Canonical Transformations and Hamiltonian Evolutionary Systems Samer Ashhab 5 0 0 Department of Mathematics, University of New Orleans, New Orleans, 2 LA 70148. n a E-mail: [email protected] J 1 2 Abstract 1 v In many Lagrangian field theories one has a Poisson bracket de- 6 finedonthespaceoflocalfunctionals. Wefindnecessaryandsufficient 5 0 conditions for a transformation on the space of local functionals to be 1 canonical in three different cases. These three cases depend on the 0 specific dimensions of the vector bundle of the theory and the associ- 5 0 ated Hamiltonian differential operator. We also show how a canonical / transformation transforms a Hamiltonian evolutionary system and its h p conservation laws. Finally we illustrate these ideas with three exam- - ples. h t a m Keywords: Hamiltonian evolutionary systems, Poisson bracket, jet bundle. AMS Subject Classification; Primary: 37K05. Secondary: 53Z05, 35Q99. : v i X r 1 Introduction a InHamiltonianevolutionarysystems onehasaPoisson bracket. ThisPoisson bracket is defined on the space of local functionals of a fiber bundle. The system is characterized by a Hamiltonian which is a local functional and the associated Poisson bracket. The Poisson bracket in turn is characterized by a Hamiltonian differential operator (see the discussion in the next section). The solutions to the evolutionary system are sections of the fiber bundle. Generally an evolutionary system has the form u = DδH t 1 where D is the associated Hamiltonian differential operator, δ the variational derivative and H the Hamiltonian of the system. The variational derivative of a local functional P = Pν has the same components as E(Pν) where M E is the Euler-Lagrange oRperator and P is a local function representing the local functional P, while ν is a volume element on the base manifold M. A good reference on evolutionary systems and their theory can be found in [5] and [8] for example. The reader may consult [6] as well. An action on the bundle induces an action/transformation on the space of local functionals. If this transformation preserves the Poisson bracket we call it a canonical transformation. This terminology was used in [1] and is in anology to the one used in [7] for the case of symplectic manifolds. In [1] canonical transformations were studied in the case the Poisson bracket is defined by a differential operator of order zero. In this paper we study canonical transofrmations in the case of higher order differential operators. We work with vector bundles with m-dimensional fibers and n-dimensional base space. We consider special cases of higher order differential operators whichwefindapplicationsforandintendtobroadenourstudyinfuturework. A canonical transformation will transform the evolutionary system, its Hamiltonian, and its conservation laws where a new system is obtained. We will show that the conditions for a canonical transformation are rather re- strictive in some cases. We illustrate by a few examples at the end of the paper. Itwouldbeinterestingtostudythesh-Liealgebrastructureandreduction as was done in [1] and [3]. However we leave this for a possible future work. 2 Background material Let π : E → M be a