Integrability and Nijenhuis Tensors - Gerdjikov

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    Integrability and Nijenhuis Tensors

    In this chapter, we link the theory of the Nijenhuis tensors and the questionsof integrability, trying to give some general view on integrability based on thetheory of Nijenhuis tensors and their fundamental fields. In the example ofseveral finite and infinite dimensional dynamical systems, we show how theseideas work. Finally, we show the natural way the Nijenhuis tensors arise whenwe have compatible Poisson tensors. The development of these ideas leads tothe notion of Poisson-Nijenhuis manifold (P-N manifold), which we introducehere. This notion is crucial for our treatment of the integrable Hamiltonian

    systems.

    14.1 Integrability Criteria and Nijenhuis Tensors

    Nowadays the inverse scattering transform method (IST) is universally recog-nized as one of the important techniques for integration of partial differentialevolution equations [1], provided the corresponding equation can be cast intoLax form. Its applications are numerous, and without exaggeration it is oneof the significant mathematical discoveries of the 20-th century. However, inspite of its success, a compact a priori criterion of complete integrability is, todate, not available. On the other hand, if we consider the soliton equations asdynamical systems on infinite-dimensional manifolds, the IST can be regardedas transformation so to say from generic coordinates (potentials) to action-angle variables [2], and this gives hopes that such criterion could be found interms of the original Lax pairs. These hopes are strengthened by the naturalway, in which, using the Lax representation, one calculates an important op-erator, (the operator, squared eigenfunctions operator, recursion operator),as has been shown in the first part of the present book. The point is that,

    as we shall see, the adjoint of this operator fits in the geometric picture as amixed tensor field N on the phase manifold M and plays on it the same roleas it does in the finite dimensional case. Such a tensor field N has usually thefollowing properties:

    Gerdjikov, V.S. et al.: Integrability and Nijenhuis Tensors. Lect. Notes Phys. 748,

    473513 (2008)

    DOI 10.1007/978-3-540-77054-1 14 c Springer-Verlag Berlin Heidelberg 2008

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    474 14 Integrability and Nijenhuis Tensors

    1. It is invariant under the dynamics (this is also known as N being a strongsymmetry, or in other words, if is the dynamical system (vector field),we have L(N) = 0.

    2. Its Nijenhuis torsion vanishes (also known as hereditary property).

    3. It has a doubly degenerate continuous spectrum (in a certain well-definedmanner), and when dealing with generic potentials it has a finite num-ber of discrete eigenvalues. The set of the discrete eigenvalues definesthe so-called soliton sector. The eigenspaces corresponding to the discreteeigenvalues are invariant with respect to the evolution and generically aretwo-dimensional. We have two possibilities here, which have been investi-gated up to now:

    (a). There exist exactly two eigenvectors in each such space. We shallcall this case a diagonalizable case.

    (b). Each eigenspace in the soliton sector is spanned by a true eigenvec-tor and a generalized one; this corresponds to a matrix with a single2 2 Jordan block. We call this case nondiagonalizable.

    In both cases, the set of the eigenvectors (true and generalized) in thesoliton sector, together with eigenvectors for the continuous spectrum,forms a complete set.

    Using the first two properties there has been constructed a geometric inte-grability scheme, see [3, 4], and in the recent years much attention has beengiven to the diagonalizable case, because when we have it, we are able to

    construct sequences of conservation laws, Abelian algebras of symmetries andhierarchies of integrable nonlinear equations [5, 6, 7, 8].

    However, in spite of the fact that the diagonalizable case occurs more fre-quently, the nondiagonalizable case can also occur. That is why we wouldlike to propound an a priori separability criterion, which can include this newspectral option. In what follows, we deal with the nondiagonalizable case; fromthe very construction, it will be clear how one can deal with the diagonalizableone. We shall see also that, as far as soliton dynamics is concerned, integra-bility can be proved without further hypotheses. For background-radiation

    dynamics (the part described by the continuous spectrum), it is still unclearhow to formulate a priori integrability criterion. The considerations we givebelow probably can be formulated directly in the terms of the correspondingLax representations (considered in terms of the bundles for which the phasemanifold is the base) [9], provided these representations could be put into theframes of the above geometric picture, but in our opinion up to now conclusiveresults in that direction have not been obtained.

    Suppose that on the manifold M one has a vector field , and Nijenhuistensor field N (that is a mixed tensor field with vanishing Nijenhuis bracket)which is invariant for the flow of , that is LN = 0. Suppose that N hasthe properties listed in the above. We propose the following.

    Integrability criterion. The dynamics defined by the vector field com-pletely separates into 1-degree of freedom dynamics. The components associated

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    14.1 Integrability Criteria and Nijenhuis Tensors 475

    to the degrees of freedom corresponding to eigenvalues which are notstationary (depend on the time) are integrable and Hamiltonian [10].

    Below, we shall give a sketch how the above criterion can be obtained,

    assuming that the calculations we perform can be justified.Denote by i the generic discrete eigenvalue of N, and just in order to fixthe ideas, assume that the continuous spectrum of N fills the real semiaxisR+ and does not overlap with the discrete spectrum. Then the vanishing ofthe Nijenhuis torsion RN associated with N means that for all 1(M)and X, Y T(M)

    RN(,Y,X) = , [(LNXN) N(LXN)]Y = 0 . (14.1)(Here

    ., .

    is the natural pairing between the 1-forms and vector fields).

    According to our assumptions there exists a basis

    {ei, i, f1,(k), f2,(k); i = 1, 2, . . . , n, k R+}ofT(M), such that

    N ei = iei

    N i = ii + ei; i = l, 2, . . . , n

    N fl,(k) = kfl,(k); l = 1, 2, and k

    R+ . (14.2)

    Let us now introduce the corresponding dual basis,

    {ti, i, 1,(k), 2,(k); i = 1, 2, . . . , n, k R+} (14.3)of 1(M), that is a basis, for which

    ti, ej = i, j = ijl,(k), fp,(h) = l,(k)p,(h)t

    i

    , j = i

    , ej = ti

    , fl,(k) = 0i, fl,(k) = l,(k), ei = l,(k), i = 0 , (14.4)

    where i, j = 1, 2, . . . , n, and l,(k)p,(h) = lp(kh), where (kh) is the Dirac

    function. The relations (14.4) written in terms of the above 1-forms read

    Nti = iti + i

    Ni = ii; i = l, 2, . . . , n

    Nl,(k) = kl,(k); l = 1, 2, and k

    R+ , (14.5)

    where N denotes the formal adjoint ofN. As we shall see, no more ingredientsare needed to prove the separability into one degree of freedom dynamics, and(except for the assumption that the is are not stationary at any point) the

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    476 14 Integrability and Nijenhuis Tensors

    integrability of the discrete part of it can also be proved. Our analysis startswith the observation that the condition (14.1) can be cast into the followingform:

    Lei

    j

    = 0, Lfl,(k)

    i

    = 0, (

    i

    j

    )Li

    j

    = 0(N i)(N i)[ei, ej ] = 0, (N i)(N j)2[ei, j ] = 0(N i)2(N j)2[i, j ] = 0, (N k)(N h)[fl,(k), fp,(h)] = 0(N i)(N k)[ei, fl,(k)] = 0, (N i)(N k)[i, fl,(k)] = 0 . (14.6)

    Then it is easily seen that the (14.6) are equivalent to

    i di = i ti dti = 1,(k) 2,(k) l,(k) = 0 , (14.7)

    this implying, by the Frobenius theorem, that without loss of generality, thes, ts and s can be considered to be closed forms, or as it is also said , thebasis

    {ei, i, f1,(k), f2,(k); i = 1, 2, . . . , n, k R+},can be chosen to be a holonomic frame.

    On the other hand, from the first line in (14.6), we get that di = (Lii)i,

    this implyingNdi = idi . (14.8)

    In particular, this means that the s can be chosen to be equal to the di

    sif, as we assumed, we have di = 0. Furthermore, the fact that the frame isholonomic implies that the set of functions 1, 2, . . . , n can be completedto form a local coordinate system

    (1, 2, . . . , n, 1, 2, . . . , n, 1,k, 2,k; k R+)

    (some of the variables may be periodic) in such a way, that

    ei =

    i, i =

    i, fl,(k) =

    l,(k). (14.9)

    Then the tensor N can be written into the following canonical form

    N =i

    i

    i di +

    i di +

    i di

    +

    2l=1

    0

    dk k

    k (k) . (14.10)

    Now, it can be checked that the condition LN = 0 we have on N and isequivalent to the following system of equations:

    di, = 0, j

    di, = 0, l,(k)

    di, = 0

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    14.2 Recursion Operators in Dissipative Dynamics 477

    (i j) j

    di, = 0, i

    l,(k), = 0

    il,(k), = 0, (k h)

    l,(h)p,(k), = 0 . (14.11)

    From the above equations separability and integrability follow. More specifi-cally, the first equation in the first line of (14.11) means the vanishing of the-components of , the second and the third equation in the first linemean the independence of the -components on the s, and on the con-tinuous coordinates. In the second line, the first equation means that eachi-component depends only on the corresponding i. The last set of equationsshows that the continuous components cannot depend on the discrete variablesand that each continuous component can only be a function of the continuousvariables with the same continuous index. The most general form of is then

    =

    ni=1

    i(i)

    i+

    2=1

    0

    dk(k)

    1(k), 2(k)

    (k). (14.12)

    The dynamic equations decouple in the following second-order systems for thecontinuous degrees of freedom (background radiation dynamics):

    1,(k) = 1,k(1,(k), 2,(k))

    2,(k) = 2,k(1,(k), 2,(k)) , (14.13)

    and the following (trivially integrable) equations:

    i = i(i), i = 0 , (14.14)

    for the discrete part (the soliton dynamics). Consequently, the discrete partof the dynamics (the soliton dynamics) is Hamiltonian with respect to thefamily of symplectic forms

    0 = i

    fi(i)di

    di , (14.15)

    where f is a function that does not vanish on the points of the discrete spec-trum.

