1 Introduction

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BIHAMILTONIAN STRUCTURES AND SEPARABILITY A. Ibort*, F. Magri**, G. Marmo*** * Depto. de Matem´aticas, Univ. Carlos III de Madrid 28911 Legan´es, Madrid, Spain. ** Dipto. di Matematica, Univ. di Milano Via C. Saldini 50, 20133 Milano, Italy. ** Dipto. di Scienze Fisiche, INFN, Univ. di Napoli Mostra d’Oltremare, Pad. 19, 80125 Naples, Italy.

Abstract It is shown that a class of St¨ ackel separable systems is characterized in terms of a GelfandZafharevich bihamiltonian structure. Such structure is a spectral extension of a Poisson-Nijenhuis structure on the phase space of the system of contact type. It is also shown that the Casimir of the bihamiltonian structure provides the family of commuting Killing tensors found by Benenti and that characterizes orthogonal separability because of Eisenhart’s theorem. It is also shown that recently found properties of quasi-bihamiltonian systems are natural consequences of the geometry of the spectral extension of the Poisson-Nijenhuis structure.

1

Introduction

The construction and characterization of separable Hamiltonian systems has an old and extraordinary history that goes back to the works of Hamilton and Jacobi [Ja84] and that is gaining a renewed interest nowadays in the realm of integrable systems (see for instance refs. [Sk95], [Fe97], [Mo97]). Early characterizations of separable systems obtained by Levy-Civita [Le04], St¨ackel [St93], etc. were brought to maturity by the work of Eisenhart that put the problem in an intrinsic and geometric form that settled the question of the separability of geodesical motion on Riemannian manifolds, characterized it in terms of the existence of a family of quadratic Killing tensors satisfying an appropriate set of conditions, and proved that they were necessarily of St¨ackel form [Ei34], [Wo75]. S. Benenti [Be92] offered a new perspective to the problem by characterizing geometrically a class of separable mechanical systems by means of a (1, 1) tensor L verifying certain conditions and the Killing tensors of the system were obtained from the tensor L by a set of recurrence conditions. These constants of the motion however are not of the type 1

obtained from recursion operators, i.e., traces of powers of a recursion operator. Thus these results providing Killing tensors for separable systems do not fit exactly the scheme of Liouville complete integrability and recursion operators put forward along recent years (see for instance refs. Magri, Marmo, etc.). More recently E. Sklyanin suggested a new relation between the method of separation of variables and integrable systems in the realm of Lax representations. The key idea on this approach is the remark about the possibility of obtaining separating variables by looking to the poles of eigenvectors of the Lax operator. This “magic recipe” as it was called originally, had to be tested in all its details case by case, lacking the theorems that would guarantee its general applicability and comprehension. We will propose in this paper a geometrical setting that not only gives a coherent geometrical framework for separability theory as it was stated by Eisenhart and refined by Benenti but also sheds some light into Sklyanin’s proposals. The main idea is that the natural framework to understand Benenti’s systems is that of bihamiltonian systems. The bihamiltonian structure arises not in the phase of the original system but on a “Hamiltonian spectral” extension of it. The emerging structure is that of a Gelfand-Zacharevich system whose Casimirs are the sought Killing tensors described by Benenti. In this way we show that the correct setting for the description of the system is not the well-known theory of recursion operators but the Gelfand-Zacharevich theorem. *****

2

St¨ ackel, Eisenhart and Benenti systems

Separation of variables. The general scheme for separation of variables consists of a 2n-dimensional symplectic manifold (M, ω) together with n independent commuting Hamiltonians µk . Then a system of local canonical coordinates (q i , pi ) are separated if there exists n relations of the form Φj (q j , pj ; µ1 , . . . µn ) = 0,

j = 1, . . . , n,

(2.1)

where the jth separation relation depends only on the variables q j , pj and such that det

∂Φj 6= 0. ∂µk

(2.2)

Thus we can solve locally the separation relations eqs. (2.1) to obtain µk = µk (q i , pj ),

k = 1, . . . , n.

(2.3)

If H is a Hamiltonian function on M , we can find locally a complete solution of the form S = S(q i ; µk ), of the Hamilton-Jacobi equation H(q i , ∂S/∂q i ) = h,

(2.4)

with (µ1 , . . . , µn ) parametrizing locally the solutions of the equation. The action function S is the generating function of the canonical transformation taking (q, p) to action-angle variables (φ, µ) defined by, pj =

∂S , ∂q j

φj = −

∂S , ∂µj 2

H = h(µ1 , . . . , µk ),

(2.5)

hence, H(q, p) = h(µ1 , . . . , µk ).

(2.6)

We will say that a set of separated variables (q i , pi ) with respect a family of Hamiltonians µk are compatible with the Hamilton-Jacobi equation (2.4), if replacing the variables µk in the r.h.s. of Hamilton-Jacobi, eq. (2.6), by means of the solved separation relations (2.3), we obtain an identity, H(q i , pj ) = h(µ1 (q i , pj ), . . . , µn (q i , pj )). Under these circumstances, the relations eq. (2.1) allow to split the complete solution S of the Hamilton-Jacobi equation into the sum of terms of the form, S(q 1 , . . . , q n ; µ1 , . . . , µn ) =

n X

Sk (q k ; µ1 , . . . , µn ).

