Separation of variables for Lax systems: a bihamiltonian point of view 1
Gregorio Falqui SISSA, Via Beirut 2/4, I-34014 Trieste, Italy E–mail:
[email protected] Abstract In this paper we will present some results on a recently developped approach to the problem of Separation of Variables for Hamilton–Jacobi equations. In particular, we will discuss a remarkable connections between the theory of separation of variables for systems admitting a Lax representation with spectral parameter and the theory of separation of variables for bihamiltonian systems expounded in [6], to which we refer also for a more complete bibliography. The paper is organized as follows: in Section 1 we will briefly discuss some points on the meaning of Separation of Variables, and sketch the main points of the classical and the more recent Lax settings. Then we will present the main points of the bihamiltonian scheme. Finally, in Section 3 we will apply the scheme to a specific example in the loop space of sl(3).
1
Introduction
The classical theory of Separation of Variables (SoV) for the Hamilton–Jacobi (H-J) equations provides the most effective way of effectively solve the equations of the motion associated with a particular Hamiltonian H. In addition to the classical results of St¨ackel and Eisenhart concerning the SoV problem for Hamiltonian functions quadratic in the momenta more recently developped by Benenti (see, e.g., [9]), in the last years this topic has received a renewed attention, due to the work of various groups (Dubrovin, Krichever and Novikov [4], Sklyanin and coworkers [12], Harnad, Hurtubise and coworkers [1], just to quote a few). The main aims of such research works are two: The first is to integrate via algebro-geometrical means Hamiltonian systems admitting a Lax representation, that are related with Soliton Equations and/or Statistical Mechanical models. The second is to apply such techniques to the quantum mechanical counterparts of these systems. 1
Work partially supported by INdAM–GNFM and the Italian M.U.R.S.T. under the research project Geometry of Integrable Systems.
1
The SoV problem one of the arenas where the applicative aspects of analytical mechanics are emphasized. Indeed, it is well known that there is no question on the possibility of solving the H-J equations locally in phase space. The real point is to provide an efficient (and possibly algorithmic way) to solve the equations of motion. Actually a sensible theory of SoV should start from the following data: 1. A class of manifolds M ; 2. A class of canonical coordinates (p, q) on M ; 3. A class of systems (i.e. a class of Hamiltonian functions on M ); and give answers to the following points: a) Give a separability test to ascertain whether the H-J equations associated with the “choosen” Hamiltonians P admit and additively separated complete integral of the form W (q, α) = Wi (qi , α); b) Give algorithms to produce separation coordinates and separation relations, that is, give algorithms to actually solve the H-J equations.
In the classical SoV theory the data are given by natural Hamiltonian systems, that is Hamiltonian vector fields defined the cotangent bundle of a configuration space Q endowed with canonical coordinates (p, q), and the separability tests are the classical Levi-Civita test and the St¨ackel test. We recall that the latter considers, in orthogonal coordinates on Q, the Hamiltonian H(q, p) =
1X 2 gi (q)p2i + V (q1 , . . . , qn ) 2
(1.1)
and claims that H(p, q) is separable if and only if there exists as invertible matrix S(q1 , . . . , qn ) and a vector U (q1 , . . . , qn ) such that the rows of S and U depend only on the corresponding coordinate (i.e., the elements S1j and U1 depend only on the first coordinate q1 , and so on and so forth), and H(p, q) is among the solutions (H1 , . . . , Hn ) of the linear system n X 1 2 Sij (q)Hj . (1.2) pi = Ui (q) + 2 j=1
In the Lax setting for a Hamiltonian system with n degrees of freedom, the cornerstone of the theory is the Lax representation ˙ L(λ) = [M (λ), L(λ)]
2
(1.3)
of the equations of motion, entailing that the associated spectral curve, that is the algebraic curve R(λ, µ) = det(µ − L(λ)) = 0 (1.4) is left invariant by the flow. One assumes that all spectral invariants of L(λ), are actually in involution, a fact essentially equivalent [2] to the existence of a r-matrix structure compatible with the Lax map L(λ). Here compatibility means that the the Poisson brackets of the matrix elements [L]ij , written symbolically as {L(λ1 ) ⊗, L(λ2 )} are given by the commutator {L(λ1 ) ⊗, L(λ2 )} = [r12 (λ1 − λ2 ), L(λ1 ) ⊗ 1 + 1 ⊗ L(λ2 )] ,