vector bundle of dimension m+n and with a base space an n-dimensional manifold M. Let J∞E be the infinite jet bundle of E. The restriction of the infinite jet bundle over an appropriate open set U ⊂ M is trivial with fiber an infinite dimensional vector space V∞. The bundle π∞ : J∞E = U ×V∞ → U U 2 then has induced coordinates given by (xi,ua,ua,ua ,...,). i i1i2 We use multi-index notation and the summation convention throughout the paper. If j∞φ is the section of J∞E induced by a section φ of the bundle E, then ua ◦j∞φ = ua ◦φ and ua ◦j∞φ = (∂ ∂ ...∂ )(ua ◦j∞φ) I i1 i2 ir where r is the order of the symmetric multi-index I = {i ,i ,...,i }, with the 1 2 r convention that, for r = 0, there are no derivatives. For more details see [2] and [6]. Let Loc denote the algebra of local functions where a local function E on J∞E is defined to be the pull-back of a smooth function on some finite jet bundle JpE via the projection from J∞E to JpE. Let Loc0 denote the E subalgebra of Loc such that P ∈ Loc0 iff (j∞φ)∗P has compact support for E E allφ ∈ ΓE with compact support and where ΓE denotes the set of sections of the bundle E → M. The de Rham complex of differential forms Ω∗(J∞E,d) on J∞E possesses a differential ideal, the ideal C of contact forms θ which satisfy (j∞φ)∗θ = 0 for all sections φ with compact support. This ideal is generated by the contact one-forms, which in local coordinates assume the form θa = dua −ua dxi. J J iJ NowletC denotethesetofcontactone-formsoforder zero. Contactone- 0 forms of order zero satisfy (j1φ)∗(θ) = 0 and, in local coordinates, they as- sume the form θa = dua−uadxi. Notice that both C and Ωn,1 = Ωn,1(J∞E) i 0 are modules over Loc . Let Ωn,1 denote the subspace of Ωn,1 which is lo- E 0 cally generated by the forms {(θa ∧ dnx)} over Loc . Let ν denote a vol- E ume element on M and notice that in local coordinates ν takes the form ν = fdnx = fdx1 ∧dx2 ∧...∧dxn for some function f : U → R and U is a subset of M on which the xi’s are defined. ∂ ∂ Define the operator D (total derivative) by D = +ua (recall we i i ∂xi iJ∂ua J assume the summation convention, i.e., the sum is over all a and multi-index J). For I = {i i ···i } where k > 0, D is defined by D = D ◦ D ◦ 1 2 k I I i1 i2 ···◦D . If I is empty I = {} then D is just multiplication by 1. We also ik I define (−D) = (−1)|I|D . Recall that the Euler-Lagrange operator maps I I Ωn,0(J∞E) into Ωn,1(J∞E) and is defined by 0 E(Pν) = E (P)(θa ∧ν) a 3 where P ∈ Loc ,ν is a volume form on the base manifold M, and the E components E (P) are given by a ∂P E (P) = (−D) ( ). a I ∂ua I For simplicity of notation we may use E(P) for E(Pν). We will also use D˜ i ∂ ∂ ∂P for +u˜a and E˜ (P) for (−D˜) ( ) so that E(P) = E˜ (P)(θ˜a∧ν) ∂x˜i iJ∂u˜a a I ∂u˜a a J I in the (x˜µ,u˜a) coordinate system. Now let F be the space of functionals where P ∈ F iff P = Pν for Z M some P ∈ Loc0. Let D be a differential operator which has components E Dab = ωabID where ωabI ∈ Loc and D is a combination of the total I E I derivatives as determined by the multi-index I, i.e., D is a composite of the I form D ◦ D ◦ ··· ◦ D . Define a Poisson bracket on the space of local i1 i2 ik functionals F by {P,Q}(φ) = [E(P)D(E(Q))◦jφ]ν, Z