(2.7)

k=1

where each function Sk satisfies the ordinary differential equation, Φk (q k , ∂Sk /∂q k ; µ1 , . . . , µn ) = 0.

(2.8)

This observation justifies the terminology “additive separation of variables” given usually to the whole process. We must point it out that this presentation differs with respect to the standard approach in most textbooks in the sense that we are not considering a distinguished cotangent bundle structure on M (that always exist locally), thus we are not assuming that the separating takes place with respect to coordinates on a particular configuration space. However, we will not insist in this point at the moment because of the local character of the notion of separation of variables. Thus, turning to the standard setting we will asume that a cotangent bundle structure is given in advance in M and thus we will be considering Hamiltonian systems in T ∗ Q for Q an n-dimensional manifold. The presentation above also differs from the ones found in standard textbooks because of the use of a given family of commuting Hamiltonians and separation relations Φi . In the usual presentations, they are obtained “a posteriori” in the process of solving the equations, an chosing them correctly is a fundamental step in the process of finding the appropriate variables. We will simply emphasize their important role by making it clear that finding them is part of the separation of variables process. St¨ ackel separibility. As it was pointed out in the introduction, St¨ackel showed how to determine all separable orthogonal geodesic motions on a Riemannian manifold. Consider thus, the kynetic energy Hamiltonian for a Riemannian metric g on Q in orthogonal coordinates q i , 1 H = g ij pi pj , 2

g ij = 0,

i 6= j.

(2.9)

We can look for separability of the previous Hamiltonian using the ansatz separation relations, eqs. (2.1), given by Φk (q k , pk ; H1 , . . . , Hn ) = p2k −

n X l=1

3

ϕlk (q k )µl .

(2.10)

The regular matrix ϕ = (ϕlk (q k )) is called the St¨ackel matrix of the system. It is immediate to show that the St¨ackel relations eqs. (2.10), are compatible with the Hamiltonian H if g kk = (ϕ−1 )k1 ,

(2.11)

and µ1 = H. Moreover it is immediate to check that the family of commuting Hamiltonians, µk are given by the quadratic forms µk =

n X

(ϕ−1 )jk p2j .

(2.12)

j=1

The Hamiltonian H is thus part of a family of n quadratic independent commuting Hamiltonians Hk = µk and the St¨ackel relations can be written again as expressing the dependence of the momentum variables on the commuting Hamiltonians Hk , p2i =

n X

ϕli (q i )Hk .

k=1

For any symmetric contravariant tensor K of order r on Q there is naturally associated ˆ written to it an homogeneous function of degree r on T ∗ Q on the p variables denoted by K in natural coordinates (q i , pi ) as ˆ p) = K i1 ···ir (q)pi1 . . . pir . K(q, If G = g ij ∂/∂q i ∧ ∂/∂q j denotes the natural contravariant tensor defined by the metric g, then we will say that the tensor K is Killing if ˆ K} ˆ = 0. {G,

(2.13)

ˆ thus St¨ackel systems have n − 1 commuting Killing tensors of order 2. But H = G, Eishenhart proved [Ei34] that the converse of this result is also true. Thus if the Hamilˆ has n − 1 commuting independent Killing tensors of order 2 it is necessarily tonian H1 = G of the St¨ackel form. Thus, St¨ackel systems are completely characterized by the St¨ackel separability conditions eqs. (2.10), or alternatively, by possesing an abelian family of quadratic constants of the motion. In this sense we see that St¨ackel systems form a subclass of orthogonal separable systems, which turn to be a subset of the family of separable mechanical systems, which are contained in the broader class of completely integrable or Liouville integrable systems. ij If we denote by K1 = K1ij pi pj , ..., Kn−1 = Kn−1 pi pj , a system of Killing tensors for the ˆ l , H1 } = Hamiltonian H1 , their involutiveness condition follow from the weaker conditions {K 0, l = 1, . . . , n−1, together with the existence of simultaneous closed eigenforms αi for them, i.e., dαi = 0. (Kl − ρil G)αi = 0, The previous results were also extended to include mechanical systems with potential terms, 1 H = g ij pi pj + V (q). 2

