obvously, X s locally Hamltonan f and only f, locally, X =df fo some functon f, whch n vew of Eq. 28 leads to the followng system of equatons fo the coeffcents X, X, X, and X of X: x n X = 1 n 1 f p, X = f p, X = 1 n f q, X = f x. Insetng ths back nto X, we ave at Eq. 29. Note also that then, X = 1 n 1 p f p + 1n 1 p f p = 1n 1 L f. Next, X wll be exact Hamltonan f and only f, n addton, f = 1 n 1 X, whch n vew of the pecedng equaton means that f must be an egenfuncton of the scalng opeato L wth egenvalue 1, ths s well known to be the case f and only f f s lnea n the multmomentum vaables p and the enegy vaable p. Now we tun to multvecto felds of tenso degee n. Hee, the man esult s the followng. Theoem 2.4: An -multvecto feld X on P, wth 0n, s locally Hamltonan f and only

9 1-9 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, f the coeffcents X 1, X, 2, X 1, and X 2 n ts standad local coodnate epesentaton (26) satsfy the followng condtons: 1 The coeffcents X 1 depend only on the local coodnates x fo M and, n the specal case N=1, also on the local fbe coodnates q fo E, 2 The coeffcents X, 2 ae antsymmetc polynomals n the multmomentum vaables of degee 1,.e., they can be wtten n the fom wth, X 2 s 1 = 1 1 X, 2 = 1 p s 1! s! 2 S 1, X 2 s 1, 2 p s s Y 2 s, s+1 s 1, whee S 1 denotes the pemutaton goup of 2,..., and the coeffcents Y 2 s, s+1 s 1 depend only on the local coodnates x fo M as well as the local fbe coodnates q fo E and ae totally antsymmetc n, 2,..., s as well as n s+1,...,. The emanng coeffcents X 1 and X 2 can be expessed n tems of the pevous ones and of new coeffcents X 1 dependng only on the local coodnates x fo M as well as the local fbe coodnates q fo E and ae totally antsymmetc n 1,...,, accodng to X 1 = p X 1 q 1 + p X1 x 1 s 1 p j s X j, 1 s 1 s+1 q (the fst tem beng absent as soon as N1) and X 2 = 1 p X 2 x 1p X,2 x p s X 1 s 1 s+1 x s=2 + X 1 q, 32 p s X,2 s 1 s+1 x 1 X 2 x. 33 It s exact Hamltonan f and only f, n addton, the coeffcents X, 2 depend only on the local coodnates x fo M as well as the local fbe coodnates q fo E and the coeffcents X 1 vansh. Poof: The poof wll be caed out by bute foce computaton. 9 We obtan fo the Le devatve of along X the expesson 1 X 3 L X = 2! x d n x X 1! X,2 + p x 2 1 X 1! p 1 X1! q 2 1 X 1! q s=2 1 1X 2 x dq d n x 2 s X,2 s 1 s+1 x dp d n x X x dp d n x 2 + X 1 dq dp d n x 1 p

10 1-10 Foge, Paufle, and Röme J. Math. Phys. 46, X1! p + 1 k X1! x 1! 1! + 1! 1 s 1 s X,1 s 1 s+1 dp p dp d n x 1 1 s 1 s X k, 1 s 1 s+1 q X 1 q j X 1 k s X 1 s 1 s+1 x dq dq j d n x 1 + 1! q j dq dq j dp d n x 1 1! X 1 p dq dp dp d n x 1. X 1 dq dp k d n x 1 p k 1 1 X l,2 p k X 1 p k dp k dp l d n x 2 dq l dp k dp l d n x 1 Note that the last thee tems would have to be omtted f =n. Let us numbe the tems n ths equaton fom 1 to 12. As we shall see, each of these tems must vansh sepaately. Tem No. 12: Afte contacton wth a sutably chosen n +2-multvecto feld, we see that X 1 cannot depend on p. Tem No. 11: Gven ndces, and mutually dffeent ndces 1,...,, we choose ndces j and 1,..., hee we use the hypothess that n such that ethe j o and, when n 1, a complementay set of ndces 1,..., n 1 to contact ths tem wth the multvecto feld j j 1 n 1 no sum ove j, concludng that X 1 cannot depend on p. Obvously, thee s one case whee ths agument does not wok, namely when N=1, =n 1, and 1,...,. Ths stuaton wll howeve be coveed n the next tem. Tem No. 6: Gven ndces k, and mutually dffeent ndces 1,..., such that 1,...,, we choose a complementay set of ndces 1,..., n 1 to contact ths tem wth the multvecto feld k 0 1 n 1, concludng that X 1 cannot depend on p k, snce n ths case the second tem n the backet gves no contbuton. In patcula, ths settles the emanng case of the pevous tem. Tem No. 10: Afte contacton wth a sutably chosen n +2 - multvecto feld, we see that X 1 cannot depend on q l f N1. Fo N=1, the whole tem vanshes dentcally, and no concluson can be dawn. Ths poves the statements n tem 1 of the theoem. Moeove, t allows to smplfy tem No. 6, as follows: 1 1 X,2 1! p dp dp d n x 2. As befoe, contacton wth a sutably chosen n +2-multvecto feld shows that X k, 2 cannot depend on p. Next we analyze tem No. 9, whch wll gve an mpotant estcton on the coeffcents X, 2. Gven ndces, j,, and mutually dffeent ndces 2,...,, we choose a set of ndces 1,..., n such that 2,..., 1,..., n = to contact ths tem wth the multvecto feld j 1 n, obtanng