    Remark 14.1. is not supposed to be a Hamiltonian system. Its admissibleHamiltonian structures are generated by the hypothesis that the eigenspacesof N are bidimensional and the requirements di = 0.

    14.2 Recursion Operators in Dissipative Dynamics

    We have seen that a nonlinear evolution equation ut = [u], defined by thevector field [u], is integrable if there exists a mixed tensor field N on M,satisfying the conditions described at the beginning of the preceding section

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    and that this even imply the existence of symplectic forms (at least on thesoliton sector) with respect to which the dynamics is Hamiltonian. In thatcase, the integrability is of the Liouville-Arnold type. On the other hand, wementioned also that there are many interesting cases, in which the dynamics is

    not an integrable Hamiltonian one and in which suitable generalization of theabove geometric scheme could still be useful. We are going to present one suchexample, in which an invariant mixed tensor field is used in order to analyzedissipative dynamics. It seems that one needs only to remove that part of theassumptions on N, which lead to the existence of constants of motion. Thuswe shall assume that N has zero Nijenhuis bracket and is invariant under but is not real-diagonalizable, that is, the eigenvalues are complex. A goodexample illustrating this situation is provided by the Burgers equation

    ut = 2uux + uxx . (14.16)

    As is well known, this equation is the simplest one in which nonlinearity anddiffusion effects compensate in such a way that we have no undesirable effectson the evolution.

    14.2.1 The Burgers Equation Hierarchy

    It is a classical result, (see [11, 12] or [13]), that the Burgers equation linearizesthrough the so-called Cole-Hopf transformation (map)

    u = h[v] =vxv

    , (14.17)

    that is, if u is as in the above, and if v satisfies the Heat Equation

    vt = vxx , (14.18)

    then, u satisfies the Burgers equation.It can be shown, (see [14]), that the Burgers Equation is a member of a

    hierarchy of nonlinear evolution equations which linearize, through the sametransformation (14.17) and which are reduced to the equations of the followingtype

    vt = Dnv; n = 1, 2, . . . . (14.19)

    where D stands for the x-derivative operator. The elements of this hierarchyhave different properties. The even elements of (14.19) are equations withdissipative dynamics. The odd elements are equations with integrable Hamil-tonian systems with respect to the following symplectic form:

    [1v, 2v] =+

    1v(x)(D12v)(x)dx , (14.20)

    where

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    14.2 Recursion Operators in Dissipative Dynamics 479

    (D1f)(x) =

    x

    f(y)dy , (14.21)

    and the corresponding Hamiltonian functions (functionals) are given by

    Hp[v] = 12

    +

    (Dpv)2dx . (14.22)

    In order that (14.20), (14.22) make sense, some assumptions on the functionalspace M must be made, for example, to assume that M consists of Schwartz-type functions on the line. Then one easily checks that

    N[v] = D (14.23)

    is a Nijenhuis -invariant tensor for the heat equation hierarchy. In the ge-ometric approach, (14.17) plays the role of a map from one manifold intoanother, and thus a Nijenhuis -invariant tensor operator for the Burgers hi-erarchy is readily obtained from N[v] (see [14, 15]). In other words, requiringN and N to be v[u]-related

    N[u] = (v

    u)1N[v](

    v

    u) , (14.24)

    easily yieldsN[u] = D + DuD1 . (14.25)

    The Burgers hierarchy is obtained by repeated applications of the operator(14.25) to the field 0 = ux, that is

    k = Nk0 . (14.26)

    The first fields of the hierarchy are

    0 = ux

    1 = 2uux

    + uxx

    2 = (3u3 + 3uux + uxx)x

    ....................................... (14.27)

    From a geometric point of view this hierarchy is v[u]-related to the linear one,and roughly speaking (the manifolds for the Burgers hierarchy and the linearhierarchy are not diffeomorphic), one can translate what can be said for(14.19) to the Burgers hierarchy. In this way, one obtains that (14.26) splitsinto the following two subhierarchies, which we call the Dissipative and the

    Hamiltonian hierarchy: Dissipative hierarchyN 0, N

    30, . . . , N 2n+10, . . . (14.28)

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    480 14 Integrability and Nijenhuis Tensors

    Hamiltonian hierarchy0, N

    20, . . . , N 2n0, . . . . (14.29)

    The equations from these hierarchies are sequences of dissipative and Hamilto-

    nian vector fields, respectively. The above situation can be better understoodexamining the spectral properties of N. Its block diagonal form is

    N =

    0

    k

    e(k) ,(k) e(k) k

    dk , (14.30)

    where

    e(k)

    [u] =

    dx(

    u cos kx

    k sin kx)

    expx

    udy u(x)

    (14.31)

    e(k)[u] =

    dx(u sin kx + k cos kx) expx

    udy

    u(x)(14.32)

    form a basis in the generic invariant subspace of N, and {(k)(k)} is thecorresponding dual basis:

    N e(k) = ke(k), N e(k) = ke(k)

    (k)

    , e

    (h) = (k)

    , e(h) = (h k)(k), e(h) = (k), e(h) = 0 . (14.33)The conditions :

    [e(k), e(h)] = [e(k), e

    (h)] = [e

    (k), e(h)] = 0 , (14.34)

    imply that the frame is holonomic, that is, ensure (at least locally) the exis-tence of coordinates (q(k), p(k)), such that:

    e(k) = q(k)

    , e(k) = p(k). (14.35)

    The operator N can be restricted to the two-dimensional integral manifoldspanned by {e(k), e(k)}, then it reads:

    J(k) (k)

    (k) J(k) (14.36)

    where

    J(k) = 12

    q(k)2 +p(k)2

    , (k) = tan1( q(k)

    p(k)) (14.37)

    are action-angle type variables. Then, one can see that operator N moves adissipative integrable field of the type

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    14.2 Recursion Operators in Dissipative Dynamics 481

    X(k)D = J

    (k)

    J(k), (14.38)

    into a Hamiltonian one

    X(k)H = J

    (k)

    (k

    )(14.39)

    and vice versa.The above alternating character of N is responsible for the splitting of(14.26) into two subhierarchies. Furthermore, we observe that

    The operator N has two-dimensional invariant spaces, but is not real di-agonalizable.

    The operator N2, generating the Hamiltonian subhierarchy, is diagonaliz-able, with doubly degenerate constant eigenvalues.

    For none of these subhierarchies the integrability criterion we have pro-pounded holds. However, we note that the projections of dissipative fieldsdefine finite degree of freedom dynamics on the two-dimensional invariantspaces, while for the Hamiltonian fields, the existence of a functional J(k)[u](integrals of motion) ensures the integrability of the corresponding projec-tion. It is worthy of remark also that the functionals J(k)[u] play the role ofa Liapunov functional for the projection of the dissipative dynamics on thetwo-dimensional invariant sub-manifolds, ensuring the asymptotic stability ofthe solutions for which J(k)[u] = 0.

    Let us consider more closely the Hamiltonian subhierarchy (14.29). First ofall, some care is needed for the appropriate choice of the functional space Mon which the dynamics is defined. It is natural to take M to be the functionalspace, whose elements u tend to constant as x , as it is the space onwhich lies the typical solitary wave of Burgers hierarchy. However, with sucha choice, it would not be possible to introduce a Hamiltonian structure on M.Indeed, if we go back to the linear hierarchy of the Heat Equation, we shallsee that the functions v (see the transformation (14.17)) behave like exp(kx)as x . But then the Hamiltonian which we had written in (14.22) isnot defined. Thus, it is natural to restrict

    Min such a way that both the

    symplectic structures and Hamiltonian function (14.22) make sense. This canbe accomplished by considering the phase space to be the space of functionsv(x) that tend to some constants fast enough as x + and x (andthen the functions u(x) vanish as x ). Also, we can circumvent thedifficulties related to the ambiguities inverting the differentiation operatorpassing to the corresponding space of equivalence classes, requiring that ineach class the integral on the whole axis of v(x) has some fixed value.

    The above construction ensures the existence of a symplectic form suchthat the subhierarchy, which we called Hamiltonian, is indeed a hierarchy

    of Hamiltonian vector fields. Note also the interesting fact that despite thefact that the eigenvalues of the operator N2 are constant, N2 generates asequence of integrals of motion. The example we have given and its analysisshows the importance of the hypothesis on the spectrum of the invariant mixed

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    482 14 Integrability and Nijenhuis Tensors

    tensor field N for the properties of the corresponding dynamical systems. Wehave seen that if we drop the hypothesis that N is diagonalizable, we canconsider also dissipative dynamics within the geometric scheme. However, thespectral properties depend strongly on the particular choice of N, and their

    investigation is not an easy matter. For example, the first part of this book isalmost entirely dedicated to this topic for the ZS. So, we shall limit ourselvesto the study of the geometric constructions permitting to obtain Nijenhuistensors N. From the next chapter onwards the book is dedicated to this.

    14.3 Noncommutative Integrability Criteria

    Let us consider again the finite dimensional case, that is, we assume that we

    have a symplectic manifold (M, ) of dimension 2n and a Hamiltonian vectorfield on it. We shall discuss now how the usual scheme of integrabilitycan be generalized. Such generalization is necessary, because if the number ofindependent first integrals of is greater than n, they cannot be anymore ininvolution. Indeed, for such a set of functions their Hamiltonian vector fieldsmust be independent and are orthogonal with respect to , that is, at eachpoint they span an isotropic space. We know, however, that the dimension ofsuch space cannot exceed n. From the other side, the Poisson bracket of twofirst integrals ia a first integral too, so it is natural to assume that possessesa set of first integrals

    {fa}1am D

    (M

    ) which close a (non-commutative)Lie algebra A over R with respect the Poisson bracket. We shall say that twoelements f, g A Poisson-commute if their Poisson bracket is zero.