4

Then V will be separable if l = 1, . . . , n − 1,

d(Kl dV ) = 0,

and a classification of St¨ackel systems of such type were given. (Chek this!!!!). Benenti systems. Recently, S. Benenti introduced an special class of Riemannian manifolds where it is possible to construct explicitely a family of Killing tensors for the geodetic flow, thus defining themselves a subclass of St¨ackel systems. Let (Q, g) be an n-dimensional Riemannian manifold and let L be a (1, 1) tensor on Q, i.e., an endomorphism L: T Q → T Q, verifying the conditions [Be92], [Tu92]: 1. L is diagonalizable in an open dense set of Q with eigenfunctions λ1 , . . . , λn . 2. The Nijenhuis torsion of L vanishes, NL = [L, L] = 0. 3. (L ◦ G)T = L ◦ G, i.e., the eigenvectors of L are orthogonal. ˆ G} ˆ = E G, ˆ where L ˆ denotes the function 4. L is a conformal Killing tensor, i.e., {L, ij ˆ L = L pi pj . Under these conditions the tensors Kl =

l X

(−1)l σl−j (λ1 , . . . , λn )Lk ◦ G,

l = 1, . . . , n − 1,

(2.14)

j=0

ˆ is St¨ackel by Eisenhat’s are Killing tensors in involution, hence the system defined by H = G theorem. We will call the triple (Q, g, L) Benenti systems.

3

The bihamiltonian structure of Benenti systems

The Poisson-Nijenhuis structure on T ∗ Q. We will look first for Poisson-Nijenhuis structures on T ∗ Q consisting on the canonical Poisson tensor π0 (or equivalently of the canonical symplectic form ω0 ) on T ∗ Q and a Nijehuis operator N compatible with it. In a neighborhood of a regular point, where the eigenvalues of N are independent, there always exist Nijenhuis-Darboux coordinates [Ma96] (λk , µk ) for the pair π0 , N , verifying ω0 = dλk ∧ dµk , N ∗ (dλk ) = λk dλk , N ∗ (dµk ) = λk dµk .

(3.15) (3.16) (3.17)

We select among the Poisson-Nijenhuis structures, those such that the canonical transformation from natural coordinates (q i , pi ) to Darboux coordinates (λk , µk ) is of contact type, i.e.,

5

we select those Poisson-Nijenhuis structures adapted to the fibration πQ : T ∗ Q → Q, then the coordinates (λk , µk ) must satisfy, λk = Fk (q 1 , . . . , q n ),

pk =

∂Fk µl . ∂q l

Thus, because eq. (3.16), N induces a (1, 1) tensor L on Q such that the eigenforms of L∗ are given precisely by dλk with eigenfunctions λk . The 2-form ω1 = ω0 ◦ N is exact, dω1 = dθ1 , and the symplectic potential θ1 has the simple expression θ1 = L∗ θ0 = Lji (q)pj dq i .

(3.18)

The second Poisson bracket {., .}1 on T ∗ Q is defined by {f, g}1 = ω1 (Xf , Xg ),

(3.19)

where Xf , Xg are the Hamiltonian vector fields correspoding to the functions f and g with respect to the symplectic structure ω0 . It is easy to show that {., .}1 is Poisson iff the Nijehuis torsion of L vanishes (which is precisely condition 2 for the tensor L arising in Benenti systems). The canonical commutation relations of the second Poisson structure {., .}1 are given by: ∂Lkj ∂Lki − pk , ∂q i ∂q j !

i

j

{q , q }1 = 0,

i

{q , pj }1 =

−Lij ,

{pi , pj }1 =

(3.20)

and the matrix expression of the Poisson tensor π1 corresponding to this bracket is given by,   0 −Lji !   ∂Lkj π1 =  i ∂Lki . − p Lj k i j ∂q ∂q Conversely, if we are given a Benenti system (Q, g, L), the (1, 1) tensor L allows us to think immediately of a Poisson-Nijenhuis structure adapted to the bundle structure of T ∗ Q. An important remark is in order here. The constants of the motion obtained from the Poisson Nijenhuis structure on T ∗ Q do not provide a system of Killing tensors in the sense of eq. (2.14). In fact, it is simple to see that the recursion tensor N = π1 ◦ ω0 is such that TrN k = 2TrLk , hence it defines constants of the motion which are just functions on Q. An alternative way of presenting the previous discussion without using Darboux coordinates for PN structures on T ∗ Q, is to think of the condition that the Nijenhuis tensor N preserves the bundle structure of T ∗ Q from the viewpoint of the canonical vector field defining such structure. The Liouville vector field ∆ = pi ∂/∂pi characterizes completely the bundle structure of T ∗ Q, thus preserving such structure is equivalent to the vanishing of the bracket [N, ∆] = 0, or equivalently, N is a tensor homogeneous of degree 0 on the p coordinates. Because the smoothness of N and the compatibility conditions we finally arrive that N has to have the form   Lji 0  N = ∂Lkj  . i − i pk Lj ∂q 6