11 1-11 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, X j, 2 p 2 1 n = X,2 p j 2 1 n. 34 Now assume the ndex to be chosen such that 2,...,, 1,..., n. Then f 2,...,, we can take = 1, say, to conclude that X j, 2 cannot depend on p, X j, 2 p =0 f 2,...,. 35 Moeove, f 2,...,, ths esult mples that applyng an opeato wth abtay to Eq. 34 gves zeo snce on the ght-hand sde hs, the -tenso klls all tems n the sum ove the ndces 2,..., n whch the ndex appeas among them, 2 X j, 2 p 1 p 2 =0 f 2,..., no sum ove. 36 The geneal soluton to Eqs. 35 and 36 can be wtten n the fom X j,2 1 1 = 1 p 2 s 1! s! j2 p s j,j js Y 2 j s, s+1 s 1, S 1 whee S 1 denotes the pemutaton goup of 2,..., and the newly ntoduced coeffcents,j Y 2 j s, s+1 s 1 ae local functons on E, they do not depend on the multmomentum vaables p k o the enegy vaable p and ae totally antsymmetc both n j 2,...,j s and n s+1,...,. Dffeentatng ths expesson wth espect to p wth = 2 gves X j, 3 p 3 1 n no sum ove = s= p s 2! s! j3 S 2 3 p s j,j js Y 3 j s, s+1 s n, whee S 2 denotes the pemutaton goup of 3,...,, whch shows that Eq. 34 wll hold povded that j,j Y 3 j s, s+1,jj s 1 = Y 3 j s, s+1 s 1. Ths poves the statements n tem 2 of the theoem. We poceed wth tems Nos. 4 and 5 whch mply X 2 p = 1 X 2 x, X 1 p = X 1 q. 37 We obseve fst of all that the hs of both equatons does not depend on the enegy vaable, so they can be mmedately ntegated wth espect to p. Fom tem No. 3 we nfe X 2 p = X,2 x + s X,2 s 1 s+1 x. 38 s=2 An explct calculaton shows that the hs of ths equaton does not depend on the p j, not only when 2,..., but even when 2,...,. Of couse, t also does not depend on p. Thus, accodng to Lemma A.2 fomulated n the appendx, we can ntegate Eq. 38 explctly to obtan ecall that 1 s the opeato that acts on polynomals n the multmomentum vaables

12 1-12 Foge, Paufle, and Röme J. Math. Phys. 46, and the enegy vaable wthout constant tem by multplyng the homogeneous component of degee s by 1/s X 2 = 1 p X 2 x 1p X,2 x s=2 p s X,2 s 1 s+1 x + Ỹ 2, 39 whee the Ỹ 2 ae local functons on E: they do not depend on the multmomentum vaables o on the enegy vaable. The same pocedue woks fo tem No. 7. Thee, we ae left wth X 1 p k = k X1 x k s X 1 s 1 s+1 x 1 s 1 s X k, 1 s 1 s+1 q. 40 Usng the same agument as befoe, we show that the hs does not depend on the p l, not only when 1,..., but even when 1,...,, and nethe does t depend on p. Theefoe, we can ntegate Eq. 40 explctly to obtan X 1 = p X 1 q 1 + p X1 x p s X 1 s 1 s+1 x 1 s 1 p k s X k, 1 s 1 s+1 q + Y 1, 41 whee the Y 1 ae local functons on E: they do not depend on the multmomentum vaables o on the enegy vaable. Fnally, we tun to tems Nos. 1, 2, and 8. They mply X 3 x =0, X 2 q = 1 1X 2 x, X 1 q j = X 1 j q, espectvely. Wth the help of 39, these educe to Ỹ 3 x =0, Ỹ 2 q = 1 1Y 2 x, Y 1 q j = Y 1 j q, whch s easly solved by settng Ỹ 2 = 1 1X 2 x, Y 1 = X 1 q. 42 Hee, the X 1 ae local functons on E: they do not depend on the multmomentum vaables o on the enegy vaable. Ths completes the poof of the statements n tem 3 of the theoem. All that emans to be shown ae the fnal statements concenng exact Hamltonan multvecto felds. To ths end, we calculate