    Let us recall now some notions we shall use. If g is a finite dimensionalLie algebra and g is a covector, then the annihilator Ann () of is thespace

    {X g : ad X = 0} . (14.40)As readily seen, Ann () is a subalgebra. Next, we define the index ind (g) ofa finite dimensional Lie algebra g.

    ind(g) = ming

    dim Ann() . (14.41)

    It can be shown that when the algebra is semisimple, the index coincides withthe rank of the algebra, that is, with the dimension of the Cartan subalgebra.

    Let us denote by r the index ind (A) of our algebra of first integrals. Now weare ready to state the noncommutative generalization of the Liouville-Arnoldtheorem; see [16, 17, 18]:

    Theorem 14.2. Suppose that Hamiltonian vector field on a symplectic

    manifoldM2n, possesses f1, f2, . . . , f k functionally independent first in-

    tegrals, which span a finite dimensional real Lie algebraA of dimension k. Inaddition, let

    dim(A) + ind(A) = k + r = dimM = 2n . (14.42)

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    14.3 Noncommutative Integrability Criteria 483

    Then the submanifolds defined by the conditions fi = const , i = 1, 2, . . . , kare invariant manifolds for . In in some neighborhood of any such manifoldone can define canonical coordinates (,,p,q) such that

    1. The s are coordinates on the invariant manifolds.

    2. The Hamiltonian equations corresponding to take the form

    i = 0, i = i(), p = 0, q

    = 0

    i = 1, 2, . . . , r; = r + 1, r + 2, . . . , n . (14.43)

    As in the commutative case, if the invariant manifolds are compact and con-nected, one can prove that they are tori, and the s can be chosen to be anglevariables. The canonical coordinates are called, in this case, generalized ac-tion angle variables.

    The Liouville-Arnold theorem can be recovered from the above formulation.Indeed, when A is Abelian, its index coincides with its dimension and thenthe condition (14.42) shows that dim (A) = n. However, it is interesting thatif the assumption (14.42) holds, and if the algebra satisfies the so-called FIcondition, (for more details see [16, 18]), on the symplectic manifold M thereexist n = 12 dim M independent first integrals of , which are in involution,that is, actually we have the situation as in the Liouville-Arnold theorem.

    The proofs of the above results can be found in [18, 19, 20, 21]; here weshall limit ourselves with a brief sketch of the ideas of the proof of the Theorem14.2.

    Let us discuss first what happens in the Liouville-Arnold case, when A iscommutative of dimension n. Then the level surfaces of the first integrals fidefine an invariant Lagrangian foliation F1 of M, that is a disjoint union ofintegral leaves, each of which is a Lagrangian manifold; see Definitions (12.17),(12.18). The Hamiltonian vector fields Xi associated with the functions fi arecommuting vector fields, tangent to the leaves, and at each point of a given leafform a basis of its tangent space. They can be used to define local coordinatesi on the leaves. The fields Xi commute with the Hamiltonian vector field

    , which is also tangent to the leaves. Consequently, in a neighborhood of apoint p M, the field can be written into the form = i i (f) Xi wherethe set (, f) are canonical coordinates, which means that the Hamiltonianequations for take the following form:

    i = i, fi = 0 . (14.44)

    In the noncommutative case, we have 2n r first integrals fa. The equationsfa = const still define a foliation F1, invariant with respect to the flow of ,but the leaves now have dimension r n, and the Hamiltonian vector fieldsXa, associated with the first integrals fa are not necessarily tangent to theleaves of this foliation. (The fields Xa of course close an algebra, isomorphicto A).

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    484 14 Integrability and Nijenhuis Tensors

    However, the condition dim (A) + ind (A) = dim M ensures that for eachleaf L, there exists a subalgebra of AL A, whose associated vector fieldscommute with all Xa on L. The Hamiltonian vector fields Xi correspondingto some basis ofAL, yield a basis of tangent vector fields for L. We have that

    Xi, XaL = 0 and therefore

    Xi, Xj

    L = 0 ifXi, Xj AL so that eachleaf will be an isotropic manifold. The foliation F1 is then called isotropic. To

    obtain a set of canonical coordinates in a neighborhood ofp L and eventuallyof the whole ofL, one needs to exploit further the properties ofF1. At each p L consider the subspace Tp(L) Tp(M) and the distribution of symplecticallyorthogonal subspaces p (Tp(L)). Since

    Xi, Xa

    L

    = 0, this distributionis generated, for all leaves, by the vector fields Xa. Furthermore, since Xasatisfies the hypotheses of the Frobenius theorem, we obtain a distributionwhich defines a second foliation a coisotropic foliation F2,1 whose leaves arethemselves foliated by those of the first foliation F1. One can now prove (atleast locally) the existence of canonical coordinates i, i, p, q, such thatthe symplectic structure and the dynamical vector field take the form

    =i

    di di +

    dp dq, =i

    i () Xi , (14.45)

    so that the equations of motion become

    i = 0, i = i, p = 0, q

    = 0 . (14.46)

    The functions i describe locally F2, and their associated Hamiltonian vec-tor fields Xi define coordinates

    i on F1. The fields Xi are independent andcommute between themselves and with . To understand better this canoni-cal coordinates, let us note that the momentum map2 J : M A, definedby J : x x A, where x(f) = f(x), f A, defines foliation of someneighborhood Uof any leaf of F2, where the leaves are Lx = J1 (x), thatis, they are the leaves of F1. Then the neighborhood Ucan be chosen to beof the type Lx P O, where O is the coadjoint orbit through x definedby the connected Lie group A, corresponding to A, and

    Pis some manifold

    transversal to O. The symplectic structure , restricted to O, coincides withthe restriction of the Poisson-Lie structure (the Kirillov structure); the co-ordinates we had in the above (p, q

    ) are canonical coordinates on O andi are coordinates on P. It can be proved [20] that all needed for the exis-tence of such local coordinates is actually the existence of the double foliation,namely, that M possesses an isotropic foliation, such that the distribution ofthe subspaces, symplectically orthogonal to the tangent spaces to its leaves,is integrable.

    1 This means that its leaves are coisotropic submanifolds.2 The reader who is not familiar with this subject can look into subsection 15.2.1

    for some relevant definitions.

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    14.3 Noncommutative Integrability Criteria 485

    14.3.1 The Invariant Tensor Fields in the Noncommutative Case

    We shall try to characterize now the noncommutative integrability in anotherway.

    Theorem 14.3. Let be a dynamical system on a 2n-dimensional manifoldM which admits a (1, 1) mixed tensor field N, having the properties: N is -invariant, that is

    LN = 0 . (14.47)

    At each point p M, Np is diagonalizable with only simple or doublydegenerate eigenvalues, whose differentials are linearly independent.

    If by S(p), p M, is denoted the sum of all the eigenspaces, associatedwith the doubly degenerate eigenvalues of N(p), then for arbitrary vectorfield, such that X(p) S(p), and for arbitrary vector field Y T(M)and 1-form , we have

    NN ( , X , Y ) = , RN(X, Y) = 0 , (14.48)

    (or equivalently, RN(X, Y) = 0), where RN is the Nijenhuis torsion of N,see (13.37). Then the vector field is Hamiltonian and defines integrableseparable dynamics.

    Proof. Let us denote by 1, 2, . . ,r the doubly degenerate eigenvalues andby 2r+1,...,2n let us denote the simple eigenvalues. According to the as-sumptions, the tensor field N can be written into the form

    N =r

    i=1

    i

    ei i + ei+r i+r

    +

    2n=2r+1

    e , (14.49)

    where the es form a basis of eigenvectors of N and the s are the elementsof the dual basis. Thus,

    N ei = iei, N ei+r = iei+r, N e = e, i r, 2r + 1Ni = i

    i, Ni+r = ii+r, N =

    , i r, 2r + 1.(14.50)

    We know, see (13.37), that

    RN (X, Y) = [N X , N Y ] + N2 [X, Y] N[NX,Y] N[X,NY] . (14.51)

    Calculating the above expression on the basis vector fields {e1, . . . , e2n} yields,RN (ei, ej) =

    (N i) (N j) [ei, ej ] + (i j)

    (Leij) ej +

    Leji

    ei

    http://-/?-http://-/?-http://-/?-http://-/?-
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    486 14 Integrability and Nijenhuis Tensors

    RN (ei, e) =

    (N i) (N ) [ei, e] + (i ) [(Lei) e + (Lei) ei] ,(14.52)

    where i, j 2r and 2r + 1, and the conditions about the spectrum of Nentail3 the following relations:

    (N i) (N j) [ei, ej ] = 0, (i j) ei (j) = 0(N i) (N ) [ei, e] = 0, ei () = e (i) = 0 . (14.53)

    It follows that for any three vector fields ei, ej , e we have

    [ei, ej ] = aei + bej + cei+r + dej+r, [ei, e] = f ei + gei+r + he . (14.54)

    Thus, any two vector fields ei and ei+r, belonging to the same eigenvalue i,satisfy the relation:[ei, ei+r] = ciei + ci+rei+r. (14.55)

    In this way, if i {1, . . . , r} the vector fields ei, ei+r form a local basis ofa 2-dimensional integrable distribution and through each point of M passes2-dimensional integral manifold of this distribution. On each such integralmanifold one can find coordinates (i, i), such that

    ei =

    i

    , ei+r =

    i

    . (14.56)

    Then the relations (14.53) ensure that the basis on which the tensor fieldN diagonalize is partially holonomic. On the other hand, using equations(14.53) we get

    di =j

    jej (i) +

    e (i) =j

    jej (i) = iei (i) , (14.57)

    and, therefore,

    Ndi = Niei (i) = iiei (i) = idi . (14.58)