(1.5)
where r12 is an N 2 × N 2 matrix, N being the order of L(λ). The class of coordinates considered in the SoV problem in this context are called algebro–geometrical coordinates. They are canonical coordinates given by projections on the coordinate axes λ, µ of points Pi lying on the spectral curve (1.4); In such a case, the separation relations are “trivially” given by the equation of the curve R(λ, µ) = 0, and the integration of the HJ equations is reduced [1] to a Jacobi inversion problem. The “magical Sklyanin’s recipe” [12] to find them is the following: Consider a Baker Akhiezer (BA) function for L, i.e a vector valued function ψ solving L(λ)ψ = µψ,a covector κ, and normalize ψ imposing that (generically),hκ, ψi = 1 and find the (candidate) algebro– geometrical canonical coordinates as the projections of the poles of such normalized BA functions on the coordinate planes in C2 . The word “candidate” above refers to the fact that (in general) the properties of such poles of being in the correct number (exactly matching the number of degree of freedom n of the problem at hand) and giving rise to canonical coordinates depends in general on the normalization, and on the particular Lax pair of the system.
2
A bihamiltonian scheme for SoV
In this section we will briefly present a further scheme for SoV, which is rooted in the theory of bihamiltonian manifolds [11, 3, 7, 5, 6]. We want to study Hamiltonian vector fields defined on symplectic manifolds (M, ω) endowed with a second compatible Poisson bracket {, }2 . It is a standard result in bihamiltonian geometry that the existence of such a compatible bracket is equivalent to the existence of a (1, 1) tensor N (the Nijenhuis tensor) such that the second bracket {, }2 is given by a closed two–form ω ′ , {F, G}2 = ω ′ (XF , XG ) = ω(N XF , XG ), (2.1) where XF is the Hamiltonian vector field associated by means of the symplectic form ω with the function F . We will call such manifolds ωN –manifolds. 3
Such a geometrical structure endows M with a class of coordinates, the Darboux– Nijenhuis (DN) coordinates. They are local coordinates (λj , µj )j=1,...,n which enjoy the following two properties: i) ω takes the canonical form ω=
n X
dλi ∧ dµi ;
i=1
ii) the adjoint Nijenhuis operator N ∗ takes the diagonal form N ∗ dλj = λj dλj ,
N ∗ dµj = λj dµj .
DN coordinates exist on a ωN manifold where N has n functionally independent eigenvalues. In this case the coordinates λj can be computed algebraically as the roots of the minimal polynomial of N , C(λ) = Det N − λ1
1
2
n
=λ −
n X
ci λn−1 .
(2.2)
i=1
The complementary coordinates µ must be computed (in general) by a method involving quadratures. A useful criterium to find them following [6]: Proposition 2.1 Let Y = −Xc1 be the Hamiltonian vector field associated with the first non trivial coefficient of the minimal polynomial of N , and let F (λ) be a function on M which depends as well on the parameter λ and satisfies Y (F (λ)) = 1.
(2.3)
Then F (λ) “interpolates” the complementary coordinates µj , that is, µj = F (λj ). So far we have introduced the class of manifolds and coordinates we are going to consider. We want now to characterize the class of Hamiltonians that are separable within this geometrical framework. Actually, the characterization is surprisingly simple, although the proof of separability requires some work. Indeed, it holds [6]: Proposition 2.2 Let H be a smooth function on a 2n dimensional ωN –manifold M . Then H is separable in DN coordinates iff there exist further n − 1 functions H2 , . . . , Hn such that the n functions {H1 ≡ H, H2 , . . . , Hn } are in involution with respect to both Poisson brackets. It is important to remark that the conditions stated in this Proposition above are tensorial, so that they can be checked in any coordinate systems, without having to compute the coordinates themselves. 4
The next obvious question we have to answer is to provide a significant class of systems which fulfill these requirements. We address the interested reader to [6] for the proof that separable systems ` a la St¨ ackel admit a reformulation in these terms. In this paper we want to discuss a newer source of separable systems, that is the so–called systems of Gel’fand–Zakharevich type, and make the connection with the Sklyanin’s theory of SoV in the case of the loop algebra of SL(3).