13 1-13 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, L X = 1! X2 X,2 x p 1 x + 1 s=2 1! X 1 x X, 2 s 1 s+1 x p p p s 1 X 2 d n x 2 X 1 s 1 s+1 x X1 p s q p 1 1 s 1Xj, s 1 s+1 q p s j X 1 dq d n x 1 + 1! X 1 p j + 1! X 1 p p + p s 1X, s 1 s+1 p p sdp j d n x 1 j 1 1 s 1X, s 1 s+1 p p sdp d n x 1, whee we have omtted fou tems that vansh because X s locally Hamltonan. Moeove, usng the expessons deved above fo locally Hamltonan multvecto felds, we see that the othe tems vansh as well f and only f we have s 1 p s X j,1 s 1 s+1 j q =0, and 1 1p X, 2 x s=2 p s X,2 s 1 s+1 x =0, X 1 X 2 q =0, x =0. But ths means that the coeffcents of the multmomentum vaables n the above expessons must be ndependent of the multmomentum vaables and that the coeffcents X 1 can wthout loss of genealty be assumed to vansh, whch completes the poof of the theoem. Poof of Theoem 1.3, pat 1: The statements of Theoem 1.3 about multvecto felds ae, n the local fom, based on the local decomposton gven n Poposton 2.1, takng nto account the scalng behavo of the coeffcent functons that follows fom Theoem 2.4, togethe wth that of the coodnate vecto felds. The global veson of these statements can be obtaned by glueng togethe such local decompostons usng appopate pattons of unty. To see that the homogeneous components X s, s= 1,..., 1, of a fbewse polynomal locally Hamltonan -multvecto feld X ae locally Hamltonan, we compute 1 0=L k d X = s +1 k d Xs, k =1,..., 1. s=0 Togethe wth d X =0, ths leads to a Vandemonde matx equaton wth entes 0,2,3,..., 1 annhlatng the vecto d X 1,...,d X 1 T. As the detemnant of a Vandemonde matx does not vansh, the above vecto must vansh. The followng poposton clafes the ntepetaton of homogeneous locally Hamltonan multvecto felds.

14 1-14 Foge, Paufle, and Röme J. Math. Phys. 46, Poposton 2.5: Let X be a locally Hamltonan -multvecto feld on P. Then (1) X s exact Hamltonan ff,x takes values n the kenel of. (2) If,X sx takes values n the kenel of, fo some ntege s between 0 and 1, then X s globally Hamltonan wth assocated Posson fom 1 1 s +1 X. (3) If,X+X takes values n the kenel of, then X =0. Poof: The fst statement follows mmedately fom Eq. 25. Smlaly, the second clam can be poved by multplyng Eq. 25 by 1 1 /s+1 and combnng t wth Eq. 1 and Eq. 4 to gve 1 1 d X s = s +1 L X + 1 s +1 X = 1 s +1,X+X, whch equals X snce, by hypothess,,x sx =0. Fnally, the thd statement follows by obsevng that the kenel of s contaned n the kenel of and hence accodng to the hypothess made, 0=,X+X = L X X L + X = L X, whee we have used the nvaance of unde. Theefoe, accodng to Poposton A.1, X s the pull-back to P of an n-fom on E va the pojecton that defnes P as a vecto bundle ove E, whch n tun can be obtaned as the pull-back to E of X va the zeo secton of P ove E. But ths pull-back s zeo, snce vanshes along the zeo secton of P ove E. It may be nstuctve to spell all ths out moe explctly fo locally Hamltonan vecto felds =1. We begn by wtng down the geneal fom of a locally Hamltonan vecto feld X: n adapted local coodnates, t has the epesentaton X = X x + X q + X + X p p, whee accodng to Theoem 2.4, the coeffcent functons X and X depend only on the local coodnates x fo M and on the local fbe coodnates q fo E the X beng ndependent of the latte as soon as N1, wheeas the coeffcent functons X and X ae explctly gven by X = p X q + p X x p X x p Xj j q + X q the fst tem beng absent as soon as N1 and 43 X X = p x p X x + X x 44 wth coeffcent functons X that once agan depend only on the local coodnates x fo M and on the local fbe coodnates q fo E. Regadng the decomposton 19, the stuaton hee s patculaly nteestng and somewhat specal snce s nondegeneate on vecto felds, so thee ae no nontval vecto felds takng values n the kenel of and hence the decomposton 19 can be mpoved. Coollay 2.6: Any locally Hamltonan vecto feld X on P can be unquely decomposed nto the sum of two tems, X = X + X +, 45

