The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Second order hamiltonian vector fields on tangent bundles Izu Vaisman Vienna, Preprint ESI 49 (1993) September 10, 1993 Supported by Federal Ministry of Science and Research, Austria

2 SECOND ORDER HAMILTONIAN VECTOR FIELDS ON TANGENT BUNDLES Izu Vaisman The Erwin Schrödinger International Institute for Mathematical Physics (ESI), Vienna, Austria October 2, 2001 Abstract. Let M be a differentiable manifold. A vector field Γ of T M which corresponds to a system of second order ordinary differential equations on M is called a second order Hamiltonian vector field if it is the Hamiltonian field of a function F C (T M) with respect to a Poisson structure P of T M. We formulate the direct problem as that of finding Γ if P is given, and the inverse problem as that of finding P if Γ is given. We show the solution of the direct problem if P is a symplectic structure such that the fibers of T M are Lagrangian submanifolds. For the inverse problem we generalize Henneaux method of looking for a solution by studying a corresponding system of linear algebraic restrictions. 1. Introduction Let M be a differentiable manifold. Then, an autonomous second order system of ordinary differential equations on M is equivalent to a certain vector field Γ called a second order vector field on the tangent manifold T M. We say that the system of equations is Hamiltonian, and that Γ is a second order Hamiltonian vector field if there exists a Poisson structure P on T M, and a function F C (T M) such that Γ is the Hamiltonian vector field of F with respect to P. If this happens, the local expressions of the equations along a symplectic leaf of P are the classical equations of Hamilton. The aim of this note is to make an introductory discussion of the following two problems: A The Direct Problem. Suppose that a Poisson structure P is given on T M. Study the existence and the generality of the second order P -Hamiltonian vector fields Mathematics Subject Classification. 58 F 05. Key words and phrases. Second order Hamiltonian vector fields. Poisson structures. It is a pleasure to express here my gratitude to the Erwin Schrödinger International Institute for Mathematical Physics in Vienna and to Professor Peter Michor for support and hospitality during my work on this subject. Typeset by AMS-TEX Typeset by AMS-TEX

3 2 Izu Vaisman B. The Inverse Problem. Suppose that a second order vector field Γ is given on T M. Study the existence and the generality of the Poisson structures P of T M such that Γ is a P -Hamiltonian vector field. These problems are generalizations of the similar problems studied in Lagrangian dynamics, and the latter could profit from the more general setting stated above. Indeed, it may happen that a Lagrangian dynamical system defined by a nondegenerate Lagrangian L will also be Hamiltonian for other, more general, Poisson structures. In this case the corresponding functions, which may be seen as pseudo-energy functions, could be used in the same way as the so-called alternative Lagrangians. The inverse problem is also a generalization of the following natural question. Let S be an autonomous system of second order differential equations on an open domain of R n. Can S be made into a Hamiltonian system in R 2n by a convenient change of variables? We refer the reader to [7] for a survey of Lagrangian dynamics, second order vector fields and the Lagrangian inverse problem, and to [10] for the theory of the Poisson manifolds. 2. Preliminaries In this section we fix the notation and describe the computational instruments. Notice that we always use the Einstein summation convention. Let M n be an n-dimensional differentiable manifold with the local coordinates (q i ) (i = 1,..., n), and let T M be its tangent manifold, and (u i ) the vector coordinates with respect to the basis (/q i ). (We shall assume that everything is C in this paper). On T M we distinguish the vertical foliation V by the fibers of the projection π : T M M. Usually, we shall complement V by a horizontal distribution H such that (2.1) T (T M) = H V, T (T M) = H V. Locally, we shall use bases of the form (2.2) V = span and their dual cobases { } u i, H = span { Q i = q i τ j i } u j, (2.3) V = span {ν i = du i + τ i jdq j }, H = span {dq i }. Then the tensor fields of T M get a natural multiple grading induced by (2.1), and which is made explicit by the use of the bases (2.2), (2.3). In particular, for the spaces of differential forms we have (2.4) Λ k (T M) = p+q=k Λ pq (T M),