2.1
GZ systems
We now apply the separability scheme we have discussed so far to a different geometric situation, that is, to ωN manifolds which are embedded into more generic bihamiltonian manifold. In particular, we shall consider the case of complete torsionless Gel’fand–Zakharevich bihamiltonian manifolds [8]. They are defined as the datum of: • a bihamiltonian manifold M, {, }λ , that is, a manifold M endowed with a linear pencil of Poisson brackets {·, ·}λ = {·, ·}Q − λ{·, ·}P or equivalently with a linear pencil Pλ = Q − λP of Poisson tensors, where {f, g}λ = hdf, Pλ dgi. • a collection of k polynomial Casimir functions H (1) , . . . , H (k) , that is, a collection of degree nj polynomials H (j) (λ) =
nj X
(j)
Hi λnj −i
i=0
such that 1. For all j = 1, . . . , k, Pλ (dH (j) (λ)) = 0. 2. n1 + n2 + · · · + nk = n, with dimM = 2n + k. (j)
3. The differentials {dHs }j=1,...k; s=0,...,nj are functionally independent. The collection of n bihamiltonian vector fields (j)
(j)
(j)
Xk = P dHk+1 = QdHk
(2.4)
associated with the Casimir polynomials H (j) is called the GZ system associated with the given GZ manifold. Standard arguments from the theory of Lenard–Magri chains (j) show that all the coefficients Hi pairwise commute both with respect to {·, ·}P and with respect to {·, ·}Q . Hence a possible strategy to fulfill the requirements of Proposition 2.2 is to try to reduce the Poisson brackets from M to a suitable submanifold where one of the two brackets is symplectic. Indeed a sufficient condition is summarized by the following: 5
Proposition 2.3 Let S be symplectic leaf of P , and suppose that in a neighbourhood of S ⊂ M we can find a decomposition T M = T S ⊕ D with D satisfying: 1. If f and g are functions invariant under D, then the functions {f, g}P and {f, g}Q are invariant as well under D. 2. D is generated by k vector fields Z1 , . . . , Zk satisfying Za (Zb (H (c) (λ)) = 0. Then: a) S is an ωN manifold, with Poisson brackets induced by those of M ; b) The determinant of the matrix Gab = Za (H (b) (λ)) is the minimal polynomial of the Nijenhuis tensor N on S So, property a) insures that the commutativity properties of the GZ Hamiltonian is preserved on S as well. But since S is obtained by fixing the values of the k (j) Casimirs of P0 , H0 , j = 1, . . . , k, the remaining ones constitute a set of n independent Hamiltonians in involution with respect to both (reduced) brackets. Then we can apply the bihamiltonian procedure for SoV, since we are indeed within the hypotheses of Proposition 2.2. In the next section we will apply this scheme to a polynomial pencil of matrices related to stationary reduction of the Boussinesq hierarchy.
3
An example in the Loop Algebra of sl(3)
In section we will present an application of the results contained in Section 2.1, concerning the case of a GZ system with two polynomial Casimirs, which can be show to represent represent the reduction to t7 stationary of the Boussinesq hierarchy. To keep the paper within a reasonable length, we will mainly report the results, skipping a lot of details. We will mainly be interested, to show how the bihamiltonian scheme of SoV, applied to such a case in which the system admits a Lax representation, gives rise to algebro–geometrical coordinates in the sense of Sklyanin. We consider the space M of matrices of the form: L(λ) = λ3 A + λ2 X2 + λX1 + X0 with
0 0 0 A= 0 0 0 1 0 0
0 1 0 X2 = p 0 1 q 0 0
−p b1,1 −1/2 q + d1 X1 = b2,1 b2,2 − b1,1 −1/2 q − d1 b3,1 b3,2 −b2,2
6
and
S + d2 T a1,1 X0 = a2,1 a2,2 − a1,1 S − d2 a3,1 a3,2 −a2,2
,
S = −b2,2 p − 1/2 b3,1 + 1/2 d1 2 + 1/2 pb1,1 + 3/8 q 2 T = −b2,1 + 3/2 pq − pd1 − b3,2 .