15 1-15 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, whee () X has scalng degee 1,.e.,,X = X, and s vetcal wth espect to the pojecton onto E. () X + has scalng degee 0,.e.,,X + =0, s exact Hamltonan, s pojectable onto E and concdes wth the canoncal lft of ts pojecton onto E. Poof: In adapted local coodnates, the two contbutons to X ae, accodng to Eqs. 43 and 44, gven by and X = X q p + X x p X + = X x + Xj X q q p j X x p + X x p + X X p q p x p + X p x p. Thus all statements of the coollay follow fom what has aleady been shown, except fo the vey last one, whch s based on the followng emak. Remak: Evey bundle automophsm of E as a fbe bundle ove M admts a canoncal lft to a bundle automophsm of ts fst ode jet bundle JE as an affne bundle ove E and, by appopate twsted affne dualzaton, to the extended multphase space P as a vecto bundle ove E. Smlaly, passng to geneatos of one-paamete goups, one sees that evey vecto feld X E on E that s pojectable to a vecto feld X M on M admts a canoncal lft to a vecto feld X JE on JE and, by appopate twsted affne dualzaton, to a vecto feld X P on P. See, fo example, Ref. 7, Sec. 4B. When N=1, lftng to P s even possble fo abtay dffeomophsms of E and abtay vecto felds on E, snce n ths case P can be dentfed wth the nth exteo powe of the cotangent bundle of E. Explctly, n tems of adapted local coodnates x,q, p, p, wemay wte X M = X x and X E = X x + X q, whee, except fo N=1, the X do not depend on the q ; then the coodnate expesson fo the lfted vecto feld, X P, s pecsely gven by the expesson fo X + above. Obvously, X P has scalng degee 0 and hence s not only locally but even exact Hamltonan. Convesely, statng wth an exact Hamltonan vecto feld X +, we can obtan X M and X E by pojecton onto M and E, espectvely. Thus, the coodnate expesson fo X + shows that pecsely all exact Hamltonan vecto felds ae obtaned by ths lftng pocedue. Smlaly, one can show that all dffeomophsms of P that peseve the multcanoncal fom ae obtaned by lftng of automophsms o, fo N=1, dffeomophsms of E, ths s the feld theoetcal analog of a well-known theoem n geometc mechancs, accodng to whch all dffeomophsms of a cotangent bundle that peseve the canoncal fom ae nduced by dffeomophsms of ts base manfold. To conclude ths secton, let us note that the defnton of pojectablty of vecto felds can be mmedately genealzed to multvecto felds: an -multvecto feld X E on the total space E of a fbe bundle ove a manfold M wth bundle pojecton :E M s called pojectable f fo any two ponts e 1 and e 2 n E, T e1 X E e 1 = T e2 X E e 2 f e 1 = e 2, o n othe wods, f thee exsts an -multvecto feld X M on M such that T X E = X M. In adapted local coodnates, ths amounts to equng that f we wte