4 Hamiltonian Vector Fields 3 where p is the H-degree and q is the V-degree, and the exterior differential d decomposes as (2.5) d = d + d +, where the components increase bidegrees by (1,0), (0,1) and (2,-1), respectively. The operator d is, in fact, the exterior differential along the fibers of T M. We refer the reader to [8] for more details. Notice that the bihomogeneous components of d 2 = 0 are (2.6) d 2 = 0, 2 = 0, d d + d d = 0, d + d = 0, d 2 + d + d = 0. Similarly, we have decompositions (2.7) V k (T M) = p+q=k V pq (T M), where V denotes spaces of multivector fields (i.e. skew-symmetric contravariant tensor fields), and p denotes the H-degree, q denotes the V-degree. Another fundamental ingredient of T M is the vertical twisting homomorphism (e.g., [7]) S : T (T M) T V defined by the conditions (2.8) SX T V, (SX) uo (< α, u >) =< α uo, π X >, where u o, u T M, X T uo (T M), α Λ 1 M. Equivalently, we have ( (2.9) S ξ i ) q i + ηi u i = ξ i u i. And, we shall extend S to a derivation of the tensor algebra of the manifold T M which is such that (2.10) Sf = 0, Sα = α S, f C (T M) and α Λ 1 (T M). An autonomous system of second order ordinary differential equations on M is equivalent to a vector field Γ on T M, called a second order vector field, such that (2.11) SΓ = E where E = u i (/u i ) is the infinitesimal generator of the homotheties of the fibers of T M (the Euler field). The local expression of Γ is (2.12) Γ = u i q i + Γi u i = ui Q i + γ i u i, where γ i = Γ i + τ i j uj, if a horizontal distribution H is chosen, and the integral paths of Γ project onto the integral paths of some local usual systems of second order ordinary differential equations on M.

5 4 Izu Vaisman It is important that a second order vector field Γ yields a Γ-canonical horizontal distribution H Γ. Namely (e.g., [7]), it turns out that (L Γ S) 2 = Id, where L denotes the Lie derivative. This shows that F def = L Γ S is an almost product structure with the associated projectors (2.13) V = 1 2 (Id + F ), H = 1 (Id F ). 2 which are such that imv = V, imv imh = {0}, and we put imh def = H Γ. There also exists a canonical almost complex structure defined by Γ on T M. def Namely, since S = S/ H is an isomorphism between H and V, we obtain a well defined almost complex structure J by asking that (2.14) J/ HΓ = S, J 2 = Id. We refer again to [7] for a survey on this structure. 3. The Direct Problem In order to show that the direct problem stated in the introduction is significant we shall give an example of a Poisson structure which has no second order Hamiltonian field. Let us take a Poisson structure with the Poisson bivector w on M itself. Then w has a natural lift w to T Mwhich has been encountered by several authors (see [11] and its references). The structure w is derived from the bracket of the 1-forms of M with respect to w in the following way: i) f, g C (M) put {f π, g π} w = 0; ii) f C (M) and α Λ 1 M, where α is seen as a fiberwise linear function on T M, put {α, f π} w = [(# w α)f] π, where # w : T M T M is such that β Λ 1 M, β(# w α) = w(α, β); iii) αβ Λ 1 M put [1], [4] {α, β} w = L #α β L #β α d(w(α, β)). That w is a Poisson structure follows from the fact that (T M, {α, β} w, # w ) is a Lie algebroid [1], [6], [11]. Let us mention in passing that w is an interesting structure. In local coordinates one has (3.1) w = w ij q i u j + 1 wij uk 2 q k u i u j, which shows that w is the complete lift of the bivector w in the sense of [12]. (The complete lift of a multivector field is the natural algebraic extension of the complete lift of a vector field X, which, in turn, is the infinitesimal generator of d(exp tx)). The structure w is nondegenerate (hence symplectic) iff w is such and, in this case, w is the pullback of the canonical symplectic structure of T M by the

6 Hamiltonian Vector Fields 5 mapping # 1 w. (A computation shows that w and the mentioned pullback yield the same Hamiltonian vector fields for f π, f C (M), and for the fiberwise linear functions α). The structure w is homogeneous in the sense of [2] since one gets w + L E w = 0 for E of (2.11). If the Poisson manifold (M, w) is integrable by the symplectic groupoid (V, σ) then (T M, w) is integrable by the symplectic groupoid (T V, σ), whose multiplication is the differential of the multiplication of V (see [1]). Etc. Now, if F C (T M), and if we use (3.1) to compute the Hamiltonian vector field X F = i(df ) w, we see that X F is a second order vector field iff ij F (3.2) w u i = uj. Hence, if we take the derivative of (3.2) with respect to u k, it follows that w must be nondegenerate and have a symmetric inverse matrix, which is absurd. On the other hand, the most important example of a second order Hamiltonian vector field is that defined by the energy of a Lagrangian dynamical system with a nondegenerate Lagrangian L C (T M). Let us recall it (see, for instance, [7] for details). The Lagrangian L has the associated 1-form (3.3) θ L = dl S = L u i dqi and the closed 2-form (3.4) ω L = dθ L = 1 2 ( 2 L q i u j ) 2 L u i q j dq i dq j + 2 L u i u j dui dq j (obtained as the pullbacks of the Liouville 1-form and the canonical symplectic form of T M by the Legendre transformation, respectively). The nondegeneracy of L means that ω L is a symplectic form. Furthermore, the energy of the system is (3.5) E L = E(L) L = u i L u i L, and the Hamiltonian vector field Γ L of E defined by (3.6) i(γ L )ω L = de L is a second order vector field. We shall say that the symplectic 2-forms (3.4) are Lagrangian symplectic forms, and we notice that such forms are V-Lagrangian (i.e., the fibers of T M are Lagrangian sub-manifolds). The solution of the direct problem for V-Lagrangian symplectic structures is given by Theorem 3.1. i) The V-Lagrangian symplectic forms of T M are given by the formula (3.7) Ω = π Θ + dζ,