We shall finally point it out the N is the complete lifting of the tensor L to T ∗ Q. The spectral extension. The geometry of the bihamiltonian structure on T ∗ Q is better captured by looking at the Poisson pencil πs = π0 +sπ1 . In fact it is even better to consider the natural extension of T ∗ Q by the real line parametrized by the scalar s, i.e., we consider the manifold M = T ∗ Q × R with coordinates (q i , pi , s). For reasons that will be apparent later, we shall denote the parameter s by E. The bihamiltonian structure on the manifold M will be constructed by extensions of the Hamiltonian structures π0 and π1 . Extensions of Poisson structures has been considered in various settings (see for instance [Ca94] for the Lie algebraic setting). Thus the extension of π0 to M will be fixed demanding on one side that T ∗ Q will be imbedded for E fixed as a symplectic submanifold of M . Then, E will be the Casimir of the first Poisson structure P0 on M and the Poisson bracket will be denoted again by {., .}0 . Thus P0 will be the direct product of the canonical Poisson structure on T ∗ Q and the trivial one in R. The extension of the second Poisson structure will be inspired by demanding that the extended one will be compatible with P0 and defining a bihamiltonian manifold of GelfandZhafarevich, i.e., a bihamiltonian manifold of maximal rank. The Casimir pencil of a GZ manifold defines an Abelian algebra of generators over a symplectic leave of one Poisson structure. Given a Hamiltonian H on T ∗ Q we can extend it as ˆ = H + f (q)E. H

(3.21)

Then we can complete the prescription for the extension of the second Poisson bracket by demanding the recurrence relation for E, that means, ˆ 0. {., E}1 = {., }H

(3.22)

The recurrence relation eq. (3.22) implies that the Casimir of P0 can be iterated, thus it becomes the first term of the Casimir of the pencil. The commutation relation for the extension of the Poisson bracket {., .}1 are, together with eq. (3.20), {q i , E}1 =

ˆ ∂H , ∂pi

{pi , E}1 = −

ˆ ∂H . ∂q i

(3.23)

We should check first Jacobi condition for {., .}1 . Because of {., .}1 is an extension of a Poisson bracket on T ∗ Q we only need to check the Jacobi condition for the following brackets {{q i , q j }1 , E}1 , {{q i , pj }1 , E}1 and {{pi , pj }1 , E}1 . Thus for the first one, we obtain, {{q i , q j }1 , E}1 + ciclyc = Lik g kj − Ljk g kj . Thus Jacobi condition for q i , q j and E amounts to the orthogonality of the tensor L (property (3)). Similar computations show that the Jacobi identity for q i , pj and E together with the second order part of Jacobi identity for pi , pj and E are equivalent to the conformal Killing 7

property of L (condition (4) above). Thus we have proved that the extended bracket {., .}1 on M is Poisson iff (Q, g, L) defines a Benenti system. The Casimir. We shall now compute the Casimir of the Poisson pencil. The structure of the Poisson tensor P1 whose associated matrix is 

P1 =

         

Lji (q)

0

∂Lj k ∂Lki pk − j ∂q ∂q i ˆ ∂H − ∂pi !

−Lij (q) ˆ ∂H ∂pi

ˆ ∂H − ∂pi ˆ ∂H ∂qi 0

          

ˆ k is quadratic in Now because of its homogeneity structure with respect to the p’s, then if H ˆ p’s and Hk+1 verifies the recurrence relation ˆ k }1 = {., H ˆ k+1 }0 , {., H

(3.24)

ˆ k+1 will also be quadratic on p’s. then H ˆ 0 = E, we will obtain P1 (dE) = P0 (dH) ˆ as desired, eq. (??). Because of this it is For H more convenient to consider E of degree 2, i.e., the dilation vector field on M will be given by ∂ ∂ ∆M = pi . + 2E ∂pi ∂E Then, it is easy to show that L∆M P1 = −P1 , hence the Poisson tensor P1 is homogeneous of degree −1 on M . An alternative way to express the previous statements is to write E as p20 , and then the spectral extension manifold M is the radial part of a complex extension of T ∗ Q1 . Summarizing the previous discussion, we have shown that the Casimir of the bihamiltonian structure defined on the spectral extension manifold M defined by a Benenti system (Q, g, L), lies in the family of quadratic functions on the p variables, thus it gives rise to a family of commuting Killing tensors and consequently the metric g is of the St¨ackel form because of Eisenhart’s theorem. Conversely, if we are given a GZ bihamiltonian structure on M homogeneous of degree −1, possessing T ∗ Q as symplectic leaves and such that the Casimir E of the first structure ˆ 0 = {., E}1 , with H ˆ the kinetic energy corresponding to a Riemancan be iterated as {., H} nian metric, then there is a family of quadratic commuting constants of the motion for the Riemannian metric and the An (almost) trivial example. We shall discuss now a simple example that serves as a guide to the construction of a quite interesting class of separable systems. We shall consider the simplest stationary reduction of 1

More details on the spectral curve associated to a bihamiltonian structure will appear elsewhere.