16 1-16 Foge, Paufle, and Röme J. Math. Phys. 46, X E = 1! X 1 x 1 x +, whee the dots denote 1-vetcal tems, the coeffcents X 1 should depend only on the local coodnates x fo M but not on the local fbe coodnates q fo E. Now we ntoduce the followng temnology. Defnton 2.7: An -multvecto feld on P s called pojectable f t s pojectable wth espect to any one of the thee pojectons fom P: to P 0, to E, and to M. Wth ths temnology, Theoem 2.4 states that fo 0n, locally Hamltonan -multvecto felds on P ae pojectable as soon as N1 and ae pojectable to E but not necessaly to P 0 o to M when N=1. Inspecton of Eq. 32 shows, howeve, that they ae pojectable to P 0 f and only f they ae pojectable to M. Consdeng the specal case of vecto felds =1, we beleve that vecto felds on the total space of a fbe bundle ove space-tme whch ae not pojectable should be egaded as pathologcal, snce they geneate tansfomatons whch do not nduce tansfomatons of space-tme. It s had to see how such tansfomatons mght be ntepeted as canddates fo symmetes of a physcal system. By analogy, we shall adopt the same pont of vew egadng multvecto felds of hghe degee, snce although these do not geneate dffeomophsms of E as a manfold, they may pehaps allow fo an ntepetaton as geneatos of supedffeomophsms of an appopate supemanfold bult ove E as ts even pat. III. POISSON FORMS AND HAMILTONIAN FORMS Ou am n ths secton s to gve an explct constucton of Posson n -foms and, moe geneally, of Hamltonan n -foms on the extended multphase space P, whee 0n. Note that Eq. 11 only makes sense fo n ths ange. A specal ole s played by closed foms, snce closed foms ae always Hamltonan and closed foms that vansh on the kenel of ae always Posson, these ae n a sense the tval examples. In othe wods, the man task s to undestand the extent to whch geneal Hamltonan foms devate fom closed foms and geneal Posson foms devate fom closed foms that vansh on the kenel of. As a wam-up execse, we shall settle the exteme cases of tenso degee 0 and n. The case =n has aleady been analyzed n Ref. 2, so we just quote the esult. Poposton 3.1: A functon f on P, egaded as a 0-fom, s always Hamltonan and even Posson. Moeove, ts assocated Hamltonan n-multvecto feld X s, n adapted local coodnates and modulo tems takng values n the kenel of, gven by Eq. (29). The case =0 s equally easy. Poposton 3.2: An n-fom f on P s Hamltonan o Posson f and only f t can be wtten as the sum of a constant multple of wth a closed fom whch s abtay f f s Hamltonan and vanshes on the kenel of f f s Posson. Indeed, f f s a Hamltonan n-fom, the multvecto feld X that appeas n Eq. 11 wll n fact be a functon whch must be locally Hamltonan and hence, by Poposton 2.2, constant. Thus df must be popotonal to and so f must be the sum of some constant multple of and a closed fom. The ntemedate cases 0n ae much moe nteestng. To handle them, the fst step s to dentfy the content of the kenel condton 13 n adapted local coodnates fo completeness, we also nclude the two exteme cases. Poposton 3.3: An n -fom f on P, wth 0n, vanshes on the kenel of f and only f, n adapted local coodnates, t can be wtten n the fom f = 1! f 1 d n 1 x ! f 0 dq d n x 0 + 1! f, 1 dp d n x ! f 0 dp d n x 0 dq dp d n x 0, 46

17 1-17 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, whee the second tem n the last paentheses s to be omtted f =n 1 wheeas only the fst tem emans f =n. Note that fo one-foms just as fo functons, the kenel condton 13 s vod, snce s nondegeneate. Also, t s n ths case usually moe convenent to eplace Eq. 46 by the standad local coodnate epesentaton f = f dx + f dq + f dp + f 0 dp. Poof: Fom the patcula expesson fo n adapted local coodnates, we see fst of all that foms of degee n vanshng on the kenel of must be n 2-hozontal snce they vansh on 3-vetcal multvecto felds and that the only tem whch s not n 1-hozontal s Futhemoe, f must vansh on the bvectos q dq dp k d n x 0. p + k k p x and p x + p whch yelds the statement of the poposton. Fo moe detals, see Ref. 9 The poposton above can be used to pove the followng nteestng and useful fact. Poposton 3.4: An n -fom f on P, wth 0n, vanshes on the kenel of f and only f thee exsts an +1-multvecto feld X on P such that x 47 Then obvously, f = X. 48 df = L X. In patcula, f s closed f and only f X s locally Hamltonan. At evey pont of P, the statement that the ncluson of the kenel of n the kenel of f mples that thee s a multvecto Y such that Y = f at ths pont, can be shown wthout efeence to the patcula fom of. 10 Howeve, the expesson fo n adapted local coodnates shows that we can even obtan a multvecto feld Y wth ths popety. Poof: The f pat beng obvous, obseve that t suffces to pove the only f pat locally, n the doman of defnton of an abtay system of adapted local coodnates, by constuctng the coeffcents of X fom those of f. Indeed, snce the elaton between f and X postulated n Eq. 48 s puely algebac,.e., t does not nvolve devatves, we can constuct a global soluton patchng togethe local solutons wth a patton of unty. A compason of X, whee X s an +1-multvecto feld! gven by Eq. 26, wth 46 shows that when n, ths can be acheved by settng 49 X 0 = 1 f 0, X, 1 = 1 f, 1, 50 X 0 = 1 +1 f 0, X 1 = f 1, 51 whle fo =n, only the last equaton s petnent fo =n 1, the same concluson can also be eached by compang 28 and 47. Coollay 3.5: An n -fom f on P, wth 0n, s a Hamltonan fom f and only f df vanshes on the kenel of and s a Posson fom f and only f both df and f vansh on the kenel of. Wth these pelmnaes out of the way, we can poceed to the constucton of Posson foms whch ae not closed. As we shall see, thee ae two such constuctons whch, taken togethe, wll be suffcent to handle the geneal case.