7 6 Izu Vaisman where Θ is a closed 2-form of M, and ζ is a 1-form of T M which vanishes on V and has rank (dζ) = 2n. ii) Such a form Ω has second order Hamiltonian vector fields Γ iff SΩ = 0, and then, Ω is a Lagrangian symplectic form iff it is exact. Proof. i) Probably, formula (3.7) is folklore. In the present proof, we use a horizontal distribution H with the bases (2.2), (2.3). The fact that Ω is V-Lagrangian means (3.8) Ω = Ω 20 + Ω 11, where the terms have indicated bidegrees. Then, using (2.5), dω = 0 becomes (3.9) d Ω 11 = 0, d Ω 20 + d Ω 11=0, d Ω 20 + Ω 11 = 0. Since d is exterior differentiation along the fibers, and the latter are contractible, d Ω 11 = 0 implies the existence of an open covering M = α U α endowed with (1,0)- forms ζ α defined on π 1 (U α ) such that Ω 11 / π 1 (U α) = d ζ α. Furthermore, if {χ α } is a partition of unity on M subordinated to {U α }, we get (3.10) Ω 11 = d ζ, where ζ = α χ αζ α is a global (1,0)-form on T M. Hence, ζ/ V = 0, and also ζ = 0. Now, using (2.6), the last two conditions (3.9) become Hence, we must have d Ω 20 + d d ζ = d (Ω 20 d ζ) = 0, d Ω 20 + d ζ = d Ω 20 d 2 ζ d ζ = d (Ω 20 d ζ) = 0. (3.11) Ω 20 d ζ = π Θ, where Θ is a closed 2-form on M. Together, (3.10) and (3.11) give us (3.7). ii) It follows from the definition of S that (3.12) S(dq i ) = 0, S(ν i ) = dq i, and, if we put ζ = ζ i (q, u)dq i, we get (3.13) SΩ = Sd ζ = ζ i u h dqh dq i. Hence, SΩ = 0 iff the unique (0,1)-form ζ which is such that S ζ = ζ ( ζ = ζ i ν i ) is d -closed. Again, since the fibers are contractible, a d -closed (0,1)-form is d - exact, say ζ = d ϕ, ϕ C (T M).

8 Thus, ζ = Sd ϕ, and we see that SΩ = 0 iff (3.14) Ω = π Θ + ω ϕ, Hamiltonian Vector Fields 7 where ω ϕ is the Lagrangian symplectic form (3.4) of the (necessarily nondegenerate) Lagrangian ϕ. Now, let us notice that def Ψ Θ = i(γ)(π Θ) is the same (1,0)-form for all the second order vector fields Γ (use (2.12)). On the other hand, let Γ ϕ be the Lagrangian dynamical system of the Lagrangian ϕ i.e., i(γ ϕ )ω ϕ = de ϕ for some energy function E ϕ. Then, Γ = Γ ϕ + Z, where Z is a vertical vector field, and the condition i(γ)ω = exact becomes (3.15) Ψ Θ + i(z)d θ ϕ = df, where θ ϕ is defined by (3.3) and, because the left-hand side of (3.15) is a (1,0)-form, f is the lift of a function on M. Since the Lagrangian ϕ is nondegenerate, (3,15) is solvable with respect to Z for any f C (M), and the solution is a vertical vector field. Hence, there are second order vector fields Γ making i(γ)ω exact, and {Γ ϕ + Z/Z satisfies (3.15)} is the set of all the solutions. Or,otherwise, if Γ 0 is one solution, all the others will be Γ = Γ 0 + Y, where Y is the Ω-Hamiltonian field of some f π, f C (M). Conversely, if there is a second order vector field Γ such that i(γ)ω = exact = de for Ω of (3.7) (even i(γ)ω = closed i.e., L Γ Ω = 0, would be enough since the result is local), then the equality of the corresponding (0,1)-components is (3.16) u i ζ i u h = E u h. By taking the derivative with respect to u j, we see that ζ j u h = u j u h 2 ζ i ui u j u h 2 E is a symmetric matrix, and, by (3.13), we get SΩ = 0. (For the case Θ = 0, this is Proposition 2.29 of [7]). Now, let us examine the exactness condition for Ω of (3.7). Any exact 2-form Ξ = dξ of T M which has no (0,2)-component is also the differential of a (1,0)-form. Indeed, if we decompose ξ = ξ 10 + ξ 01, we must have d ξ 01 = 0 and, therefore, ξ 01 = d ϕ for some ϕ C (T M) (because the fibers are contractible). This implies Ξ = dξ 10 + dd ϕ = dξ 10 + d(d d )ϕ = d(ξ 10 d ϕ) as announced. In particular, if Ω is exact so is π Θ, and some equality π Θ = dµ 10 must hold. Since π Θ is of bidegree (2,0) this also gives d µ 10 = 0, and µ is projectable to M. Hence, Ω is exact iff Θ is exact. Finally, if SΩ = 0 and Θ is exact and if we put Θ = d(α h (γ)dq h ) in (3.14), we get Ω = ω L for L = ϕ + α h u h, and we are done.