M where φ ∈ ΓE, ν is a volume form on M,P = Pν,Q = Qν, and Z Z M M P,Q ∈ Loc0. In this expression one may represent D as a matrix differential E operator and E as a row/column vector (as appropriate) consisting of the components E . We assume that D is Hamiltonian so that we have a genuine a Poisson bracket that is antisymmetric and satisfies the Jacobi identity (e.g. see [8]). Using local coordinates (xµ,ua) on J∞E, observe that for φ ∈ ΓE I such that the support of φ lies in the domain Ω of some chart x of M, one has {P,Q}(φ) = ([E (P)Dab(E (Q))]◦jφ◦x−1)(x−1)∗(ν) a b Z x(Ω) where x−1 is the inverse of x = (xµ). The functions P and Q in our definition of the Poisson bracket (of lo- cal functionals) are representatives of P and Q respectively, since gener- ally these are not unique. In fact F ≃ Hn(J∞E), where Hn(J∞E) = c c Ωn,0(J∞E)/(imd Ωn,0(J∞E)) and imd is the image of the differential c H c H T 4 d defined by d = dxiD . We refer the reader to [1] for more details and H H i for the notation. The interested reader may also consult [2] and [4] for more on the de Rham complex and its cohomology. Let ψ : E → E be an automorphism, sending fibers to fibers, and let ψ : M → M be the induced diffeomorphism of M. Notice that ψ induces M an automorphism jψ : J∞E → J∞E where (jψ)((j∞φ)(p)) = j(ψ ◦φ◦ψ−1)(ψ (p)), M M for all φ ∈ ΓE and all p in the domain of φ. In these coordinates the indepen- dent variables transform via x˜µ = ψµ (xν). Local coordinate representatives M of ψ and jψ may be described in terms of charts (Ω,x) and (Ω˜,x˜) of M, M and induced charts ((π∞)−1(Ω),(xµ,ua)) and ((π∞)−1(Ω˜),(x˜µ,u˜a)) of J∞E. I I Observe that the total derivatives satisfy D (F ◦jψ) = ((D˜ F)◦jψ)D ψj (2.1) i j i M where x˜j = ψj (x). M Example Let M = R2,E = R2 × R1 and consider the transformation ψ defined by u˜ = xu+3yu2,x˜ = xcosθ+ysinθ,y˜= −xsinθ+ycosθ. Now let F = u˜ then F ◦jψ = (xu +u+6yuu )cosθ+(xu +3u2+6yuu )sinθ so x˜ x x y y that D (F ◦jψ) = (xu +2u +6yu2 +6yuu )cosθ+(xu +u +6uu + x xx x x xx yx y x 6yu u + 6yuu )sinθ. On the other hand the right-hand side of equation x y yx (2.1) yields [u˜ cosθ+u˜ (−sinθ)]◦jψ which when evaluated and simplified x˜x˜ x˜y˜ yields the same expression as above. Remark For simplicity we may skip writing the tilde’s, so in the above example one simply writes F = u instead of F = u˜ , ...etc. x x˜ 3 Canonical transformations of the Poisson structure Let L : J∞E → R be a Lagrangian in Loc (generally we will assume E that any element of Loc is a Lagrangian). Let Lˆ = L ◦ (xµ,ua)−1 and let E I 5 L˜ = L ◦ (x˜µ,u˜a)−1. Then, in local coordinates, L˜ is related to Lˆ by the I equation (L˜ ◦jψ¯)det(J) = Lˆ, where jψ¯ = (x˜ν,u˜b )◦(xµ,ua)−1 and J is the Jacobian matrix of the transfor- K I mationψ = x˜ν◦(xµ)−1. Withabuseofnotationwemayassume coordinates M and charts are the same and write x˜ν = ψ (xµ). For simplicity, we have also M assumed that ψ is orientation-preserving. In this case the functional M L˜= L˜dnx˜ ZΩ˜ is the transformed form of the functional Lˆ= Lˆdnx Z Ω ˆ ˜ ˜ where L and L are related as above, Ω is the domain of integration and Ω ¯ is the transformed domain under jψ (see [8] pp.249-250). Notice that both of