8

KdV, i.e., KdV3 . The stationary manifold of the t3 KdV hierarchy is given by the manifold M3 of dimension 3 parametrized by u, ux and uxx as the KdV equation is given by ∂u = uxxx + 6uux . ∂t3 The submanifold M is defined inside the state space of the KdV hierarchy by the equations uxxx = −6uux and all its differential consequences. The first KdV flow, ∂u = ux , ∂t1 will induce the flow in M3 defined by u˙ = ux ,

u˙ xx = uxxx = −6uux .

u˙ x = uxx ,

We will use the variables v to denote ux and w = uxx . The vector field X in M3 thus obtained is given by ∂ ∂ ∂ +w − 6uv . (3.25) X=v ∂u ∂v ∂w It is well-known (even if proved by hand) that the stationary KdV flows are bihamiltonian. In our case, it is easy to check that the following is a Poisson tensor for X, 



0 1 0  6u  0 P0 =  −1 , 0 −6u 0

(3.26)

or equivalently, we have the following non-zero fundamental commutation relations {u, v}0 = 1,

{v, w}0 = 6u.

ˆ = v 2 /2 − uw. The second Poisson structure is given by the The Hamiltonian for X is H tensor P1 ,   0 u v  0 6u2 + w  P1 =  −u (3.27) , −v −6u2 − w 0 or equivalently, the non-zero commutation relations defined by P1 are, {u, v}1 = u,

{u, w}1 = v,

{v, w}1 = 6u2 + w.

It is a simple exercise to prove that P0 and P1 are compatible. The Hamiltonian of the vector field X with respect to the second Poisson structure is given by E = w + 3u2 ,

(3.28)

i.e., X = P0 (dH) = P1 (dE). Moreover E is a Casimir for the Poisson tensor P0 . Hence we have obtained a bihamiltonian structure of maximal rank with the recurrence relation {., E}1 = {., H}0 .

9

Using better coordinates q, p, E, (q = u, p = v, E = w + 3u2 ) to reconcile our notations with the ones used previously, we have M = T ∗ R × R, and 







0 q p  0 E − 3q 2  P1 =  −q , 2 −p −E + 3q 0

0 1 0   P0 =  −1 0 0  , 0 0 0 and the Hamiltonian

1 H = p2 + 3q 2 − qE. 2 Notice that this system is an example of the bihamiltonian structure above with the tensor (1, 1) on Q = R given by ∂ L(q) = −q ⊗ dq. ∂q The tensor L is provides a simplest example of a (1, 1) tensor verifying conditions (1-4). It is also remarkable that the hierarchy obtained from the Casimir E is given by the vector field X alone because P1 (dH) = 0 and H is a Casimir for P1 . E P0

. 0

H P1

P0

&

.

P1

&

X

0

Moreover denoting by π0 and π1 the Poisson tensors on T ∗ R and Y = π0 (dh) = ρπ1 (dh), with ρ = q, thus the Hamiltonian vector field on the cotangent bundle is quasi-bihamiltonian in the terminology of [Mo97]. We will see that this is a common feature of the theory.

4

Bihamiltonian structures and St¨ ackel separability

The general setting. From the previous discussion a geometrical picture emerges that contains the basic ingredients of St¨ackel separability. This picture consists on a GZ bihamiltonian manifold M with Poisson brackets P0 and P1 of maximal rank. A symplectic submanifold S for the Poisson bracket P0 a the vector field Z transverse to S such that LZ P1 = X ∧ Z. The vector field Z can be (at least in a tubular neighborhood of S) written as Z = ∂/∂E where is E is the Casimir of P0 Now we shall consider a Lagrangian submanifold Q of S. By Darboux-Weinstein theorem there is a neighborhood of Q on S which is symplectically equivalent to a neigborhood of the zero section of T ∗ Q. Thus, we can restrict our atttention to such neighborhood and we can think that S = T ∗ Q without losing generality. The Casimir E will be iterabile on M and from it we will obtain a family of commuting Hamiltonians H1 , . . . , Hn in T ∗ Q that will be the generalized St¨ackel family of constants of the motion for the Hamiltonian system defined by H1 . 10

If the Poisson tensor P1 is homogeneous of degree −1 with respect to the momentum coordinates of T ∗ Q we are again in the situation described in §2. The family of commuting Hamiltonians will have the form ˆ0 = E H ˆ 1 = 1 g ij pi pj + V1 (q) + f1 (q)E H 2 (1) ˆ 2 = 1 g ij pi pj + V2 (q) + f2 (q)E . H 2 (2) ············ ˆ n = 1 g ij pi pj + Vn (q) + fn (q)E H 2 (n)

(4.29)

The homogeneity condition imply that there exists a (1, 1) tensor field L with vanishing ˆ 1 of the natural Poisson Nijenhuis torsion such that P1 is precisely the spectral extension by H ∗ bracket on T Q defined by L. We reproduce in this way all the results discussed so far. Notice however that the choice of the symplectic submanifold S and the Lagrangian submanifold is largely undetermined. That means that different choices of the submanifold Q could lead to families that will give separable systems or not. We do not know in advance for which S ˆ a are quadratic, if any. Moreover, if we find a couple Q ⊂ S such and Q the Hamiltonians H that P1 is homogeneous then automatically the hamiltonians will be quadratic and of St¨ackel form in these coordinates, hence the system will be orthogonal separable and eventually it could be solved completely. The recurrence relations. We shall compute now the family of quadratic constants of the motion obtained by this method, eqs. (4.29). Computing the different components of the vector fields Xa+1 = ˆ a+1 ) = P1 (dHa ) we will obtain the recurrence relations, P0 (dH ˆa ∂H ˆ1 ∂H ˆa ˆ a+1 ∂H ∂H = −Lji + . ∂pi ∂pj ∂pi ∂E