18 1-18 Foge, Paufle, and Röme J. Math. Phys. 46, The fst constucton s a genealzaton of the unvesal multmomentum map of Ref. 2, whch to each exact Hamltonan -multvecto feld F on P assocates a Posson n -fom JF on P defned by Eq. 52 below. What emaned unnotced n Ref. 2 s that ths constucton woks even when X s only locally Hamltonan. In fact, we have the followng genealzaton of Poposton 4.3 of Ref. 2. Poposton 3.6: Fo evey locally Hamltonan -multvecto feld F on P, wth 0n, the fomula JF = 1 1 F defnes a Posson n -fom JF on P whose assocated Hamltonan multvecto feld s F+,F, that s, we have 52 djf = F+,F. 53 Poof: Obvously, JF vanshes on the kenel of snce ths s contaned n the kenel of. Moeove, snce L F s supposed to vansh, we can use the algebac elatons fo the Le devatve along multvecto felds and = to compute djf = 1 1 d F = 1 1 L F F d = 1 L F + L F + F = F, + F. The second constucton uses dffeental foms on E, pulled back to dffeental foms on P va the taget pojecton : P E. Chaactezng whch of these ae Hamltonan foms and whch ae Posson foms s a smple execse. Poposton 3.7: Let f 0 be an n -fom on E, wth 0n. Then () * f 0 s a Hamltonan fom on P f and only f df 0 s n -hozontal. () * f 0 s a Posson fom on P f and only f f 0 s n 1-hozontal and df 0 s n -hozontal. Poof: In adapted local coodnates x,q fo E and x,q, p, p fo P, we can wte f 0 = 1! f 1 0 d n 1 x ! f 0 0 dq d n x 0 +, 54 whee the dots denote hghe ode tems contanng at least two dq s. Now applyng Poposton 3.3 to * f 0, we see that * f 0 wll vansh on the kenel of f and only f the tems denoted by the dots all vansh,.e., f f 0 can be wtten n the fom f 0 = 1! f 1 0 d n 1 x ! f 0 0 dq d n x But ths s pecsely the condton fo the n -fom f 0 to be n 1-hozontal. Note that ths equvalence holds even when = n 1, povded we undestand the condton of beng 0-hozontal to be empty. Smlaly, snce Poposton 3.4 mples that a fom on P s Hamltonan f and only f ts exteo devatve vanshes on the kenel of, the same agument appled to d * f 0 = * df 0 shows that, espectvely of whethe * f 0 tself vanshes on the kenel of o not and hence whethe we use Eq. 54 o Eq. 55 as ou statng pont, * f 0 wll be Hamltonan f and only f 1 1 f 2 df 0 = 0 1! x d n x 2 +! 1 f 0 q f 0 1 x dq d n x 1. But ths s pecsely the condton fo the n +1-fom df 0 to be n -hozontal. Moeove, t s easy to wte down an assocated Hamltonan -multvecto feld X 0,

19 1-19 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, X 0 = f 0! q f 0 1 x p 1 x 2 x 1 f 2 0 1! x p x 2 x. Note also that f f 0 s n 1-hozontal and thus has the fom stated n Eq. 55, df 0 would contan just one addtonal hghe ode tem, namely Its absence means that 1 f 0 0 j +1! q dq dq j d n x 0. f 0 0 j q = f 0 0 q j, so thee exst local functons f 0 0 on E such that f 0 0 = f 0 0 q. Ths mples that f 0 can be wtten as the sum f 0 = f h + f c of a hozontal fom f h and a closed fom f c,defned by settng 56 f h =!f f 1 0 x d n x 1 and f c = 1 f 1 0! x d n 1 f 0 x ! q dq d n x 0. The same knd of local decomposton nto the sum of a hozontal fom and a closed fom can also be deved f f 0 s abtay and thus has the fom stated n Eq. 54; ths case can be handled by deceasng nducton on the numbe of dq s that appea n the hghe ode tems denoted by the dots n Eq. 54. We shall efan fom wokng ths out n detal, snce unfotunately the decomposton 56 depends on the system of adapted local coodnates used n ts constucton: unde coodnate tansfomatons, the tems f h and f c mx. Theefoe, ths decomposton has no coodnate ndependent meanng and s n geneal vald only locally. Fnally, we note that n the above dscusson, we have delbeately excluded the exteme cases =0 n-foms and =n functons. Fo n-foms, the equvalences stated above would be ncoect snce f f 0 has tenso degee n and hence X 0 has tenso degee 0, X0 would by Poposton 2.2 be a constant multple of wheeas d * f 0 would be educed to a lnea combnaton of tems of the fom dq d n x, mplyng that * f 0 can only be Hamltonan f t s closed. Fo functons, the constucton s unnteestng snce accodng to Poposton 3.1, all functons on P ae Posson, and not just the ones lfted fom E. Now we ae eady to state ou man decomposton theoem. In what follows, we shall smply wte f 0 nstead of * f 0 when thee s no dange of confuson, the man excepton beng the poof of Theoem 3.8 below. Theoem 3.8: Any Hamltonan n -fom and, n patcula, any Posson n -fom f on P, wth 0n, admts a unque decomposton