M is subspace of the affine hyperplane A of degree 3 matrix polynomials with highest coefficient equal to A, contained in the full loop space of sl(3); it inherits a bihamiltonian structure Pλ = Q − λP as a reduction [10] of the two Poisson structures defined on A via their Hamiltonian vector fields: ∂F ∂X0 [X1 , ·] [X2 , ·] [A, ·] ∂F XF = (3.1) := PdF 0 [X2 , ·] [A, ·] ∂X1 ∂F [A, ·] 0 0 ∂X2 and
YF =
−[X0 , ·] 0 0 0 [X2 , ·] [A, ·] 0 [A, ·] 0
∂F ∂X0 ∂F ∂X1 ∂F ∂X2
:= QdF.
(3.2)
One can prove the following facts: 1. M is a manifold of dimension 14, with a global set of coordinates {p, q, d1 , d2 , a1,1 , . . . , a3,1 , b1,1 , . . . b3,1 }. 2. The coefficients of the characteristic polynomial of L ∈ M are polynomial Casimirs of the reduced pencil Pλ on M; The bihamiltonian vector fields they generate can be cast in a Lax form: ∂H ]. L˙ = [L, ∂L
(3.3)
3. The coefficients of the expansion in powers of µ of the characteristic polynomial det(µ − L) of L are: 1 1 H (1) = TrL2 , H (2) = TrL3 , 2 3 7
(3.4)
If we subtract the trivial term −λ7 from H (3) , we see that they have the expansion (1)
(1)
(1)
(2)
H (1) = H0 λ2 + H1 λ + H2 ;
(2)
(2)
H (2) = H0 λ4 + H1 λ3 + · · · + H4
(3.5) (1)
(2)
A direct inspection shows that the eight coefficients {H0 , . . . , H4 } are functionally independent. Since (generically) rk(P ) = rk(Q) = 12 we can conclude that M is a complete GZ manifold of dimension 14 and rank 2, endowed with two sequences of bihamiltonian vector fields of length 2 and 4. 4. The pencil Pλ can be reduced with respect to the distribution D generated by the vector fields Z1 := L˙ = [Z1 ]i,j = δ2,i δ1,j
Z2 := L˙ = [Z2 ]i,j = δ3,i δ1,j .
(3.6)
5. The distribution D is transversal to the symplectic leaves of the reduced Poisson tensor P , and the second derivatives Zi (Zj (H (l) (λ))) vanish. So we know that the GZ system at hand is separable in DN coordinates, once we fix (1) (2) the values of H0 and H0 which are Casimirs of P . The next topic deals with the computation of the DN coordinates, and the determination of the separation equations, making use of Propositions 2.3 and 2.1 The problem of giving closed expressions to the µ and λ coordinates can be solved together with the determination of the separation relations by the following arguments. Developing Det(µ1 − L(λ)) along the first column we have: Det(µ1 − L(λ)) = (µ − L1,1 )∆1,1 − L2,1 ∆2,1 − L3,1 ∆3,1 .