    Next,

    d = d =

    2rk=1

    iei () +

    2n=1

    e () =

    2n=1

    e () , (14.59)

    Using the above relations, it is now possible to choose a holonomic basis insuch a way, that N has the following expression

    N =

    rj=1

    j

    ej j + er+j dj

    +

    n,=2r+1

    C e d , (14.60)

    3 See the proof of the Haantjes theorem.

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    14.3 Noncommutative Integrability Criteria 487

    with

    C =2n

    =2r+1e (

    ) [e ()]1

    and di = 0 . (14.61)

    In addition, in a neighborhood of each 2-dimensional submanifold, we canchoose coordinates (,,), such that the tensor N can also be written intothe form:4

    N =

    rj=1

    j(

    i di +

    i di) +

    n,=2r+1

    C

    d . (14.62)

    On the basis we have constructed, the vector field can be written as

    =r

    i=1

    (i

    i+ i

    i) +

    n=2r+1

    Ee , (14.63)

    so that the condition LN = 0 implies i = 0, E = 0. It follows that

    =

    ri=1

    i

    i, i

    i. (14.64)

    Finally, we note that one can find symplectic structures with respect to whichthe vector field is Hamiltonian. Indeed, the closed 2-form

    =r

    k=1

    Gk

    k, k

    dk dk +2n

    ,=2r+1

    f (, ) d d , (14.65)

    will be invariant if

    i

    Gii

    = 0 . (14.66)

    The nondegeneracy condition for can be cast into the form

    det(f)

    rk=1

    Gk = 0 . (14.67)

    This is equivalent to the requirement that if i

    i, i

    vanishes at some point,

    then it also vanishes on the integral curve of i through that point.

    If the vector field has no singular points, one can immediately point outa class of symplectic structures with respect to which it is Hamiltonian:

    4 The notation has been chosen to correspond to the geometric structure wepreviously described.

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    488 14 Integrability and Nijenhuis Tensors

    =r

    k=1

    gk (k)

    k (k, k)dk dk +

    2n,=2r+1

    f (, ) d d , (14.68)

    where gk and f are arbitrary functions such that

    det(f)r

    k=1

    gkk

    = 0 . (14.69)

    Remark 14.4. If k is identically zero for some index k, we can define

    =

    igi (i) di di +

    jgj (j)

    j (j , j)dj dj

    +2n

    ,=2r+1

    f (, ) d d , (14.70)

    where the index i in the sum runs over those eigenspaces, for which j = 0.When has zeroes but does not vanish identically, we have to exclude fromthe original manifold the subset of the zeroes of . The resulting manifold isinvariant with respect to the flow of , and we can proceed as before.

    If the submanifold = const is compact and connected, we can introduce

    as usual action-angle coordinates (J, ) so that the vector field and thesymplectic structure , in the coordinates (J,,) take the following form:

    =r

    i=1

    i (Ji)

    i(14.71)

    =r

    k=1fk (Jk) dJk dk +

    2n

    ,=2r+1f (, ) d d . (14.72)

    In this case, the family of symplectic structures with respect to which isHamiltonian is described in [20, 22, 23]. The tensor field N can be used togenerate compatible invariant symplectic structures according to

    N (X, Y) = 1 (NX,Y) + 1 (X,NY) + 2 (X, Y) (14.73)

    with

    1 =r

    k=1

    fk (Jk) dJk

    dk, 2 =

    1

    2

    2n

    ,=2r+1

    f (, ) d

    d .

    (14.74)Let us outline now the construction of mixed tensor fields in the case

    of noncommutative integrable systems. Suppose we have a noncommutative

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    14.3 Noncommutative Integrability Criteria 489

    integrable system with properties as in Theorem 14.2. By the analysis we hadin the above, we can assume that they have the symplectic structure

    = i

    di di +

    dp dq

    and the dynamics is defined by the equations of motion

    i = 0, i = i, p = 0, q

    = 0

    i = 1, 2, . . . , r; = r + 1, r + 2, . . . , n . (14.75)

    Denoting by the collection of the ps and qs, we have

    i = 0, i = i, = 0 . (14.76)

    Consider now the following tensor field

    N =r

    j=1

    j(

    i di +

    i di) +

    2n,

    C ()

    d , (14.77)

    where C () = is diagonal matrix. It can be verified that N has a vanish-

    ing torsion and is invariant, provided the Hamiltonian function is separable,that is, if H can be written into the form:

    H = K1() + K2() , (14.78)

    with

    K1() =r

    i=1

    Hi(i) . (14.79)

    If K1 is not separable but

    det 2K1

    ji = 0 , (14.80)

    the construction of the invariant tensor field can be done as in [4]. This showsthat in the noncommutative case, an invariant torsion free tensor field canalso be found. Of course, such a tensor field always generates (by repeatedapplication) Abelian algebras of symmetries. Thus, regardless of the vanish-ing of the torsion on the whole space the noncommutative features are re-lated to the nondegenerate eigenvalues and are still described by the term

    , C ()

    d.

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    490 14 Integrability and Nijenhuis Tensors

    14.3.2 The Kepler Dynamics

    The Kepler problem is one of the celebrated problems of Classical Mechanics,and all theoretical constructions in Mechanics and Symplectic Geometry aretested on it; see for example the recent monograph [24]. It is interesting toobtain for it a recursion operator. We believe that its construction deserves aspecial subsection, since it is not elementary and has been done quite recently.

    As is known, the phase space for the Kepler problem is the symplecticmanifold (M0, ), where M0 = T(R3 \ {0}) = (R3 \ {0}) R3 and =3

    i=1 dpi dxi. The dynamical system (the Kepler dynamics) is defined bythe Hamiltonian function

    H =1

    2mp2 k

    r. (14.81)

    Here p = (p1, p2, p3) is the linear momentum; x = (x1, x2, x3) is the positionvector which length is denoted by r; k is a positive constant and m is themass of the particle. Through this section || || will denote the usual R3 normdefined by the standard inner product x, y = 3i=1 xiyi, so r=||x||. It is wellknown that the angular momentum L = x p is a vector constant of motion:

    {H, L} = 0 , (14.82)which is a short way to write the relations {H, Li} = 0; i = 1, 2, 3. The aboveimplies that the trajectory lies in the plane orthogonal to L = const , passingthrough the origin. The Kepler problem, however, possesses another vectorfirst integral, the so-called Laplace-Runge-Lenz vector, given by:

    B =1

    mp L k x

    r. (14.83)

    In other words, {H, B} = 0. Of course, since we cannot have 3 additional in-dependent integrals of motion, there must be a relation between the integrals.Indeed, one can observe that

    B, L = 0 , (14.84)which implies that the Laplace-Runge-Lenz vector lies in the plane of themotion. We also have

    {Li, Bj} =3

    s=1

    ijsBs , (14.85)

    where ijs is the Levi-Civita symbol. One verifies also that

    {Bi, Bj} = 2H

    m

    3s=1

    ijsLs . (14.86)

    It is well known that for a negative energies the motion is bounded and theorbits are ellipses. We shall be interested in these kind of motions, so we

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    14.3 Noncommutative Integrability Criteria 491

    restrict our symplectic manifold to the set M, where H < 0. Since M is anopen subset ofM0 it inherits all its structures. On M, we have H < 0, and,therefore, we can introduce the vector

    A =2Hm

    12

    B , (14.87)

    and the Poisson brackets we had before become:

    {Li, Aj} =3

    s=1

    ijsAs, {Ai, Aj} =3

    s=1

    ijsLs . (14.88)

    As a consequence, for the components of the vector-functions

    I =1

    2(L + A), J =

    1

    2(L

    A) (14.89)

    we have the following Poisson brackets

    {Ih, Is} = hslIl{Jh, Js} = hslJl{Ih, Js} = 0 . (14.90)

    The above shows that the Lie algebra of the symmetries for the Kepler dy-namics is the algebra o (3)o(3) o (4) or, which is the same, su (2) su (2).In particular, the Hamiltonian H can be written into the form

    H = mk2

    2 (L2 + A2)= mk

    2

    4 (I2 + J2). (14.91)

    In terms of the generators of o (4): Lhk = Lkh, (1 k, h 4), k = h definedby:

    Lhs = hsiLi; h, s = 1, 2, 3

    Lh4 = L4h = Ah; h = 1, 2, 3 (14.92)the Hamiltonian H becomes

    H = mk2

    C1, (14.93)

    where C1 =4

    i,j=1 LijLij is the so-called first Casimir element of o (4).Our next step is to construct the action-angle variables (Js,

    s); s = 1, 2, 3.The idea to find them is the following. Let us assume that the Liouville form =

    3i=1pidxi in the action-angle coordinates equals

    =3

    s=1

    Jsds (14.94)

    where 0 s 2.

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    492 14 Integrability and Nijenhuis Tensors

    Remark 14.5. Since d(3

    s=1 Jsds) = = d, the form written in the right-

    hand side of (14.94) and the form differ by a closed form . Assuming that

    is exact, we have that3

    s=1 Jsds = + dF, where F is some function. Then

    dF will give no contribution in the integration by which we find the action

    variables (see (14.95) below), and we can simply assume that (14.94) holds.If s is the curve obtained when we vary s from 0 to 2, while the othervariables remain constant, then

    Js =1

    2

    s

    . (14.95)

    We now note that we shall obtain some linear combinations of the Jis ifinstead of the curves i we use the curves i = n

    1i 1 + n

    2i 2 + n

    3i 3 with

    some integers nji so it remains to guess some closed curve on the torus where

    the motion occurs. This is relatively simple, since the motion defined by ourHamiltonian is always periodic. After the variables Ji are obtained, we mustcheck that the Hamiltonian H depends only on Jis as should be. The nextstep will be to find i. This would be easy if we are using the Hamilton-Jacobiequation formalism. How this idea works will be seen below; see also [25].