(4.30)

ˆ a+1 ˆ ˆa ∂H ˆ1 ∂H ˆa ∂Lkj ∂H ∂Lki ∂H j ∂ Ha = −L + − p − . k i ∂q i ∂q j ∂q i ∂q j ∂pj ∂q i ∂E

(4.31)

ˆ1 ∂H ˆa ˆ1 ∂H ˆa ∂H ∂H = . ∂q i ∂pi ∂pi ∂q i

(4.32)

!

Particularizing for quadratic Hamiltonians of the form given by eqs. (4.29), the previous eqs. (4.30-4.32) become the family of recurrence and compatibility conditions, ∂fa+1 ∂fa ∂f1 = −Lij i + fa i . j ∂q ∂q ∂q

(4.33)

∂Va+1 ∂Va ∂V1 = −Lij i + fa i . j ∂q ∂q ∂q

(4.34)

kj ij ij g(a+1) = −Lik g(a) + fa g(1) ,

(4.35)

together with the compatibility conditions, ∂fa kl ∂f1 kl g(1) = k g(a) , k ∂q ∂q 11

(4.36)

∂Va kl ∂V1 kl g(1) = k g(a) , k ∂q ∂q ij ∂g(a)

g kl ∂q k (1)

and

ij ∂g(a+1)

∂q k

=−

ij ∂g(a)

Ll ∂q l k

=

− fa

ij ∂g(1)

∂q k

ij ∂g(1)

∂q k



(4.37)

kl , g(a)

lj 2g(a)

(4.38) ∂Lil ∂Lij − ∂q k ∂q l

!

.

(4.39)

The consistency of the compatibility conditions is easily proved by induction as follows. It is obviously true for a = 1. Let us consider eq. (4.36) for a + 1, then we get using the recurrence relation eqs. (4.33), (4.35) !

 ∂f1  ∂f1 kl kj ij − k −Lik g(a) + fa g(1) + fa i g(1) ∂q ∂q ∂q ∂fa ik l ∂f1 ik l = − i (g(1) Lk ) + i (g(a) Lk ) ∂q ∂q ! ∂fa ik ∂f1 ik = − i g(1) + i g(a) Llk ) = 0 ∂q ∂q

∂fa+1 kl ∂f1 kl − = g g (1) ∂q k ∂q k (a+1)

∂fa −Lij i

The Leverrier-Newton method. As we pointed it out in the previous paragraph, the recurrence relations eqs. (3.24) can be solved using an adapted version of the Leverrier-Newton recurrence formulae for computing the characteristic polynomial of a given matrix [Wi65]. We shall adapt first this method to compute the spectral curve of a Nijenhuis torsionless (1, 1) tensor field L on a manifold Q. Let L be a diagonalizable endomorphism of the tangent bundle of the manifold Q possessing a real spectrum given by the eigenfunctions λ1 , . . . , λn . We define the canonical symmetric 1-forms αk = dσk , where σk are the canonical symmetric functions of the eigenfunctions λa . If the tensor L has vanishing Nijenhuis torsion, it is well-known that in the neighborhood of points wehere the eigenfunctions are independent, it can be brought into the normal form L = λa

∂ ⊗ dλa . ∂λa

If we denote by pk = TrLk the traces of powers of the tensor L, then the Leverrier method for finding the characteristic polynomial pL (z) = z n + a1 λn−1 + · · · + an−1 z + an of L, allows to compute recursively its coefficients ak by means of Newton’s formula, a1 = −p1 ,

, 2a2 = −(p2 + a1 p1 ),

kak = −(pk + a1 pk−1 + · · · + ak−1 p1 ),

A simple computation shows that 1 L∗ (α1 ) = dTrL2 , 2

(L∗ )k (α1 ) = 1/kdTr(L∗ )k , 12

...,

...

(4.40)

thus because ak = (−1)k σk , we have α1 = dσ1 = −da1 = dp1 = d Tr L, 1 1 α2 = dσ2 = da2 = − d(p2 + a1 p1 ) = − (dTrL2 − 2p1 dTrL) = −L∗ α1 + σ1 α1 , 2 2 and, in general we have, αk = −L∗ αk−1 + σk−1 α1 . (4.41) Formula (4.41) allows to compute iteratively αk , to obtain, (−1)a αa+1 = [(L∗ )a − σ1 (L∗ )a−1 + σ2 (L∗ )a−2 + · · · + (−1)a σa ]α1 ,

(4.42)

which is the form that Newton’s formula eq. (4.40) takes now. We will recognize in the formulas above, eq. (4.41), the recurrence relation eq. (4.33). If we write it as a recurrence relations for the family of 1-forms dfa =