(3.7)
The crucial point is that, thanks to the specific form of the matrices Zk , the equality ∆k+1,1 = Zk (H (1) )µ + Zk (H (2) ). holds. Indeed, deriving the expression Det(µ1 − L(λ)) = µ3 − λ7 − H (1) µ − H (2) with respect to Zk we get ∂ Det µ1 − L(λ) Zk (µ) − Zk (H (1) )µ + Zk (H (2) ) , ∂µ 8
(3.8)
while, taking the same derivative of its development along the first column (3.7), we get ∂ Det µ1 − L(λ) Zk (µ) − ∆k+1,1 ∂µ
since Zk (Lj,1 ) = δj,k+1 , and all the other terms do not explicitly depend on the coordinates a3,1 , a3,2 . Taking the development in powers of µ of the system {∆11 = 0, ∆1,2 = 0, ∆1,3 = 0}, whose solutions {µi , λi } (if any) clearly lie on the spectral curve (3.7) we arrive at the linear system A1 A0 1 0 Z (H (1) ) Z (H (2) ) 1 1 0 Z2 (H (1) ) Z2 (H (2) )
2 µ µ = 0, 1
(3.9)
where the Ai ’s are the coefficients of µi of ∆1,1 (whose explicit expression is irrelevant in this context). So the condition for this system to admit non vanishing solutions is exactly DetG(λ) = 0; that is, the projection on the λ–plane of the solutions to the system (3.9) are the eigenvalues of the Nijenhuis tensor; The expansion in λ of the matrix G is the following: 2 O(λ3 ) λ + O(λ) G∼ , O(λ) λ4 + O(λ)
so its determinant is given by the expression Det(G) = λ6 −
6 X
ci λ6−i .
(3.10)
i=1
One can check that the coefficients ci are functionally independent, and so are the roots λi of Det(G), which provide the first half of DN coordinates (the λ coordinates). If we remove the one of the dependent equations (say the second) from this system and apply the Cramer’s rule, we easily find for that the pairs {λj , muj } solving the system (3.9) have the expression {λi , µi }, with DetG(λj ) = 0 and Z2 (H (2) ) µi := µ(λi ) = − . Z2 (H (1) ) λ=λi 9
(3.11)
To finish our job, we have to show that actually the rational function that interpolates the µi coordinates gives rise to the second half of the DN coordinates. To this end we rely on Proposition 2.1. Writing explicitly formula (3.8) as −L12 −L13 ∆3,1 = Z2 (H (1) )µ + Z2 (H (2) ) = Det (3.12) µ − L22 −L23 and taking into account that the action of the vector field Y = −Xc1 on the Lax matrix L(λ) is −2 0 0 Y (L(λ)) = (3.13) 0 1 0 , 0 0 1 it follows that indeed, if µ(λ) is the rational expression (3.11), then Y (µ(λ)) = 1.
Acknowledgments This paper is an outgrowth of joint work with F. Magri and M. Pedroni.
10
References [1] M. R. Adams, J. Harnad, J. Hurtubise, Darboux Coordinates and LiouvilleArnold Integration in Loop Algebras. Commun. Math. Phys. 155 (1993), 385413. [2] O. Babelon and C. M. Viallet, Hamiltonian Structures And Lax Equations, Phys. Lett. B237 (1990) 411–420. [3] M. Blaszak, On separability of bi–Hamiltonian chain with degenerated Poisson structures, Jour. Math. Phys. 39 (1998), 3213–3235 [4] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, Integrable Systems I, Enciclopædya of Mathematics, Dynamical System IV (V. I. Arnold, ed.), SpringerVerlag, Berlin, 1990. [5] G. Falqui, F. Magri, G. Tondo. Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy. Theor. Math. Phys., 122, 176–192 (2000). [6] G. Falqui, F. Magri, M. Pedroni, Separation of variables and bi-Hamiltonian manifolds SISSA Preprint (2001). [7] G. Falqui, F. Magri, M. Pedroni, J. P. Zubelli A Bi-Hamiltonian theory for stationary KdV flows and their separability, Reg. Chao. Dyn., 5 33-51, (2000) [8] I. M. Gel’fand, I. Zakharevich, Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures math-ag/9903080, to appear in Selecta Mathematica. [9] E.G. Kalnins, Separation of variables for Riemannian spaces of constant curvature. Longman (Harlow, UK), 1986. [10] J. E. Marsden, T. Ratiu, Reduction of Poisson Manifolds. Lett. Math. Phys. 11 (1986), 161–169. [11] C. Morosi, G. Tondo, Quasi–Bi–Hamiltonian systems and separability, J. Phys. A.: Math. Gen. 30 (1997), 2799-2806. [12] E. Sklyanin, Separations of variables: new trends, Progr. Theor. Phys. Suppl. 118 (1995), 35–60.
11