    Let us we consider the Hamilton-Jacobi equation in spherical coordinates(r,,) whose origin coincides with (0, 0, 0). The spherical coordinates arechosen of course because the Hamilton-Jacobi equation for the Kepler prob-lem in spherical coordinates allows separation of variables, and we can find thesolution explicitly. In spherical coordinates, the symplectic form, the Hamil-tonian, and the Hamiltonian vector field corresponding to H take the form:

    = dpr dr + dp d + dp d (14.96)

    H =1

    2m(p2r +

    p2r2

    +p2

    r2 sin2 ) k

    r(14.97)

    =1

    m pr

    r+

    pr2

    +

    p

    r2 sin2

    1r2

    p2 +

    p2

    sin2

    pr p

    2 cos

    r2 sin3

    p k

    r2

    p

    . (14.98)

    Since the Hamiltonian does not depend explicitly on time, the Hamilton-Jacobi equation

    S

    t+

    1

    2m

    S

    r

    2+

    1

    r2

    S

    2+

    1

    r2 sin2

    S

    2 k

    r= 0 (14.99)

    can be reduced, setting S = W

    Et, to the form

    1

    2m

    W

    r

    2+

    1

    r2

    W

    2+

    1

    r2 sin2

    W

    2 k

    r= E . (14.100)

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    14.3 Noncommutative Integrability Criteria 493

    We look now for a solution W (r,,) of the type

    W (r,,) = Wr (r) + W () + W () , (14.101)

    where Wr, W and W, depend only on the corresponding variable r, and. In this way, the Hamilton-Jacobi equation for W becomes

    1

    2m

    dWr

    dr

    2+

    1

    r2

    dW

    d

    2+

    1

    r2 sin2

    dW

    d

    2 k

    r= E . (14.102)

    Standard reasoning leads to the conclusion that the above equation is equiv-alent to the system:

    dW

    d=

    dWd

    2= 2

    2

    sin2 (14.103)

    dWr

    dr

    2= 2m

    E+

    k

    r

    2

    r2.

    where , are constants of integration. The constant has clear physicalmeaning it is simply the projection of the angular momentum on the z-axis.The constant is equal to the length of the angular momentum, because

    ||L||2 = ||x v||2 = p2 +p2

    sin2 . (14.104)

    If denotes the angle between the plane of the orbit and the (x, y) plane, wehave

    p = ||L|| cos = cos . (14.105)

    Therefore, the solution of the Hamilton-Jacobi equation is S = Et + W:

    W = +

    d

    2

    2

    sin2 +

    dr

    2mE+

    2mk

    r

    2

    r2. (14.106)

    As a consequence,

    p =

    p =

    2 2

    sin2 (14.107)

    pr =

    2mE+

    2mk

    r

    2

    r2.

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    494 14 Integrability and Nijenhuis Tensors

    Now we implement the scheme for finding the action-angle variables we out-lined in the above. First we find the action variables

    Js =1

    2 s

    ; i = 1, 2, 3 (14.108)

    where = pd +pd +prdr (14.109)

    and the closed curves 1, 2, 3 define 3 independent cycles in the phase space the space of the variables (r,,,, , r), for which the Hamiltonian hasnegative values. These expressions will become most simple if 1 is givenparametrically by p = p(), while the other variables except p, remainconstant; 2 is given parametrically by p = p(), while the other variablesexcept p

    , remain constant and the same for

    3. In this case, we have the

    action variables

    J =1

    2

    1

    d

    J =1

    2

    2

    2

    2

    sin2 d (14.110)

    Jr =1

    2 32mE+

    2mk

    r

    2

    r2dr.

    To guess 1 is easy; we can choose it to be parameterized by , since when varies from 0 to 2, while the other variables remain constant, we have aclosed curve. Thus we obtain

    J = . (14.111)

    As to the curves 2, 3, since the movement is periodic, they can be chosento join the turning points, that is, they are fixed by the requirement that thecorresponding velocities vanish or, better, the corresponding momenta p andp

    r(expressed, of course, in terms of variables

    and

    ) vanish. We have:

    p2 = 2

    2

    sin2 = 0

    p2r = 2mE+2mk

    r

    2

    r2= 0 (14.112)

    and the first equation defines two angles 1, 2, the second two values of r -r1, r2. Then the closed curve 2 parameterized by is defined by p()when goes from 1 to 2 and

    p() when goes back from 2 to 1,

    while the rest of the variables remain constant. In the same way we define thecurve 3. In order to find J, we must integrate from 1 to 2 and multiplythe result by 2. The same is true for the integral over r, but we integrate herefrom r1 and r2. Using (14.105) we reduce the equation for 1, 2 to

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    14.3 Noncommutative Integrability Criteria 495

    sin2 =22

    = cos2 . (14.113)

    Since always lies between 0 and , we have sin > 0 and sin 1 = sin 2 =cos . We find that

    1= /2

    ,

    2= /2 + . But the integral from /2

    to /2 equals the integral from /2 to /2 , and we obtain

    J =42

    /2/2

    1

    sin

    sin2 cos2 d (14.114)

    =2

    sin2

    /20

    cos2

    1 sin2 sin2 d , (14.115)

    where the new variable is introduced, related to the old one by

    cos = sin sin . (14.116)

    Putting x = tan , we transform the last integral into the form

    J =2

    +0

    1

    1 + x2 cos

    2

    1 + x2 cos2

    dx (14.117)

    =2

    2

    2

    cos

    = (1 cos ) . (14.118)

    and finally, using again (14.105), we get

    J = . (14.119)The integral giving Jr can be calculated using the Residue Theorem. We firstremark that the roots r1 and r2 of the equation

    2mE+2mk

    r

    2

    r2= 0 , (14.120)

    which gives the integration limits, are positive if E < 0 (as we have seen

    they correspond to the radii of the turning points of the motion). Next, thefunction

    f(z) =

    2mE+

    2mk

    z

    2

    z2, (14.121)

    allows analytic continuation from the real axis to the complex plane. It hastwo branch points at

    z = k2E

    1

    1 +

    22E

    mk2

    (14.122)

    and a simple pole at z = 0. We cut the complex plane on the line segment[r1, r2], and we choose the branch of f+(z) of f(z), which on the upper sideof the cut (z = x + i0+) equals

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    496 14 Integrability and Nijenhuis Tensors

    f(x) =

    2mE+2mk

    x

    2

    x2, (14.123)

    and on the lower side (z = x i0+) equals

    f(x) =

    2mE+

    2mk

    x 2

    x2. (14.124)

    The function f+(z) is analytic in C \ ([r1, r2] {0}) and has a simple pole atz = 0. The integral over r, first integrating from r1 to r2 and then integratingagain from r2 to r1, changing the sign of the integrand, can be consideredas complex integral of the function (1/2)f+(z), first going on the lower sideof the cut from r1 to r2 and then back on the upper side from r2 to r1.The Cauchy theorem ensures that the same value will be obtained, if we

    take the complex integral (1/2)f+(z)dz over a closed contour , oriented

    anticlockwise, which encompasses the segment [r1, r2] and does not encompassthe pole z = 0. Such integral can be calculated using the Residue theorem,and this gives:

    Jr = i (Res(f; 0) + Res(f;)) . (14.125)Now, since Res(f+; 0) =

    2 and Res(f+; ) = mk/2mE, we obtainJr = + mk2mE . (14.126)

    Then from J + J = , we get

    J + J + Jr =mk2mE . (14.127)

    The above relation shows that the Hamiltonian function can be written interms of the action-angle variables in the following way:

    H = mk2

    2 (J + J + Jr)2 . (14.128)

    Finally, replacing in (14.106) the variables E, , with their expressions interms of action coordinates

    E = mk2

    2 (J + J + Jr)2

    = J J = J , (14.129)

    we get the function W as a function of the J

    s. This allows us to define thecorresponding angle variables:

    h =W

    Jh; h = 1, 2, 3 , (14.130)

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    14.3 Noncommutative Integrability Criteria 497

    where in the above and below we write J1 = Jr, J2 = J, J3 = J in order tocast the expressions in compact form. After performing the integrations, weobtain a symplectic map from the original coordinates (pr, p, p,r ,,) tothe action-angle coordinates Jh,

    h

    ; h = 1, 2, 3. This map is implicitly givenby:

    J1 =

    p2 +p2

    sin2 + mk

    2mk

    r p

    2

    r2 p

    2

    r2 sin2 p2r

    1(14.131)

    J2 =

    p2 +

    p2

    sin2 p (14.132)

    J3 = p (14.133)

    1

    = J2 m2k2r2 + 2mkJ2r J2(J + J)2

    1

    2

    (14.134)

    + arcsinmkr J2

    J

    J2 (J + J)2(14.135)

    2 = 1 arcsin [mkr (J + J)2]J

    J2 (J + J)2(14.136)

    arcsin (J + J)cos

    (J + J)2 J

    (14.137)

    3

    = 2

    + arcsin

    Jcot (J + J)2 J +

    1

    , (14.138)

    whereJ = J1 + J2 + J3 = Jr + J + J . (14.139)

    In order to define the map in an explicit way, in the last three relationsJr, J, J must be expressed in terms of (pr, p, p,r ,,), using the firstthree of the above equations.