∂fa i dq , ∂q i

they become, dfa+1 = −L∗ (dfa ) + fa df1 ,

(4.43)

Notice that d(dfa ) = 0, implies that d(L∗ df1 ) = 0. Then a simple computation shows that if df1 = ai dλi , then ∂ai /∂λj = 0 and all solutions of eqs. (4.43) can be computed iteratively from f1 = a1 λ1 + · · · + an λn . We will take for what follows that choice of a1 = · · · = an = 1, that gives the solution fa = σa . This solution could have been obtained directly comparing eq. (4.41) with (4.43). With respect to the recurrence relations defining the potentials Va for the St¨ackel system, converting them again as before as recurrence relations for 1-forms, dVa =

∂Va i dq , ∂q i

we get again, dVa+1 = −L∗ dVa + σa dV1 ,

(4.44)

and iterating we obtain the analog of eq. (4.42) for dVa , (−1)a dVa+1 = [(L∗ )a − σ1 (L∗ )a−1 + . . . + (−1)a σa ]dV1 =

a X

(−1)k σk (λ1 , · · · , λn ) (L∗ )a−k dV1 .

k=0

(4.45)

Similarly, the recurrence relations for the metrics ga , eqs. (4.35), written as recurrence relations for symmetric contravariant 2-tensors, Ga = gaij ∂/∂q i ⊗ ∂/∂q j , look like, G(a+1) = −LG(a) + σa G(1) , 13

thus, iterating the formula we will obtain a

(−1) G(a+1) =

a X

(−1)k σk (λ1 , · · · , λn )La−k G(1) ,

(4.46)

k=0

which apart from a overall sign, are precisely the Killing tensors Ka obtained by S. Benenti, eq. (2.14). We have solved, the recurrence relations, and found that the extended Hamiltonians are given explicitely by ˆa = 1 G ˆ (a) + Va + σa E. H (4.47) 2

Quasi-bihamiltonian systems and separability. ˆ n+1 = 0. In fact, in the recurrence An important observation arises from the fact that H relations eq. (4.33), we obtain dfn+1 = [(L∗ )n − σ1 (L∗ )n−1 + · · · + (−1)n σa ]α1 which vanishes identically by the Cayley-Hamilton theorem. A similar statement holds for ˆ n is a Casimir for P1 because dVn+1 and Gn+1 . Thus we can conclude that H ˆ n ) = P0 (dH ˆ n+1 ) = 0. P1 (dH We can then complete the picture of the Casimirs of the pencil of Poisson structures P0 , P1 . ˆ0 E=H P0

. 0

ˆ1 H P1

P0

&

. X1

ˆ n−1 H

··· P0

& X2

. · · · Xn−1

ˆn H P1

P0

&

. Xn

P1

& 0

ˆ n is a Casimir of the Poisson tensor P1 has another interesting conseThe fact that H quence. In fact the following computation ˆ n ) = (π1 + X ˆ ∧ ∂ )(dHn + Edσn + σn dE)) = 0 = P1 (dH H1 ∂E π1 (dHn ) + Eπ1 (dσn ) + XHˆ 1 (dHn ) ∧

∂ − σn XHˆ 1 , ∂E

but XHˆ 1 = XH1 + EXσ1 . Thus, restricting the previous equation to the symplectic submanifold E = 0 in T ∗ Q × R, this is restricting the previous equation to the original phase space, we obtain, 0 = π1 (dHn ) + XH1 (dHn ) ∧

14

∂ − σ n XH 1 , ∂E

(4.48)

but {H1 , Hn }0 = 0, hence eq. (4.48) becomes π1 (dHn ) = σn π0 (dH1 ),

(4.49)

and the system with Hamiltonian H1 is Pfaffian quasi-bi-hamiltonian on T ∗ Q in the terminology of [Mo97]. Furthermore, the previous discussion also explains why certain classes of quasi-bi-hamiltonian systems are separable. In fact, what we have proved is that the bihamiltonian systems of St¨ackel type we have been describing above, that in particular include Benenti systems, are orthogonal separable in an appropriate cotangent bundle and in addition the Hamiltonians H1 and Hn of the hierarchy are quasi-bi-hamiltonian.

5

More examples

The separability of the stationary reduction of KdV. It is convenient to close this paper exhibiting a non-trivial example combining all the features we have discussed before. It seems particularly appropriate because of the discussion in the preceeding section and the example of KdV3 discussed above, to bring the attention to higher order stationary reduction KdV flows. Thus for instance if we consider the seventh-order KdV stationary reduction, we arrive to the KdV-Novikov system defined on the 7-dimensional manifold M with coordinates (q1 , q2 , q3 , p1 , p2 , p3 , E). This system is a GZ bihamiltonian system with Poisson structures P0 , P1 defined by the matrices 



0 I 0   P0 =  −I 0 0  , 0 0 0 with





q1 −1 0   L =  2q2 q1 q3  , q3 0 0

and Hamiltonian





0 −L a   P1 =  Lt M b  , −at −bt 0 

(5.50)