9 8 Izu Vaisman Remark 3.2. Since a closed form is always locally exact, it follows that any V-Lagrangian symplectic form which is in the kernel of S (or, equivalently, which admits second order locally Hamiltonian vector fields) is locally Lagrangian symplectic. More exactly, it is Lagrangian symplectic on sets π 1 (U) where U is a convex neighborhood in M. This is Proposition 3.3 of [7] where, also the crucial argument of the contractibility of the fibers of V is used. Furthermore, it is easy to see that the second order global Hamiltonian vector fields Γ constructed for Ω of (3.14) in the proof of Theorem 3.1 are Lagrangian dynamical fields locally but, in general, not globally. (See the example at the end of Section 3.1 of [7]). We end this section by a few words on the case of a general Poisson structure P. In view of (2.7), if we use a horizontal distribution H, we can write (3.17) P = P 20 + P 11 + P 02, where (3.18) P 20 = 1 2 pij Q i Q j, P 11 = s ij Q i u j, P 02 = 1 2 rij u i u j (p ij = p ji, r ij = r ji ). P must satisfy the well known Poisson condition (3.19) [P, P ] = 0 where the bracket is that of Schouten-Nijenhuis (e.g., [5]). In view of (2.11), P has second order Hamiltonian fields iff F C (T M) such that (3.20) S(i(dF )P ) = E = u i u i. Since S is a derivation, (3.20) gives (3.21) i(sdf )P + i(df )SP = E, where SdF = θ F as given by (3.3). Here, if we use (3.17) and then separate the (1,0) and (0,1)-components, we see that the (1,0)-component vanishes identically, and we remain with the condition (3.22) i(d F )SP 20 + S(i(d F )P 11 ) = E. The point to be noticed is that the (0,2)-component of P does not enter in (3.22). Then, (3.22) simplifies if P 20 = 0 when, using (3.18), we remain with the partial differential system ij F (3.23) s u i = uj with the unknown function F. The Poisson structures with P 20 = 0 are those for which the mapping π : (T M, P ) (M, 0) is a Poisson morphism [10], and we shall call them zero-related Poisson structures of T M. For instance, the V-Lagrangian symplectic structures satisfy this condition. An important class of zero-related Poisson structures of T M will be described in the next section.

10 Hamiltonian Vector Fields 9 4. The Inverse Problem We begin by a simple example which shows that the solution of the inverse problem may not be unique and, also, provides a pseudo-energy function (see Introduction). On T R n = R 2n, take the Lagrangian (4.1) L = 1 2 (δ iju i u j + α ij q i q j ). where (δ ij ) is the unit matrix and (α ij ) is a constant symmetric matrix (a modified harmonic oscillator). The corresponding dynamical field is (4.2) Γ L = u i q i + α ijq j u i. Now, consider the bivector field (4.3) P = 1 ( 2 kih δ jh q i q j α jh u i ) u j, where (k ij ) is a nonsingular constant skew-symmetric matrix such that kα is again a skew-symmetric matrix. (Hence, we must assume that n is even). Since it has constant coefficients, P is a Poisson bivector, and we can see that Γ L is the P - Hamiltonian vector field of the function (4.4) F = κ ij q i u j. where κ ij k ih = δ h j. Notice that, if det(α jh) = 0, P is not a symplectic structure. Generally, in the inverse problem we start with a fixed second order vector field Γ and, as indicated in Section 2, we shall have a canonical horizontal distribution H Γ which is the one that we shall use hereafter. And, we shall look for Poisson bivectors P, decomposed as in (3.17), (3.18) with respect to H Γ and satisfying (3.19), such that Γ is a P -Hamiltonian vector field. A necessary condition for this to happen is (4.5) L Γ P = 0, and it is this condition which we exploit first. Lemma 4.1. i) The distribution H Γ of Γ of (2.12) has the local basis Q i of (2.2) where (4.6) τ j i = 1 Γ j 2 u i. ii) L Γ satisfies the local formulas (4.7) u i = Q i + τ j i u j, L Γ q i = Γh q i L Γ Q i = τ j i Q j (Γτ ji + τ hi jh Γj τ + q i L Γ ) u j, u j,