    The Hamiltonian H, the symplectic form , and the Hamiltonian vector

    field can be written through the action-angle variables as follows:H = mk2J2

    =

    3h=1

    dJh dh

    = 2mk2J3

    1+

    2+

    3

    . (14.140)

    As already seen, the analysis of a given Hamiltonian system and the search for

    alternative symplectic forms and Nijenhuis tensors is much easier in action-angle variables which was the reason we calculated them here for the Keplerproblem. As one can check, (see [25]), the vector field is Hamiltonian alsowith respect to the symplectic form 1:

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    498 14 Integrability and Nijenhuis Tensors

    1 =h,k

    Rhk dJh dk , (14.141)

    where the matrix R = (Rhk)1k,h3 is defined by:

    R =1

    2

    Jr J JJ J Jr + J J

    J J J Jr + J

    . (14.142)

    (We prefer to write Jr, J, J instead of J1, J2, J3 in the final expressions).The form 1 generates new Poisson brackets

    {f, g}1 =k,h

    (R1)kh

    f

    Jh

    g

    k f

    kg

    Jh

    , (14.143)

    and corresponds to the new Hamiltonian function H1:

    H1 = 2mk2J1 . (14.144)

    In the original coordinates (p, q) the symplectic form 1 is simply written as:

    1 =i

    dKi di (14.145)

    where the functions Ki(p, q) and i(p, q) are given by:

    K1 =1

    4[Jr + (J J)2]

    K1 =1

    4[Jr + (J J)2]

    K3 =1

    2J[Jr + J] . (14.146)

    (Recall that Ji = Ji(p, q), i = i(p, q)). Since is nondegenerate, one

    can construct a mixed invariant tensor field N0, such that (N0(X), Y) =1(X, Y). The tensor N0 has the form

    N0 =h,k

    Rhk dJh

    Jk+ Rkh d

    h k

    , (14.147)

    As easily verified, N0 has double degenerate eigenvalues and vanishingNijenhuis torsion, the last property being equivalent to the compatibility ofthe symplectic structures and 1. However, one can check, see [25], that the

    action of N0 does not give new integrals of motion as one would expect, forexample

    N0 dH = d

    k

    m

    H

    . (14.148)

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    14.4 Compatible Poisson Structures and Poisson-Nijenhuis Structures 499

    This is not what we would like to obtain, and so we must look for anotherNijenhuis tensor. It can be found changing the action-angle variables to someothers, closely related to them. As already seen, the Kepler dynamics possessesfive independent first integrals, for example, those chosen among Hamiltonian

    and the components of the angular momentum and the components of theLaplace-Runge-Lenz vector. In action-angle coordinates (J, ) we can choosethe following independent five first integrals

    Jr, J, J, 1 2, 2 3 . (14.149)

    Now, using the so-called Delauney action-angle variables, (Ii, i); i = 1, 2, 3,(see [24]), which are given by

    I1 = Jr + J + J = 1

    I2 = J + J = 3

    I3 = J = 4

    1 = 1 = 1

    2 = 2 1 = 5

    3 = 3 2 = 6 , (14.150)

    and introducing the new variables 1, 1, ; = 3, 4, 5, 6, we can constructthe following invariant tensor field

    N = 1

    1 d1 +

    1 d1

    +

    6=3

    d. (14.151)

    As easily checked, the Nijenhuis torsion of this tensor field is zero. This tensorfield can be used as Nijenhuis tensor for the Kepler Dynamics Its eigenvaluesgive all the first integrals of the field .

    The above example illustrates the general concept that the integrability isclosely related to the existence of a mixed invariant tensor field with vanishing

    Nijenhuis torsion, though sometimes it is hidden, and it is not an easy taskto find it.

    14.4 Compatible Poisson Structures

    and Poisson-Nijenhuis Structures

    Following [26] we shall say that on the manifold M is defined P-N struc-ture (Poisson-Nijenhuis structure) if on

    Mare defined simultaneously Poisson

    tensor P and Nijenhuis tensor N, satisfying the following coupling conditions:

    (a) N P = P N

    (b) P LN(X)() P LX(N) + LP()(N)(X) = 0 , (14.152)

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    500 14 Integrability and Nijenhuis Tensors

    for arbitrary choice of the vector field X and the 1-form . We also introducethe following (2, 1) type tensor field [P, N] through the property

    [P, N](X, ) = P LN(X)() P LX(N) + LP()(N)(X) . (14.153)

    A manifold endowed with a P-N structure will be called Poisson-Nijenhuismanifold or for shortness P-N manifold.

    We present here the condition (b) exactly as it has been initially introducedby Magri in the first work that mentioned P-N structures [26] (for more re-cent developments see [27, 28]). In order to understand its meaning, let usassume that P is invertible, then B = (P)1 : 1(M) T(M) defines asymplectic form through B(X, Y) = B(X), Y. The coupling condition (b)then is a consequence from dB = 0, the first coupling condition, and the re-lation dN(B) = 0. For symplectic form that satisfies (NX,Y) = (X,NY)

    the second coupling condition is equivalent to the requirement dN(B) = 0.Recall that the operation dN has been introduced earlier in (13.19), and ac-cording to its definition for a symplectic form, the condition dNB = 0 isequivalent to diN(B) = 0 which means that the form iN is closed. SinceiN(X, Y) = (NX,Y) + (X,NY), all this simply shows that the secondcoupling condition ensures that N(X, Y) = (NX,Y) is a closed 2-form.

    The second coupling condition has also natural interpretation in termsof the so-called Lie bi-algebroid structures [29], which we are not going tointroduce here.

    The structure we have introduced seems very specific. However, it turnsout that for the so-called soliton equations it arises in a natural way. Thepoint is that almost in each approach to the theory of completely integrablesystems one can notice the crucial role played by the so-called compatiblePoisson tensors (see [26, 30, 31, 32, 33]), or as also called Hamiltonian pairs(see [34, 35]).

    Definition 14.6. Two Poisson tensors P and Q are compatible if the tensorP + Q is Poisson tensor too.

    It is evident that for this it is necessary and sufficient that[P, Q]S = 0 . (14.154)

    If P, Q are compatible it easily follows that for arbitrary constants a, b thefield aP + bQ is also a Poisson tensor.

    After these preliminaries we prove now theorem [26], showing how P-Nstructures arise when we have a compatible Poisson pair.

    Theorem 14.7. Let P and Q be Poisson tensors on M. Let Q1 exist, thatis, we assume that there exists a smooth field of linear maps m

    Q1

    m. Then

    the tensor fields N = P Q1 and Q endow the manifold with P-N structure.Proof. First of all, let us note that if N = P Q1, then the first couplingcondition is already satisfied, and it remains to prove the second coupling

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    14.4 Compatible Poisson Structures and Poisson-Nijenhuis Structures 501

    condition and the fact that N is Nijenhuis tensor. Let us begin with thesecond coupling condition.

    Let P = NQ and Q be compatible Poisson tensors. This means thatsimultaneously [Q, Q]S = 0, [NQ,NQ]S = 0 and [Q,NQ]S = 0. In more

    detail

    QLQ12, 3 + QLQ23, 1 + QLQ31, 2 = 0 (14.155)for any three 1-forms 1, 2, 3;

    N QLNQ12, 3 + N QLNQ23, 1 + N QLNQ31, 2 = 0 (14.156)for any three 1-forms 1, 2, 3;

    QLNQ12, 3 + QLNQ23, 1 + QLNQ31, 2 +N QLQ12, 3 + N QLQ23, 1 + NQLQ31, 2 = 0 (14.157)for any three 1-forms 1, 2, 3. Making use of (14.155) we have:

    N QLQ12, 3 = QLQ12, N3 =QLQ2(N3), 1 QLQN31, 2 =QLQ2(N3), 1 QLNQ31, 2 .

    Analogously,

    N QLQ23, 1 = QLQ3(N1), 2 QLNQ12, 3N QLQ31, 2 = QLQ1(N2), 3 QLNQ23, 1.

    Inserting the above expressions into (14.157) we get the following equation

    QLQ1(N2), 3+QLQ2(N3), 1+QLQ3(N1), 2 = 0 . (14.158)Thus we see that provided N Q = QN and Q = Q are true the relations[Q, Q]S = 0 [Q,NQ]S = 0 lead to the equation (14.158) and conversely, thisequation, together with [Q, Q]S = 0 leads to [Q,NQ]S = 0.

    Remark 14.8. It is easy to see that we can introduce the (3, 0) tensor field[Q, N]S by requirement:

    [Q, N]S(1, 2, 3) = (14.159)

    QLQ1(N2), 3 + QLQ2(N3), 1 + QLQ3(N1), Q2 ,then the calculation we have done show that for arbitrary 1-forms 1, 2, 3

    we have:

    [NQ,Q]S(1, 2, 3) + [Q, N]S(1, 2, 3) = (14.160)

    + ([Q, Q]S(N1, 2, 3) + cycl (1, 2, 3)) .

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    502 14 Integrability and Nijenhuis Tensors

    We shall show now that if Q is invertible, the relation (14.158) is equivalentto the coupling condition (14.152 b). Now, in order to obtain (14.152 b) wetransform the three terms in the left-hand side of (14.158). For the secondone, using some simple identities, we get

    QLQ2(N3), 1 = LQ2(N3), Q1 =LQ2N3, Q1 + N3, LQ2(Q1) =LQ23,NQ1 + 3, N[Q1, Q2] =LQ23,NQ1 LQ2(N)(Q1), 3 =QLQ23, N1 LQ2(N)(Q1), 3 . (14.161)

    If we put in (14.155) 3 instead of 1, N1 instead of 2 and 2 instead of

    3, the third term can be cast into the form

    QLQ3(N1), 2 = QLQ23, N1 QLQN12, 3 . (14.162)Finally, inserting (14.161), (14.162) into (14.158) we arrive at:

    QLQ1(N2) LQ2(N)(Q1) QLNQ12, 3 = 0 . (14.163)It remains to take into account that 1, 2, 3 are arbitrary and that Q isinvertible to obtain that for each 1-form and each vector field X we have

    QLX(N

    ) LQ(N)(X) QLNX() = 0 , (14.164)which is none but the coupling condition.

