Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems

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1 Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems Artur Sergyeyev 1 and Maciej B laszak 2 1 Mathematical Institute, Silesian University in Opava, arxiv: v3 [nlin.si] 26 Sep 2007 Na Rybníčku 1, Opava, Czech Republic 2 Institute of Physics, A. Mickiewicz University Umultowska 85, Poznań, Poland Artur.Sergyeyev@math.slu.cz and blaszakm@amu.edu.pl September 26, 2007 We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized transform preserves the Liouville integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stäckel transform. Keywords: multiparameter generalized Stäckel transform, integrable systems, separation curves, reciprocal transformation Introduction The Stäckel transform [9], also known as the coupling-constant metamorphosis [15], cf. also [17, 18, 19, 28, 29] for more recent developments, is a powerful tool for producing new Liouville integrable systems from the known ones. This is essentially a transformation that maps an n-tuple of functions in involution on a 2n-dimensional Poisson manifold into another n-tuple of functions on the same manifold, and these n new functions are again in involution. In its original form the Stäckel transform affects just one coupling constant which enters the Hamiltonian linearly and interchanges this constant with the energy eigenvalue, see [9, 15]. In the present paper we introduce a multiparameter generalization of the classical Stäckel transform, which, just like its known counterpart, enables us to generate new Liouville integrable systems from the known ones or bring known integrable systems into a simpler form. This multiparameter generalized Stäckel transform allows for the Hamiltonians being nonlinear functions of the parameters. This property considerably increases the power of the transform in question; for instance, it turns out that the separation of variables in the Hamilton Jacobi equation can be considered as a particular case of the multiparameter generalized Stäckel transform, see the discussion in the next section for details. 1

2 Moreover, we show that the induced transformations for equations of motion are nothing but reciprocal transformations. This generalizes to the multiparameter case the earlier results of Hietarinta et al. [15] on the one-parameter Stäckel transform. The significance of reciprocal transformations in the theory of integrable nonlinear partial differential equations is well recognized. These transformations were intensively used in the theory of dispersionless (hydrodynamic-type) systems as well as the theory of soliton systems, see e.g. [23, 25] and references therein. On the other hand, some particular examples of transformations of this kind for finite-dimensional Hamiltonian systems are also known, for instance the Jacobi transformation, see [21] and a recent survey [29]. The reciprocal transformations of somewhat different kind have also appeared in [15, 30, 28]. In the present paper we consider reciprocal transformations for the Liouville integrable Hamiltonian systems in conjunction with the generalized Stäckel transform and, in contrast with the earlier work on the subject, we concentrate on the multi-time version of these transformations. In fact, as we show below, these transformations, when applied to the equations of motion of the source system, in general do not yield the equations of motion for the target system, unless we restrict the equations of motion onto the level surfaces of the corresponding Hamiltonians, see Propositions 2 and 3 below for details. We show that two Liouville integrable systems related by an appropriate multiparameter generalized Stäckel transform for the constants of motion are related by the reciprocal transformation for the equations of motion restricted to appropriate Lagrangian submanifolds, see e.g. Ch.3 of [10] and references therein for more details on the latter. Moreover, we present a multitime extension of the original reciprocal transformation from [15], and study the applications of this extended transformation to the integration of equations of motion in the Hamilton Jacobi formalism using the separation of variables, cf. [9]. As a byproduct, we present reciprocal transformations for a large class of dispersionless, weakly nonlinear hydrodynamic-type systems. These Killing systems [8] are intimately related to the Stäckelseparable systems [13, 14, 5]. In the rest of the paper we consider the relations among classical Liouville integrable Stäckel systems on 2n-dimensional phase space. In [7] infinitely many classes of the Stäckel systems related to the so-called seed class, namely, the k-hole deformations of the latter, were constructed. Here we show that any k-hole deformation can be obtained from the Benenti-type system through a suitably chosen multiparameter generalized Stäckel transform. 1 Main results Let (M, P) be a Poisson manifold with the Poisson bracket {f, g} = (df, Pdg). Consider r functionally independent Hamiltonians H i, i = 1,...,r, on M, and assume that these Hamiltonians further depend on k r parameters α 1,...,α k, so H i = H i (x, α 1,...,α k ), i = 1,..., r, (1) where x M. Note that in general r is not related in any way to the dimension of M. 2