20 1-20 Foge, Paufle, and Röme J. Math. Phys. 46, f = f 0 + f + + f c wth f + = f s, 57 whee (1) f 0 s (the pull-back to P of) an n -fom on E whose exteo devatve s n -hozontal and whch s othewse abtay f f s Hamltonan wheeas t s estcted to be n 1-hozontal ff f s Posson. (2) f + s of the fom f + = JF = 1 1 F wth F = 1+L 1 X +, 58 and coespondngly, fo,...,, f s s of the fom 1 1 f s = Xs 1, 59 s whee X s any fbewse polynomal Hamltonan -multvecto feld assocated wth f, decomposed accodng to Eq. (19). (3) f c s a closed n -fom on P whch vanshes on the zeo secton of P (as a vecto bundle ove E) and whch s othewse abtay f f s Hamltonan wheeas t s estcted to vansh on the kenel of ff f s Posson. We shall efe to Eq. (57) and to Eq. (60) below as the canoncal decomposton of Hamltonan foms o Posson foms on P. Poof: Let f be a Posson n -fom and X be a Hamltonan -multvecto feld assocated wth f. As aleady mentoned n the ntoducton, we may wthout loss of genealty assume X to be fbewse polynomal and decompose t nto homogeneous components wth espect to scalng degee, accodng to Eq. 19, X = X + X + + Then defnng F as n the theoem, o equvalently, by wth X + = X s 1. we obtan F = F s 1 wth F s 1 = 1 s X s 1, F +,F = X +, and hence accodng to Eq. 53, the exteo devatve of the dffeence f JF s gven by df JF =df djf = X X+ = X. Applyng the equvalence stated n Eq. 23, we see that snce X has scalng degee 1, X must have scalng degee 0 and hence, accodng to Poposton A.1, s the pull-back to P of some n -fom f 0 on E, df JF = X = * f 0. Next, we defne f 0 to be the estcton of f JF to the zeo secton of P, o moe pecsely, ts pull-back to E wth the zeo secton s 0 :E P,

21 1-21 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, and set Then and f 0 = s 0 * f JF, f c = f * f 0 JF. df c =df JF d * s 0 * f JF =df JF * s 0 * df JF = * f 0 * s 0 * * f 0 =0, s 0 * f c = s 0 * f JF s 0 * * f 0 = f 0 s 0 * * f 0 =0, showng that ndeed, f c s closed and vanshes on the zeo secton of P. Poof of Theoem 1.3, pat 2: The statements of Theoem 1.3 about dffeental foms ae mmedate consequences of Theoem 3.8. Remak: It should be noted that despte appeaances, the decompostons 57 of Theoem 3.8 and 21 of Theoem 1.3 ae not necessaly dentcal: fo,...,, the f s of Eq. 57 and the f s of Eq. 21 may dffe by homogeneous closed n -foms of scalng degee s. But the decomposton 57 of Theoem 3.8 seems to be the moe natual one. Theoem 3.8 mples that Posson foms have a athe ntcate local coodnate epesentaton, nvolvng two locally Hamltonan multvecto felds. Indeed, f we take f to be a geneal Posson n -fom on P, wth 0n, we can apply Popostons 3.4 and 3.6 to ewte Eq. 57 n the fom f = f F + 1 Fc, 60 whee f 0 s as befoe whle F and F c ae two locally Hamltonan multvecto felds on P of tenso degee and +1, espectvely, satsfyng F =0 and F c =0. The condton F c =0 wll guaantee that Fc vanshes on the zeo secton of P. In tems of the standad local coodnate epesentatons 46 fo f, 55 fo f 0, 26 fo F and fo F c, and fo and, Eqs. 2 and 3, we obtan f 1 = 1 1 pf s p s F,1 s 1 s+1 + f F c 1, 61 f 0 = 1 s p s F 0 s 1 s+1 + f 0 0 F c 0, s=0 62 f, 1 = F c, 1, 63 f 0 = F c 0, whee the coeffcents of F and of F c ae subject to the constants lsted n Theoem 2.4; n patcula, the coeffcents F c 0 and F c 1 can be completely expessed n tems of the coeffcents F c 0 and F c, 1, accodng to Eqs. 32 and 33 wth eplaced by +1, X eplaced by F c, and X eplaced by 0. In patcula, we see that the coeffcents f 1 ae antsymmetc polynomals n the multmomentum vaables of degee. Moe explctly, we can ewte Eq. 61 n the fom 64