    Remark 14.9. With the help of the tensor field [N, Q]S we have introduced inthe remark (14.8), the calculations that we have preformed can be written as

    [Q, N]S(1, 2, 3) = [Q, Q]S(N1, 2, 3) + [Q, N](Q1, 2), 3 . (14.165)

    Now we must show that N is Nijenhuis tensor. For this, we shall consider(14.156) but first we shall make some preparations. First

    N QLNQ1(2), 3 = LQN1(2), QN3 =LQN12, QN3 2, [N Q3,NQ 1] =LQN1N2, Q3 2, [N Q3,NQ 1] =QLQN1(N2), 3 2, N[N Q1, Q3] 2, [N Q3,NQ 1].

    Next, using (14.155), (14.158) we obtain:

    N QLNQ2(3), 1

    =

    QLQN2(3), N

    1

    =

    QLQ3(N1), N2 QLQN1(N2), 3 =QLQ1((N)22), 3 + QLQN2(N3), 1QLQN1(N2), 3 = LQ1((N)22), Q3

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    14.4 Compatible Poisson Structures and Poisson-Nijenhuis Structures 503

    +QLNQ2(N3), 1 QLQN1(N2), 3 =LQ1(N)22, Q3 + (N)22, LQ1Q3LNQ2(N3), Q1 QLQN1(N2), 3 =

    L

    Q1

    2, N2Q

    3

    2, N2[Q

    3, Q

    1]LNQ2(N3), Q1 QLQN1(N2), 3.

    We transform also the last term in (14.156) with the help of (14.155) as follows:

    N QLNQ3(1), 2 = QLQN3(1), N2 =QLQ1(N2), N3 QLQN2(N3), 1 =LQ1(N2), QN3 + LQN2(N3), Q1 =LQ1

    (N2), QN

    3

    N2, LQ1(QN

    3)

    +LQN2(N3), Q1 = LQ12, N2Q3+2, N[N Q3, Q1] + LQN2(N3), Q1.

    Finally, we insert these expressions in (14.156) and get:

    2, [N X3, N X1] + N2[X3, X1] N[N X3, X1] N[X3, N X1] = 0,

    where we have put X1 = Q1, X3 = Q3. As Q is invertible and 1, 2, 3 arearbitrary, we obtain exactly the zero Nijenhuis bracket condition

    [N X3, N X1] + N2[X3, X1] N[NX3, X1] N[X3, N X1] = 0.

    The theorem is proved.

    A construction similar to the one we have used in the above theoremcan be applied also in the following situation. Suppose on the manifold M wehave simultaneously a Poisson tensor P and a closed 2-form (not necessarilynondegenerate), for example a presymplectic form. Let be the correspondingfield of linear maps

    m m : Tm(M) Tm(M) . (14.166)

    Then the following theorem (see [26]) holds:

    Theorem 14.10. If the form that corresponds to P is closed, then thetensor fields P and N = P define P-N structure on the manifold M.The theorem can be proved using simple calculations, similar to what we haveused in the above.

    An interesting situation arises on symplectic manifold M with symplecticform , if in addition there is a nondegenerate Nijenhuis tensor N for whichwe have:

    N = N . (14.167)

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    504 14 Integrability and Nijenhuis Tensors

    This condition is an analog of the coupling condition (a) for the P-N structure.Of course, as usual, here is the field of maps m m : Tm(M) Tm(M)that corresponds to .

    In this case, if the eigenvalues of N are smooth functions on M theygenerate a system of integrable vector fields, without the additional require-ments usually imposed on and N, see for example [26]. In order to provethis other version of the Liouville theorem is used, called the Liouville-Cartantheorem,5 [36].

    Theorem 14.11. Let(M, ) be a symplectic manifold of dimension 2n, letbe a closed 1-form onM and let X be the Hamiltonian field that correspondsto it (iX = ). Suppose that1, 2, . . . , n1 are closed 1-forms that are firstintegrals of X (s(X) = 0, 1 s n 1). Let us assume that the forms shave the properties:

    1. The forms s are in involution (that is, {i, j} = 0).2. The forms, 1, 2, . . . , n1 are linearly independent for each point m of

    an open set U M.Then

    1. On the set U there exist n linear forms (1-forms) , 1, 2, . . . , n1, suchthat

    U =

    +

    n1

    s=1

    s

    s . (14.168)

    2. The 2-forms d,d1, d2, . . . , d n1 belong to the ideal generated by theforms , 1, 2, . . . , n1 in the exterior algebra (M).

    The 1-forms , 1, 2, . . . , n1 generate on U M integrable Pfaffian sys-tem. Let N be integral manifold for it and let j : N M be the inclusionmap. Then

    1. The field X is tangent to N and induces on it a vector field Z (X and Zare j-related).

    2. The forms j, j1, j2, . . . , jn1 (restrictions of , 1, 2, . . . , n1on N) are linearly independent at each point of N.

    3. The formsj1, j2, . . . , j

    n1 are first integrals of Z (js(Z) = 0 and

    djs = 0).4. dj = 0 and j(Z) = 1.

    The theorem states that one can find a submanifold N of M such that thefield X is tangent to it and on N we have n 1 independent integrals (thes) for X. Thus X is integrable in quadratures (completely integrable).

    Let us return now to the situation when we have a Nijenhuis tensor Non a symplectic manifold (M, ) coupled with the symplectic structure by5 Actually in the formulation below, we united two theorems, the so-called theorem

    of Gallissot and the Liouville-Cartan theorem.

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    14.4 Compatible Poisson Structures and Poisson-Nijenhuis Structures 505

    (14.167). In order to simplify the notations, and to avoid repeating all the timethat we have some property pointwise, we shall use the following conventions.Let N be a (1, 1) tensor on the manifold M, that is, N is a field of linearoperators Nm : Tm(M) Tm(M). We shall call it semisimple if Nm issemisimple for each m. Next, let us suppose that the eigenvalues i(m) ofNm are smooth functions and the eigenspaces Si(m) corresponding to i(m)have constant dimensions. Then we shall refer to the distributions m Si(m)as the eigenspaces and shall denote them by Si. We shall call the dimensionof Si(m) the dimension of Si and we shall call the functions i eigenvaluescorresponding to Si. Following the same logic, we shall denote the distributionm Si(m)Sj(m) by SiSj . If is symplectic form on M we shall say thatSi and Sj are orthogonal with respect to ifSi(m) and Sj(m) are orthogonalwith respect to m. If X is a field we shall say that X Si if for each mX(m) Si(m) and so on, each property that holds pointwise for some fieldsof objects will be stated as property of the corresponding fields.

    Now we have the following proposition (see [37] and [26]).

    Proposition 14.12. Let (M, ) be 2n-dimensional symplectic manifold onwhich there exists Nijenhuis tensor N, such that N = N. Let Nbe semisimple and let its eigenvalues i be smooth functions on M. Let thedimension of the eigenspaces Si corresponding to i be constant onM. Then:

    1. The eigenspaces Si, corresponding to the eigenvalues i are orthogonalwith respect to and have even dimension.

    2. If none of the functions i is locally equal to a constant, that is, thereis no open subset V M such that i|V = const, then the forms diare independent and are in involution. The Hamiltonian vector fields Xicorresponding to i (iXi = di) belong to the subspaces Si.

    3. If for each i dim(Si) = 2, that is, if each eigenvalue is exactly doubledegenerate, and if these eigenvalues are not locally equal to constants then:

    (a) {i}ni=1 is complete set of functions in involution and each vector fieldXi is completely integrable Hamiltonian system.

    (b) The 2-form can be locally expressed in the following way:

    =n

    i=1

    i, i = |Si , i = di i,

    where i are 1-forms on M. If Yi are the vector fields correspondingto i (i = iYi) then Xi, Yi span the subspaces Si.If the vector fieldsXi, Yi can be chosen in such a way that, [Xi, Yi] = 0,then LXiN = 0.

    (c) If the eigenvalues i have no zeroes on M, then the 2-forms (k)

    corresponding to the tensor field

    (k) = Nk; k = 0, 1, 2, . . . ,are again symplectic forms on M.

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    506 14 Integrability and Nijenhuis Tensors

    Proof. At the beginning we recall that a Nijenhuis tensor N, whose eigenspaceshave constant dimension holds the Nijenhuis theorem. Now we start with theproof.

    1. First of all, for m M let us consider, Zi Si(m), Zj Sj(m), i = j.We have

    m(NmZi, Zj) = i(m)m(Zi, Zj) = m(Zi, NmZj) = j(m)m(Zi, Zj).

    From here it follows that m(Zi, Zj) = 0, and therefore Si and Sj are orthog-onal w.r.t. the symplectic form . Naturally, we have also Si Sj = {0}.

    2. Let iXi = di. Then if i = j from the Nijenhuis theorem, it followsthat (Xi, Sj) = 0. This entails that Xi Si and therefore {di} are ininvolution. As di = 0 then Xi; i = 1, 2, . . . , n belong to different subspacesSi and are linearly independent. It follows that di; i = 1, 2, . . . , n are linearly

    independent too.3. Let now

    dim Si = 2, =n

    i=1

    i, i = |Si .

    Then {di}ni=1 is a complete involutive set of closed 1-forms. Moreover, foreach distribution

    S = Si1 Si2 . . . Sikthe 2-form S =

    |S is symplectic form on the corresponding integral sub-

    manifolds of S and the forms

    di1 , di2 , . . . , d ik

    form a complete set in involution on that submanifold. According to theLiouville-Cartan theorem, (14.11), each field Xi is then completely integrableHamiltonian system and

    S = di1 i1 + di2 i2 . . . + dik ik .