11 10 Izu Vaisman and it has a decomposition (4.8) L Γ = L 1 Γ + L Γ + L 1 Γ, where the components increase the bidegree of a multivector field by (-1,1), (0,0) and (1,-1), respectively. Proof. i) The vectors H(/q i ) where H is defined by (2.13) form a basis of H Γ. A straightforward computation gives ( ) H q i = q i + 1 Γ j 2 u i u j, which is exactly the announced result. ii) Formulas (4.7) are just the result of the corresponding local calculations, and (4.8) is obtained if we look at L Γ S for S = 1 Q p!q! σi1...ipj1...jq i1 Q ip u j1 u, jq if we remember that L Γ is a derivation, and use (4.7) Proposition 4.2. Let P be a bivector field, and let (3.17), (3.18) be its decomposition with respect to H Γ. Then, if L Γ P = 0, we have { (4.9) P = Id SL Γ SL 1 Γ + 1 } { 4 (SL Γ) 2 Q 20 + Id + 1 } 2 SL Γ Q 11, where Q 20, Q 11 are bivectors of the indicated bidegrees, and SQ 11 = 0. Proof. From (3.17) and (4.8), it follows that L Γ P = 0 is equivalent to (4.10) L ΓP 20 + L 1 ΓP 11 = 0, L ΓP 11 + L 1 Γ P 20 + L 1 ΓP 02 = 0, L 1 Γ P 11 + L ΓP 02 = 0. We shall also use the notation (3.18). which uses (4.7) yields Then, a straightforward computation (4.11) SL 1 ΓP 02 = 2P 02, whence, if we apply S to the second relation (4.10), we get (4.12) P 02 = 1 2 {SL ΓP 11 + SL 1 Γ P 20}. Furthermore, if we write P 11 = 1 2 (sij + s ji )Q i u j (sij s ji )Q i u j,

12 we obtain the unique decomposition Hamiltonian Vector Fields 11 (4.13) P 11 = Q 11 + Θ 11 which satisfies SQ 11 = 0. Then, computing again locally and using (4.7), we shall obtain (4.14) SL 1 ΓP 11 = 2Θ 11. Accordingly, if we apply S to the first condition (4.10), we deduce (4.15) Θ 11 = 1 2 SL ΓP 20. Finally, if we put Q 20 (4.9). Remarks 4.3. def = P 20, and insert (4.12) and (4.15) into (3.17) we obtain 1) In order to identify the bidegrees in (4.9) we shall notice that S is an operator which increases bidegrees by ( 1, 1). 2) In order to check whether a bivector (4.9) satisfies indeed L Γ P = 0 we must come back and check the relations (4.10). 3) For P to be zero-related, we must take Q 20 = 0 in (4.9). 4) If one is interested in the inverse problem for symplectic structures only, one can write down similar formulas for 2-forms Ω. Then, the independent term will be Ω 02. The continuation of the discussion of the general inverse problem will require stronger means of mathematical analysis. However, simple algebra can provide more information in particular situations, and, here, we describe how the method of Henneaux [3], [7] extends to the search of solutions of the inverse problem by zero-related Poisson structures of T M (see Section 3). The zero-related structures satisfy the linear algebraic condition P 20 = 0, which in view of (3.12) is equivalent to (4.16) P (Sα, Sβ) = 0, α, β Λ 1 (T M). This implies (4.17) Γ(P (Sα, Sβ)) = 0, which, modulo (4.16) and L Γ P = 0, means (4.18) P ((L Γ S)α, Sβ) + P (Sα, (L Γ S)β) = 0. In (4.18) we use the extension of F = L Γ S to a derivation of the tensor algebra of T M, similar to the extension of S. In fact, we shall define such an extension for every bundle morphism T (T M) T (T M). Now, (4.18) is again a linear equation for P, and it can be treated like (4.16).