these are local coordinate expressions of the equation L = Lν, for Z M appropriately restricted charts. Now suppose that ψ is an automorphism of E, jψ its induced automorphism on J∞E, and ψ its induced (orientation- M preserving) diffeomorphism on M. Also suppose that Lˆ and L˜ are two La- grangians related by the equation (L˜ ◦jψ)det(ψ ) = Lˆ. We have: M Lemma 3.0.1 Let P be a Lagrangian as above, then ∂ψc E ((P ◦jψ)det(ψ )) = det(ψ ) E(E˜ (P)◦jψ). (3.2) a M M ∂ua c ∂ψc Proof FirstnoticethatEua(Lˆ) = det(ψM)∂uEa (Eu˜c(L˜)◦jψ)(see[8]pp.250). ˜ ˆ ˜ But (L ◦ jψ)det(ψ ) = L. The identity 3.2 follows by letting P = L. M Notice that this is justified since L˜ is arbitrary in the sense that given any L′ there exists an Lˆ derived from a Lagrangian L as above such that (L′ ◦ jψ)det(ψ ) = Lˆ since jψ is an automorphism. M ˆ Let ψ denote the mapping representing the induced action of the auto- morphism on sections of E, i.e., ψˆ : ΓE → ΓE where ψˆ(φ) = ψ◦φ◦ψ−1 and M φ is a section of E. This induces a mapping on the space of local functionals given by 6 Ψ(P) = (P ◦ψˆ)(φ) = P(ψ ◦φ◦ψ−1) M = [P ◦j(ψ ◦φ◦ψ−1)]ν Z M M = [P ◦jψ ◦jφ◦ψ−1)]ν Z M M = [P ◦jψ ◦jφ](detψ )ν, M Z M where P(φ) = (P ◦jφ)ν, Z M and φ is a section of E. Now we find conditions on those automorphisms of the space of local functionals under which the Poisson structure is preserved for a few special cases of the bundle E and differential operators D. 3.1 Case I: dim(M)=1, dim(E)=2; D = ωD x We begin with the case where M is 1-dimensional, E is 2-dimensional and consider first order differential operators of the form D = ωD with ω ∈ x Loc . For simplicity we will skip the tilde’s in our notation. Recall that E {P,Q} = D(E(Q))E(P)ν, and hence Z M {P,Q}(φ) = [D(E(Q))E(P)◦jφ]ν. (3.3) Z M InthiscasethePoissonbracketispreserved: {Ψ(P),Ψ(Q)}(φ) = Ψ({P,Q})(φ) if and only if ([ωD E((Q◦jψ)detψ )E((P ◦jψ)detψ )]◦jφ)ν = x M M Z M ([(ωD E(Q)E(P))◦jψ ◦jφ]detψ )ν, x M Z M 7 but since this latter equation holds for all sections φ of E it is equivalent to ωD E((Q◦jψ)detψ )E((P ◦jψ)detψ ) = [(ωD E(Q)E(P))◦jψ]detψ x M M x M up to a divergence. Using Lemma 3.0.1 this is equivalent to ∂ψ ∂ψ E E ωD (detψ E(Q)◦jψ)detψ (E(P)◦jψ) = x M M ∂u ∂u (ω ◦jψ)(D (E(Q))◦jψ)(E(P)◦jψ)detψ x M or ∂ψ ∂ψ E E ω[D (detψ )(E(Q)◦jψ)+detψ (D (E(Q))◦jψ)D ψ ] x M M x x M ∂u ∂u ∂ψ E (E(P)◦jψ) = (ω ◦jψ)(D (E(Q))◦jψ)(E(P)◦jψ) x ∂u up to a divergence and where we have used equation (2.1). (Notice that detψ cancelled from both sides of the equation since it is nonzero.) Finally M ∂ψ E sincethelastequationistrueforallP andQwemusthaveD (detψ ) = x M ∂u ∂ψ 0 and ω◦jψ = ω(detψ )2( E)2. (One may factor out (D (E(Q))◦jψ) over M x ∂u appropriate domains from the left-hand side of the equation to verify that we ∂ψ E must have D (detψ ) = 0 since Q is arbitrary, otherwise the equation x M ∂u may not hold for all P and Q. Also notice that by the arbitrariness of P and Q the divergence term must be zero.) Observe that the first condition ∂ψ E implies that detψ is a constant, while the second condition was sim- M ∂u plified using D ψ = detψ since M is one-dimensional and so areits fibers. x M M Theorem 3.1.1 Let E → M be a vector bundle with dim(M)=1, dim(E)=2 and