0 p2 0   M =  −p2 0 −p3  , 0 p3 0

ˆ 1 = 1 (p1 p2 + p2 ) + V1 (q1 , q2 , q3 ) − 2q2 E, H 3 2

(5.51)

(5.52)

with potential function, 5 5 1 1 V1 = − q14 + q12 q2 + q1 q32 − q22 . 8 2 2 2 The St¨ackel family of Hamiltonians are obtained after a tedious computation and are given by ˆ 2 = H2 + (q 2 + 2q2 )E, H ˆ 3 = H3 − q 2 E, H (5.53) 1 3 and

1ˆ H2 = G 2 + V2 , 2

1 ˆ3 H3 = G + V3 , 2

(5.54)

with G2 = LG1 − 2q1 G1 ,

G3 = L2 G − 2q1 LG + (q12 + 2q2 )G, 15

(5.55)

dV2 = L∗ dV1 − 2q1 dV1 , Hence H2 = H3 =

dV3 = [(L∗ )2 − 2q1 L∗ + (q12 + 2q2 )]dV1 .

(5.56)

 1 2 p1 + 2q1 p1 p2 − 2q2 p22 − 2q3 p2 p3 + 2q1 p23 + V2 (q1 , q2 , q3 ), 2

 1 2 2 q3 p2 − 2q3 p1 p3 − 2q2 q3 p2 p3 + (q12 + 2q2 )p23 + V3 (q1 , q2 , q3 ). 2

Hydrodynamic systems. Recently it has been noted the relation between the integrability of a system of hydrodynamic type qti = vji (q)qxj , (5.57) and the St¨ackel separability of certain finite dimensional Hamiltonian systems [Fe97]. Tsarev [Ts85] proved Novikov’s hypothesis on the integrability of systems of hydrodynamic type stating that the system (5.57) is integrable if there exist Riemann invariants for it. If we think of the functions vji as the component functions of a (1, 1) tensor L on the manifold Q, Novikov’s condition is equivalent to the diagonalization of L in a local chart. Thus if L has vanishing Nijenhuis torsion, it will define integrable systems of hydrodynamical type. But such a tensor L will also define a Benenti system for a suitable metric g. Then considering the Hamiltonians H given by the kinetic energy of the metric g and F the second Killing tensor in the family defined by L we will obtain the St¨ackel separable Hamiltonians used in [Fe97] used to integrate the system (5.57). Moreover, the system of hydrodynamic type before, will belong to a family of n − 1 commuting flows defined by the systems of hydrodynamic type corresponding to the pairs of Hamiltonians H1 , Ha in the family of quadratic Killing tensors (2.14).

Ackwnoledgements The author AI wishes to ackwnoledge the partial financial support provided by CICYT under the programme PB95-0401 as well as the NATO collaborative research grant 940195.

16

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L. P. Eisenhart. Separable systems of St¨ackel. Ann. Math., 35, 284-305 (1934).

[Fe97]

E.V. Ferapontov, A.P. Fordy. Separable Hamiltonians and integrable systems of hydrodynamic type. J. Geom. Phys., 21, 169-182 (1997).

[Ja84]

C.G. Jacobi. Vorlesungen u ¨ber dynamic, Gesammelte Werke (Suplementband) (1884).

[Ma96] F. Magri, T. Marsico. Some developments of the concepts of Poisson manifolds in the sense of A. Lichnerowicz, Gravitation, Electromagnetism and Geometrical Structures. Ed. G. Ferrarese, Pitagora, Bologna, 207-222 (1996). [Mo97] C. Morosi, G. Tondo. Quasi-bi-Hamiltonian systems and separability. J. Phys. A: Math. & Gen., 30, 2799-2806 (1997). [Wo75] N. M. J. Woodhouse. Killing tensors and the separation of the Hamilton-Jacobi equation. Commun. Math. Phys., 44, 9-38 (1975). [Le04]

T. Levi-Civita. Sulla integraziones della equazione di Hamilton-Jacobi per separazione di variabili, Math. Ann. 59, 383-397 (1904).

[St93]

¨ P. St¨ackel. Uber die Bewegung eines Punktes in einer n-fachen Manigfaltigkeit. ¨ Math. Ann. 42, 537-563 (1893); ibid. Uber Quadratizche Integrale der Differentialgleichungen der Dynamik. Ann. Math. Pura Appl., 26, 55-60 (1897).

[Sk95]

E. Sklyanin. Separation of Variables, Progr. Theor. Phys., **, 35-60 (1995).

[To95] G. Tondo. On the integrability of stationary and restricted flows of the KdV hierarchy. J. Phys. A: Math. & Gen., 28, 5097-5115 (1995). [Ts85]

S.P. Tsarev. On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Dokl. Soc. Acad. Sci. 282, 3 (1985).

[Tu92] F.J. Turiel. C.R.A.S. Paris, 315, 1085-88 (1992). [Wi65] J. H. Wilkinson. The algebraic eigenvalue problem. Oxford Univ. Press (1965).

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