13 12 Izu Vaisman Thus, modulo L Γ P = 0, the sequence of conditions (4.19) Γ (k) (P (Sα, Sβ)) = 0, α, β Λ 1 (T M) yields a sequence of linear equations for P. If too many independent equations result, P does not exist. But, if k 0 such that (4.19) for k = k is a linear consequence of (4.19) for k k 0, then the continuation of the sequence adds no independent conditions, and we have a chance of finding solutions. Afterwards we shall have to see which of these algebraic solutions are indeed solutions of the inverse problem for Γ. The method actually works in applications, particularly in small dimensions [3], [7]. Probably, it can also be transposed into a computer program which would provide solutions of the inverse problem, at least if M is an open subset of R n. An interesting class of zero-related Poisson structures of T M can be obtained as follows [11]. Take a Lie algebroid structure [6] of T M defined by a new Lie bracket of vector fields of M, say [X, Y ], and an anchor bundle morphism A : T M T M which satisfies the conditions (4.20) [X, fy ] = ((AX)f)Y + f[x, Y ], A[X, Y ] = [AX, AY ]. Then, T M gets a natural Poisson structure which can be pulled back to T M by any Legendre mapping of a nondegenerate Lagrangian. The result is a zero-related Poisson structure of T M. In particular, the Lagrangian symplectic structures ω L of Section 3 are obtained if [X, Y ] = [X, Y ], A = Id. This class of Poisson structures of T M is characterized by (4.21) {f π, g π} = 0, {θ L ( X), f π} = [(AX)f] π, {θ L ( X), θ L (Ỹ )} = θ L( [X, Y ] ), where f, g C (M), X, Y are vector fields on M, and denotes the complete lift recalled in Section 3; for X = ξ i (q)(/q i ), this lift is [12] (4.22) X = ξ i ξi + uh qi q h u i. In fact, we may just use (4.21) as a definition and check the properties of a Poisson bracket. Let us also add that the first relation (4.20) implies [9] (4.23) [X, Y ] = [AX, Y ] + [X, AY ] A[X, Y ] + Ψ(X, Y ), where Ψ is an adequate T M-valued 2-form on M. In particular, if A is a Nijenhuis tensor (e.g., [7]) i.e., [AX, AY ] A[X, AY ] A[AX, Y ] + A 2 [X, Y ] = 0, we may take Ψ = 0 and get a Nijenhuis algebroid structure. The Poisson structures (4.21) are called Lagrangian Poisson structures of T M [11].

14 Hamiltonian Vector Fields 13 Proposition 4.4. If P is a Lagrangian Poisson structure, and L Γ P = 0 for some second order vector field Γ, then SP = 0. Proof. We give Γ by (2.12) and P by (3.17), (3.18). Then, the first relation (4.21) gives P 20 = 0 and, also, since L Γ P = 0, {Γ(f π), g π} + {f π, Γ(g π)} = 0. For f = q i, g = q j, the last relation becomes {u i, q j } + {q i, u j } = P (du i, dq j ) + P (dq i, du j ) = s ji + s ij = 0, where s ij are those of (3.18). Hence, in the notation of (4.13), Θ 11 = 0. Therefore, SP = SP 11 = 0. As a consequence of Proposition 4.4, the Hennaux sequence of linear equations (4.19) has to be supplemented by the sequence (4.24) (L (k) Γ S)(P ) = 0 (modulo L ΓP = 0), which is also a sequence of linear equations in P, and can be used just like (4.19). For k = 0 we get (4.25) (SP )(α, β) = P (Sα, β) + P (α, Sβ) = 0, for k = 1, we have, modulo L Γ P = 0, (4.26) P ((L Γ S)α, β) + P (α, (L Γ S)β) = 0, where, in both cases, α, β are arbitrary 1-forms on T M. Notice that the sequence (4.24) is also significant for the original case of [3], where we have to use it for a Lagrangian symplectic 2-form ω L instead of P. In the case of non zero-related Poisson structures P, other conditions of a similar algebraic nature may be treated in the same way. For instance, we might look for the case P 11 = 0, which has an invariant meaning because the horizontal distribution H Γ has one. P 11 = 0 means (4.27) P (Sα, Hβ) = 0, α, β 1 (T M), and we have the corresponding sequence of linear equations in P (4.28) Γ (k) (P (Sα, Hβ)) = 0 (modulo L Γ P = 0). For k = 1, this is (4.29) P ((L Γ S)α, Hβ) + P (Sα, (L Γ H)β) = 0, etc. Other interesting properties that we might consider are F (P ) = 0, J(P ) = 0, etc. (The F and J associated to Γ were defined in Section 2).