let the Poisson bracket {·,·} be defined by D = ωD . Then the in- x duced transformation Ψ on the space of local functionals is canonical, i.e., {Ψ(P),Ψ(Q)} = Ψ({P,Q}) for all P,Q ∈ F if and only if ∂ψ E (i) D (detψ ) = 0, and x M ∂u ∂ψ (ii) ω ◦jψ = ω(detψ )2( E)2. M ∂u 8 abi 3.2 Case II: dim(M)=n, dim(E)=m + n; D = ω D i Suppose that M is n-dimensional with fibers of dimension m, and consider first order differential operators of the form D = ωabiD . Preserving the i ˆ ˆ ˆ Poisson bracket: {P ◦ψ,Q◦ψ}(φ) = ({P,Q}◦ψ)(φ) is equivalent to ([ωabiD E ((Q◦jψ)detψ )E ((P ◦jψ)detψ )]◦jφ)ν = i b M a M Z M ([(ωabiD E (Q)E (P))◦jψ ◦jφ]detψ )ν, i b a M Z M but since this latter equation holds for all sections φ of E it is equivalent to ωabiD E ((Q◦jψ)detψ )E ((P◦jψ)detψ ) = [(ωabiD E (Q)E (P))◦jψ]detψ i b M a M i b a M up to a divergence. Using Lemma 3.0.1 this is equivalent to ∂ψd ∂ψc ωabiD (detψ EE (Q)◦jψ)detψ E(E (P)◦jψ) = i M ∂ub d M ∂ua c (ωabi ◦jψ)(D (E (Q))◦jψ)(E (P)◦jψ)detψ i b a M or ∂ψd ∂ψd ωabi[D (detψ E)(E (Q)◦jψ)+detψ E(D E (Q)◦jψ)D ψj ] i M ∂ub d M ∂ub j d i M ∂ψc E(E (P)◦jψ) = (ωabi ◦jψ)(D (E (Q))◦jψ)(E (P)◦jψ) ∂ua c i b a up to a divergence and where we have used equation (2.1). Finally this can be rewritten as ∂ψb ∂ψb ωcdj[D (detψ E)(E (Q)◦jψ)+detψ E(D (E (Q))◦jψ)D ψi ] j M ∂ud b M ∂ud i b j M ∂ψa E(E (P)◦jψ) = (ωabi ◦jψ)(D (E (Q))◦jψ)(E (P)◦jψ) ∂uc a i b a up to a divergence. As before for this to hold for all P and Q we must ∂ψb have D (detψ E) = 0 for all d,j satisfying ωcdj 6= 0 for some c, and j M ∂ud ∂ψa ∂ψb ωabi ◦jψ = detψ ωcdj E ED ψi . M ∂uc ∂ud j M 9 Theorem 3.2.1 Let E → M be a vector bundle with dim(M)=n, dim(E)= n+m, and let the Poisson bracket {·,·} be defined by D = ωabiD . Then the i induced transformation Ψ on the space of local functionals is canonical, i.e., {Ψ(P),Ψ(Q)} = Ψ({P,Q}) for all P,Q ∈ F if and only if ∂ψb (i) ωcdj 6= 0 for some c ⇒ D (detψ E) = 0 for all b, and j M ∂ud ∂ψa ∂ψb (ii) ωabi ◦jψ = detψ ωcdj E ED ψi . M ∂uc ∂ud j M I 3.3 Case III: dim(M)=1, dim(E)=2; D = ω D I Lastly we consider the case where we have a one-dimensional manifold M with one-dimensional fibers and n-th order differential operators of the form D = ωID = ωnD +ωn−1D +···+ω1D +ω0. For simplicity we assume I n n−1 1 that I is a number rather than a multi-index since in this case the base manifold is one-dimensional so, for example, D means (D )n and D = n x I (D )I. Following similar analysis as before and skipping a few steps the x Poisson bracket is preserved if and only if ∂ψ ∂ψ [ω0(detψ EE(Q)◦jψ)+ωID (detψ EE(Q)◦jψ)]detψ M I M M ∂u ∂u ∂ψ E(E(P)◦jψ) = [(ω0E(Q)+ωID E(Q))E(P)]◦jψdetψ I M ∂u or ∂ψ I ∂ψ [ω0(detψ EE(Q)◦jψ)+ ωI[D (detψ E)D (E(Q)◦jψ)] M ∂u (cid:18) k (cid:19) k M ∂u I−k ∂ψ E(E(P)◦jψ) = [(ω0E(Q)+ωID E(Q))E(P)]◦jψ I ∂u up to a divergence. Now observe that using equation (2.1) we have D (F ◦jψ) = (D F ◦jψ)(detψ )I +(D F ◦jψ)(I,I −1)+ I I M I−1 (D F ◦jψ)(I,I −2)+···+(D F ◦jψ)D (detψ ) I−2 1 I−1 M I −1 where D = (D )I and we use (I,j) for the permutations of j I x (cid:18) j −1 (cid:19) detψ ’s and I −j D ’s with detψ in the rightmost slot (this may be de- M x M rived by induction). For example (3,2) = D ((detψ )2)+detψ D (detψ ). x M M x M 10