22 1-22 Foge, Paufle, and Röme J. Math. Phys. 46, f 1 = 1 1 pf 1 + f 1 s + f F c 1, whee nsetng the expanson 31 wth X eplaced by F, X s 1 eplaced by F s 1 and Y s 1 eplaced by G s 1 =1/sg s gves, afte a shot calculaton, f 1 s = p 1 s! s! 1 p s s g 1 s, s+1 s. S Fnally, we want to clafy the elaton between Posson foms and Hamltonan multvecto felds n tems of the standad local coodnate epesentatons. Theoem 3.9: Let f be a Posson n -fom and X be a Hamltonan -multvecto feld on P assocated wth f. Assume that, n adapted local coodnates, f and X ae gven by Eqs 46 and 26, espectvely. Then X 1 = 1 1 f 1 p f 1 x, 65 X,2 1 f 2 =, 66 n +1 p X 1 = 1 f 1 q f 1 x, 67 X 2 = f 2 x, 68 that s, locally and modulo tems takng values n the kenel of, X s gven by X = + 1 1! f2 x p f 1! 2 n +1 p q 1 f 2 p x + 1 f 2 x f 2 q p + 1 x f 2 x x 2 x p x 2 x. If, n the canoncal decomposton 57 and 60 of f, the closed tem f c = 1 Fc s absent, then f 0 =0. If f s hozontal wth espect to the pojecton onto M, then f 0 =0. In these cases, the above fomulas smplfy accodngly. Poof: Thee ae seveal methods fo povng ths, wth cetan ovelaps. Let us begn wth the tval case of closed foms f, fo whch we must have X=0. Assumng f to be of the fom f c = 1 Fc and usng Eqs to ewte the expessons on the hs of the above equatons n tems of the components of F c, we must show that 69 F c 1 p + 1 F c 1 x =0, F c 2 p =0,

23 1-23 Hamltonan multvecto felds and Posson foms J. Math. Phys. 46, F c 1 q 1 F c 1 F c 2 x =0, x =0. But ths follows dectly fom the petnent elatons fo locally Hamltonan multvecto felds deved n the poof of Theoem 1.3 whch hold snce F c s locally Hamltonan. To handle the emanng cases whee f s of the fom f = f F, t s ease to poceed by dect nspecton of Eq. 11. Indeed, we may fo a geneal Posson fom f apply the exteo devatve to Eq. 46 and compae the esult wth the expesson fo X. In ths way, Eqs. 68, 67, and 65 can be obtaned dectly by equatng the coeffcents of d n x 2, of dq d n x 1 and of dp d n x 1, espectvely. The only case whch eques an addtonal agument s Eq. 66, snce collectng tems popotonal to dp d n x 1 leads to 1 1 1! X, 2 dp d n x 2 = 1! f 1 p dp d n 1 f,2 x 1 1! x dp d n x 2 1 f,1! x dp d n x 1. But when f s of the fom f = f F, Eq. 63 mples that the last two tems on the hs of the equaton above vansh. Moeove, snce F s Hamltonan, we know fom Theoem 2.4 that the F 1 depend on the p only f 1,...,, and hence accodng to Eq. 61, the same s tue fo the f 1. Ths educes the fst tem on the hs of the above equaton to an expesson whch, when compaed wth the lhs, leads to the concluson that fo any choce of mutually dffeent ndces and 2,...,, we have X,2 = f 2 p f 2,..., no sum ove. Summng ove gves Eq. 66. IV. POISSON BRACKETS In the chaactezaton of locally Hamltonan multvecto felds and of Posson foms deved n the pecedng two sectons, the decomposton nto homogeneous tems wth espect to scalng degee plays a cental ole. It s theefoe natual to ask how ths decomposton comples wth the Schouten backet of Hamltonan multvecto felds and wth the Posson backet of Posson foms. To ths end, let us fst ecall the defnton of the Posson backet between Posson foms gven n Ref. 1 fo n 1-foms and n Ref. 2 fo foms of abtay degee. Defnton 4.1: Let f and g be Posson foms of tenso degee n and n s onp, espectvely. The Posson backet s the Posson fom of tenso degee n s+1 on P defned by f,g = 1 s 1 Y X +d 1 1s 1 Y f X g 1 1s Y X, whee X and Y ae Hamltonan multvecto felds assocated wth f and g, espectvely. We fnd the followng popetes of the two mentoned backet opeatons wth espect to scalng degee. Poposton 4.2: Let X and Y be homogeneous multvecto felds on P of scalng degee k and l, espectvely. Then the Schouten backet X,Y s of scalng degee k+l, L X = kx, L Y = ly L X,Y = k + lx,y. 71 Poof: The poposton s a consequence of the gaded Jacob dentty fo multvecto felds, 8 70