15 14 Izu Vaisman 5. Final Remarks We end by indicating some further questions. First, it would also be interesting to look for Nijenhuis tensors N on T M such that L Γ N = 0 for a given second order vector field Γ, and this for the following reason. Should there exist a pair (P, N) which is a Poisson-Nijenhuis structure of T M, and such that L Γ P = 0, L Γ N = 0, it would follow that L Γ P (k) = 0 for all the Poisson structures of the hierarchy of Poisson-Nijenhuis structures of (P, N) [4]. Second, in [11], A. Weinstein extends the Lagrangian dynamics to arbitrary Lie algebroids π : A M, and, in the process, he defines second order vector fields on A. In the case A = T M with a new Lie bracket [ ] and an anchor A : T M T M, already encountered in Section 4, these new second order vector fields represent again a system of differential equations namely, one of the following form [11] (5.1) dq i dt Ai j(q) dqj dt = 0, d 2 q i dt 2 = Γi ( q, dq ). dt These are the second order systems whose solutions belong to the leaves of the generalized foliation im A. The vector field of T M whose orbits project to the solutions of (5.1) is (5.2) Ξ = (A i ju j ) q i + Γi u i. Now, it makes sense to say that (5.1) is a Hamiltonian system if Ξ is a Hamiltonian vector field with respect to some Poisson structure of T M, and, we should study the direct and the inverse problems for Ξ of (5.2). The Lagrangian dynamical fields of (T M, [ ], A) in the sense of [11] provide examples of differential equations of type (5.1) which are Hamiltonian in this general sense. A third natural question is that of studying the original direct and inverse problems formulated in Section 1 for the particular class of the quadratic second order fields or sprays Γ on T M. The vector field Γ of (2.12) is called quadratic or a spray if it is such that (5.3) [E, Γ] = Γ (E = u i u i ). This condition means that the local components Γ i of (2.12) are homogeneous of the second degree with respect to (u i ). In particular, we may take (5.4) Γ i = α i jku j u k (α i jk = α i kj), which turns out to be an invariant condition, where αjk i are the coefficients of a linear connection on M, and the integral paths of Γ project to the geodesics of this linear connection. It is well known that in the case of a Riemannian connection the geodesics are given by a Lagrangian dynamical field of T M. Another natural particularization is that of M = R n, T M = R 2n and P = a linear Poisson structure on R 2n. (We may also take M to be an open subset of R n

16 Hamiltonian Vector Fields 15 only). It is well known that, then, P is the Lie-Poisson structure of an identification of R 2n with the dual space G of a Lie algebra G [10]. For instance, it is easy to see that if P = P 1 P 2, where P 1 and P 2 are the Lie-Poisson structures of G1, G2 and G 1, G 2 are n-dimensional Lie algebras, then there are no P -Hamiltonian second order vector fields on R 2n. Indeed, in this case we have (5.5) P = 1 2 i,j,k c k ijq k q i q j + 1 c ij k u k 2 u i u j, where c k ij and c ij k are the structural constants of G 1 and G 2, respectively. Now, i(df )P (F C (R 2n )) is a second order vector field iff i,j,k (5.6) i,k c k ijq k F q i = uj. and this condition has the consequence (5.7) ( i k 2 F c k ijq k ) u h q i = δj h. Therefore, ( k ck ij qk ) must be a nonsingular matrix i.e., the Lie-Poisson structure of G 1 must be symplectic. Of course, this is impossible for the entire space G 1. But, even if there exists an open domain D of G 1 where the Lie-Poisson structure P 1 is symplectic (for instance, such a domain exists if G is the Lie algebra of the 1- dimensional affine group), there are no functions F C (T D) which satisfy (5.6). Indeed, let θ be the symplectic form of P 1 on D, with the local components θ sj defined by (5.8) θ sj (q)( k c k ijq k ) = δ i s. Then (5.6) yields (5.9) F q s = θ sju j, with the differential consequence (5.10) θ sj q k = θ kj q s. Since θ is a closed form, we also have (5.11) θ sj q k + θ jk q s + θ ks q j = 0,

17 16 Izu Vaisman and, because θ is skew-symmetric, (5.10) and (5.11) imply θ ks = const. (locally). Therefore, also k ck ij qk = const., and the only possible case is c k ij = 0, when (5.6) never holds. On the other hand, there are examples where second order Hamiltonian vector fields exist. To get one, we notice first that a Lie-Poisson structure on T R n G is zero-related iff G has an n-dimensional abelian subalgebra (provided that we choose the isomorphism correctly). Now, let us assume that G has an n-dimensional abelian ideal A endowed with a monomorphism of vector spaces ϕ : A G such that: i) A ϕ(a) = {0}; ii) X, Y A one has (5.12) [X, ϕ(y )] = [Y, ϕ(x)]. Also, assume that (5.13) D = {γ A / γ G with γ = γ/ A and (X A, X 0) one has dim(coad X γ)(ϕ(a)) = n}. is a nonvoid open subset of A. Then, we can see that the direct problem for the restriction of the Lie-Poisson structure of G to T D has solutions. Indeed, let X i (i = 1,..., n) be a basis of A, and put X i = ϕ(x i ), i = i + n. Then, (X i, X i ) is a basis of G such that (5.14) [X i, X j ] = 0, [X i, X j ] = b k ijx k, [X i, X j ] = c k ijx k + c k ij X k, where b, c are the structural constants of G, and we also have (5.15) b k ij = b k ji, because of (5.12). (If c k ij = 0 ϕ(a) is also a Lie subalgebra of G, and G is the semidirect product ϕ(a) ρ A, where ρ is defined by ρ ϕ(x) (Y ) = [ϕ(x), Y ], X, Y A). Accordingly the Lie-Poisson bivector of G is (5.16) P = b k ijq k q i u j k,i,j + 1 (c k 2 ijq k + c k ij u k ) u i u j. k,i,j Now, the functions F (q, u) which have a second order P -Hamiltonian vector field are characterized by (3.23), which, in our case, is (5.17) i,k and it has the differential consequence (5.18) ( i,j k (b k ijq k ) F u i = uj, 2 F b k ijq k ) u i u s = δj s.