    Here il are 1-forms, such that dil belong to the ideal generated by dil inthe algebra of differential forms on the corresponding submanifold. It is notdifficult to check that (at least locally) one can extend these forms to forms onM and choose them in such a way that di = dii, where i is some 1-formand di i = i. Let Yi be the vector field corresponding to the 1-form i(i = iYi). Then

    (Xi, Yi) = 1, (Xi, Yj) = 0; i = j.

    For this reason Yi belongs to Si and Xi, Yi span Si.Let us prove now that LXiN = 0. Since we have

    (LXiN)(Y) = [Xi, N Y] N[Xi, Y],

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    14.5 Principal Properties of Poisson-Nijenhuis Manifolds 507

    then taking into account that Xi, Yi span Si it is enough to prove that theright-hand side of the above equation is zero for Y = Xj , Yj . By assumption[Xi, Yi] = 0 and then

    (LXiN)(Xj) = [dj(Xi)]Xj = 0; i = j(LXiN)(Xi) = [di(Xi)]Xi = (Xi, Xi)Xi = 0(LXiN)(Yj) = [dj(Xi)]Yi = 0 . (14.169)

    Finally, it is clear that

    (k) =

    ni=1

    ki di i

    and therefore d(k) = 0. The proposition is proved.

    The proposition shows that spectral properties are sometimes so restrictive,that they ensure some of the requirements that we usually impose a priori.

    14.5 Principal Properties of Poisson-Nijenhuis Manifolds

    Let us return again to the P-N structures. The application of these structuresare motivated by the interesting features of their fundamental fields.

    Definition 14.13. The field X is called fundamental for the P-N structure if

    LXN = 0, LXP = 0 . (14.170)

    In other words, X is fundamental field for the P-N structure if it is funda-mental for both the tensors P and N.

    In the following theorem are collected the most essential properties of thefundamental fields, see for example [26]:

    Theorem 14.14. Let

    Mbe P-N manifold. Let N be the set of 1- forms

    satisfying the conditions:

    d = 0, dN = 0 . (14.171)

    N(M) will be called the set of fundamental forms. Then the set of vectorfields PN(M)

    X = {P() : 1(M), d = 0, dN = 0} (14.172)are fundamental for the P-N structure. The vector spaces PN(

    M) and

    N(M) are Lie algebras (with respect to the Lie bracket and Poisson bracketrespectively) and P is homomorphism between these algebras:

    [X, X ] = P{, }P . (14.173)

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    508 14 Integrability and Nijenhuis Tensors

    Moreover, the above algebras are invariant under the action of N and N,respectively, and N (N) commute with the Lie algebra operation:

    N{, }P = {N, }P = {, N}P; , N(M)

    N[X, X ] = [N X, X ] = [X, N X ]; X, X PN(M) . (14.174)Proof. Let be 1-form such that d = dN = 0. As already seen, thecondition d = 0 is equivalent to the condition

    LX = d[(X)] = d, X , (14.175)for arbitrary vector field X. Then

    LN(X) LX(N) = d,NX dN, X = 0.

    But from the coupling condition

    P(LN(X) LX(N)) + LP()(N)(X) = 0it follows that LP()N = 0. Thus, the vector field P() is fundamental forthe tensor N. It is also fundamental for the tensor P (see Proposition 12.24),and therefore P() is fundamental field for the P-N structure. Next, if , N(M) we have the following chain of relations:

    N

    {,

    } {N,

    }= Nd

    , P

    d

    N , P

    =

    NLP() LP(N) = [LP(N)] = 0 . (14.176)Taking into account that {, } = {, }, we easily get the first line of(14.174). Thus we have also proved that N{, } is closed. What remains toprove is that N(M) is invariant under the action of N. For this it is enoughto prove that if N(M), that is, if d = dN = 0, then d(N)2 = 0too. But this is one of the properties of a Nijenhuis tensor, see proposition(13.24). Consider now the fundamental fields P . For the Poisson bracket oftwo closed forms , , we have

    [X, X ] = P{, } . (14.177)Therefore, if N(M)

    N[X, X ] = N P{, } = P N{, } = P{N, } . (14.178)From the other hand

    [N X, X ] = [N P , X ] = [P N , P ] = P{N, } (14.179)

    and finally we getN[X, X ] = [N X, X ] = [X, N X ] . (14.180)

    The theorem is proved.

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    14.6 Hierarchies of Poisson Structures 509

    14.6 Hierarchies of Poisson Structures

    It is remarkable that P-N structure generates also a hierarchy of Poissonstructures. More specifically, we have the following.

    Theorem 14.15. Let the tensors P and N endow the manifold M with P-Nstructure. Then on M there exist infinite number of Poisson structures Pk =NkP = P(N)k, k = 1, 2, . . . and there are infinitely many P-N structures,defined by the pairs (Pk, N

    s), k, s = 1, 2, . . ..

    The proof of the above theorem readily follows from some remarkable iden-tities, introduced in [26], which we shall present below. Let P and N be twofields of the type Pm : T

    m(M) Tm(M); Nm : Tm(M) Tm(M) such that

    NP = P N. Recall that we defined the following tensors (tensor fields):

    The (3, 0) type tensor field [P, P]S (Schouten bracket):

    [P, P]S : 1(M) 1(M) 1(M) D(M)

    [P, P]S(1, 2, 3) = 1, P LP23 + cycl (1, 2, 3) , (14.181)

    for 1, 2, 3 1(M).The (2, 1) type tensor field [P, N]:

    [P, N] :

    T(

    M)

    1(

    M)

    T(

    M)

    [P, N](X, ), =P[LNX() LX(N)] + LP()(N)X, , (14.182)

    for , 1(M); X T(M).The (1, 2) tensor field [N, N] (Nijenhuis bracket):

    [N, N] : T(M) T(M) T(M)[N, N](X1, X2) = LNX1(N)X2 N LX1(N)X2 (14.183)

    for X1, X2 T(M), or

    [N, N](X1, X2), =N2([X1, X2]) + [N(X1), N(X2)], N[X1, N(X2)] + N[N(X1), X2], (14.184)

    for 1(M); X1, X2 T(M).Then we have

    Proposition 14.16. For the tensor fields P (of type (2, 0)) and N (of type(1, 1)) such that N P = P N and for 1, 2, 3, , 1(M) and X T(M), hold the following identities:

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    510 14 Integrability and Nijenhuis Tensors

    [N P , N P ]S(1, 2, 3) =

    [P, P]S(N1, N

    2, 3) + 2, [N, N](P 1, P 3)+ [P, N](P 3, 1), N2, (14.185)

    [NP,N](X, ),

    =

    [P, N](X, ), N

    +

    [N, N](P , X ),

    .

    (14.186)

    Proof. Let us consider [N P , N P ]S:

    [N P , N P ]S(1, 2, 3) =

    N1, P LPN23 + N2, P LPN31 + N3, P LPN12 =[P, P]S(N

    1, N2, 3) 3, P LPN1(N2)

    +

    N2, P LP3(N

    1)

    N2, P LPN3(1)

    +N3, P LPN1(2) =[P, P]S(N

    1, N2, 3) + 3, N P LPN1(2) P LPN1(N2)

    +2, N P[LNP3(1) LP3(N1) =[P, P]S(N

    1, N2, 3) + [P, N](P 3, 1), N2

    2, N LP1(N)(P 3) 3, P LPN1(N)2 =[P, P]S(N

    1, N2, 3) + [P, N](P 3, 1), N2

    +[N, N](P 1, P 3), 2 2, LNP1(N)P 3

    3, P LNP1(N

    )2 . (14.187)But,

    3, P LNP1(N)P 2 = P 3, LNP1(N)P 2 =[LNP1(N)]P 3, 2 = 2, LNP1(N)P 3 , (14.188)

    and the last two terms in (14.187) cancel. This proves (14.185).In order to prove (14.186) we consider the following chain of equalities:

    [N P , N ](X, ), =N P[LNX() LX(N)] + LNP(N)X, =N P[LNX() LX(N)] + [N, N](P , X ) + N LP(N)X, =N{P[LNX LX(N)] + LP(N)X}, + [N, N](P , X ), =[P, N](X, ), N + [N, N](P , X ), . (14.189)

    This completes the proof of the proposition. Finally, taking into account The-orem 13.25 the proof of the Theorem 14.15 is easily obtained by induction.

    From the above considerations, we get two important corollaries, whichshow how the P-N structure generates hierarchies of commuting Hamiltonianvector fields and hierarchies of Poisson structures for them:

    http://-/?-http://-/?-
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    References 511

    Corollary 14.17. If, N(M) are in involution (X andX commute),then for arbitrary natural numbersk andn the forms(N)k, (N)n are alsoin involution (the fields (N)kX and (N)

    nX commute).

    Corollary 14.18. The fields of the type (N)k, where

    N

    (M

    ), areHamiltonian with respect to a hierarchy of Poisson structures P , N P , . . . N kP.If N1 exists the hierarchy is infinite and consists of the Poisson tensors ofthe type NrP where r is an integer.

    Remark 14.19. Naturally, when the manifold M is finite dimensional as a re-sult of the Caley-Hamilton theorem, we can obtain principally new structuresonly up to some number p. After this number (for k > p) the Poisson tensorsNkP are linear combinations of the Poisson tensors NsP; s = 0, 1, 2, . . . p.

    As we shall see, the corollaries (14.17), (14.18) have special applicationin the theory of soliton equations. They explain the fact that the solitonequations occur in hierarchies, have commuting flows, and are Hamiltonianwith respect to a hierarchy of Poisson structures.

    We finish the review of the properties of the P-N manifolds with the remarkthat the identities (14.160), (14.165) we established in the proof of theorem(14.7) show that we can invert it in the following way.

    Theorem 14.20. If M is a P-N manifold, endowed with Poisson tensor Pand a Nijenhuis tensor N, then P and N P are compatible Poisson tensors.

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