18 Hamiltonian Vector Fields 17 Thus F cannot exist unless ( k bk ij qk ) is a nonsingular matrix. But, this is just the coordinate expression of the property (5.13) which defines D. Then, if we define the (symmetric) matrix θ sj (q) on D by (5.19) θ sj ( b k ijq k ) = δs, i j k the integration of (5.17) gives (5.20) F = 1 2 θ sj(q)u s u j + ψ(q), and the P -Hamiltonian vector fields of the functions (5.20) are of the second order. In order to have a concrete example, take G = R n ρ R n = a semidirect product of abelian Lie algebras, where ρ : R n End (R n ) = Derivations (R n ) is a Lie algebra homomorphism i.e., (5.21) 0 = ρ [X,Y ] Z = ρ X ρ Y Z ρ Y ρ X Z (X, Y, Z R n ). Then, we may take the second factor of G as A, and, with respect to the natural basis of R n, the Lie-Poisson bivector of G will be (5.22) P = (ρ s jkq s ) q k u j. j,k,s This bivector is of the form (5.16) iff ρ s jk = ρs kj, and the meaning of this condition is that the product defined on R n by X Y = ρ X Y is commutative. Now, (5.21) means that this product is also associative. Therefore, for every associative and commutative n-dimensional R-algebra, there is an associated Poisson bivector (5.22) in the class (5.16). For instance, for n = 2, and for the algebra of the complex numbers, we obtain (5.23) P = q 1 ( q 1 q 2 ( q 1 u 1 + u 2 + q 2 q 2 ) u ) 2. u 1 The corresponding domain D of (5.13) is R 2 \{0}, and P defines a symplectic structure of (R 2 \{0}) R 2 such that the P -Hamiltonian vector fields of the functions (5.20), which in our case are 1 (5.24) F = 2[(q 1 ) 2 + (q 2 ) 2 ] [q1 ((u 1 ) 2 (u 2 ) 2 ) + 2q 2 u 1 u 2 ] + ψ (q), are of the second order. Of course, if the reader so wishes, he can just check the example (5.23), (5.24) by a straightforward computation, and skip the general construction which led to this example. The case of a general Lie-Poisson structure of R 2n is still to be studied. Finally, we mention the equivariant version of the direct and the inverse problems. Namely, we shall say that a system of the second order on M has a symmetry group G (a Lie group) if the corresponding vector field Γ on T M is invariant by the group T G. If this happens we should also ask for T G-invariant Poisson structures in our problems. The general questions mentioned in this section are open.

19 18 Izu Vaisman References 1. A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, Dept. Math.Univ. Lyon 2/A (1987), P. Dazord, A. Lichnerowicz and Ch.-M. Marle, Structures locale des variétés de Jacobi, J. Math. Pures et Appl. 70 (1991), M. Henneaux, On the inverse problem of the calculus of variations, J. Physics A,15 (1992), L Y. Rosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis Structures, Ann. Inst. H. Poincaré série A (Physique Théorique) 53 (1990), A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geometry 12 (1977), K. MacKenzie, Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Notes series 124, Cambridge University Press, Cambridge, G.B., G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Physics Reports 188 (1990), I. Vaisman, Cohomology and differential forms, M. Dekker Inc., New York, I. Vaisman, The Poisson-Nijenhuis manifolds revisited, Preprint (1993). 10. A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geometry 18 (1983), A. Weinstein, Lagrangian mechanics and groupoids, Preprint, Univ. of Calif., Berkeley (1992). 12. K. Yano and S. Ishihara, Tangent and cotangent bundles, M. Dekker Inc., New York, I. Vaisman, Department of Mathematics, University of Haifa, Haifa 31905, Israel address: RSMA753@UVM.HAFA.AC.IL