3 Suppose that there exists a k-tuple of pairwise distinct numbers s i {1,..., r} such that det ( H si / α j i,,...,k ) 0. (2) Now fix a k-tuple {s 1,...,s k } such that (2) holds and consider the system H si (x, α 1,...,α k ) = α i, i = 1,...,k, where α i are arbitrary parameters, as a system of algebraic equations for α 1,..., α k. By the implicit function theorem, the condition (2) guarantees that the solution of this system exists and is (locally) unique. We can write this solution in the form α i = A i (x, α 1,..., α k ), i = 1,...,k. Now define the new Hamiltonians H si, i = 1,..., k, by setting H si = A i (x, α 1,..., α k ), i = 1,...,k. In other words, the Hamiltonians H si, i = 1,..., k are defined by means of the relations H si = α i, i = 1,..., k. (3) Here and below the subscript means that we have substituted H si for α i for all i = 1,...,k. Next, let H i = H i, i = 1,...,r, i s j for j = 1,...,k. (4) Note that the Hamiltonians H j involve k parameters α i, i = 1,..., k for all j = 1,...,r: H i = H i (x, α 1,..., α k ), i = 1,..., r. We shall refer to the above transformation from H i, i = 1,...,r, to H i, i = 1,...,r, as to the k-parameter generalized Stäckel transform generated by H s1,..., H sk. In analogy with [9] we shall say that the r-tuples H i, i = 1,...,r, and H i, i = 1,..., r, are Stäckel-equivalent. The condition (2) guarantees that the above transformation is invertible. Indeed, consider the dual of the identity (3), that is, H si [ Φ] = α i, i = 1,..., k, (5) where the subscript [ Φ] means that we have substituted H si for α i for all i = 1,...,k. It is readily seen that by the implicit function theorem the condition (2) guarantees that we can solve (5) with respect to H sj, j = 1,...,k. If we do this and define the remaining Hamiltonians H i by the formulas H i = H i [ Φ], i = 1,...,r, i s j for j = 1,...,k, (6) then it is straightforward to verify that (3) and (4) hold identically. In other words, the formulas (5) and (6) define the inverse of the transformation defined using (3) and (4). Clearly, these two transformations are dual, with the duality transformation swapping H i and H i for all i = 1,..., r and swapping α j and α j for all j = 1,...,k. 3

4 Note that in the special case when the Hamiltonians H i are linear in the parameters α j, the above formulas undergo considerable simplification, and we can explicitly express H i via H i. Namely, let Then equations (3) take the form H i = H (0) i + α j H (j) i, i = 1,...,r. (7) H (0) s i + H sj H (j) s i = α i, i = 1,...,k, (8) and we can readily solve them for H si : H si = det W i / detw, (9) where W is a k k matrix of the form W = H s (1) 1 H s (k) H s (1) k H s (k) k, and W i are obtained from W by replacing H s (i) j by H s (0) j α j for all j = 1,...,k. By (4) we have H i = H (0) i + H sj H (j) i, i = 1,...,r, i s j for j = 1,...,k, (10) where H si are given by (9). It is straightforward to verify that if we set k = 1 then the transformation given by (9) and (10) becomes nothing but the standard Stäckel transform [9], also known as the coupling-constant metamorphosis [15]. It turns out that the k-parametric generalized Stäckel transform preserves the commutativity of the Hamiltonians H i. More precisely, the following assertion holds. Proposition 1 Let H i, i = 1,...,r and H i, i = 1,...,r be related by a k-parameter generalized Stäckel transform generated by H s1,...,h sk. Then the following assertions hold: i) if {H si, H sj } = 0 for all i, j = 1,..., k then { H si, H sj } = 0 for all i, j = 1,...,k; ii) if {H si, H j } = 0 for all i = 1,...,k and all j = 1,...,r then { H si, H j } = 0 for all i = 1,...,k and all j = 1,..., r; iii) if {H i, H j } = 0 for all i, j = 1,...,r then { H i, H j } = 0 for all i, j = 1,...,r. Proof. Prove i) first. For any smooth functions f and g on M that further depend on the parameters α 1,...,α k, we have the following easy identities: {f, g} = {f, g} + ( f/ α j ) { H sj, g}, (11) 4

5 {f, g } = {f, g} + k ( f/ α j ) { H sj, g} + k ( g/ α j ) {f, H sj } + k ( f/ α i ) ( g/ α j ) { H si, H sj }. i, Using the assumption {H si, H sj } = 0 and (3), we find that (12) 0 = { α i H si, α j H sj } = {H si H si, H sj H sj }, whence {H si H si, H sj H sj } = 0. Writing out the Poisson bracket on the left-hand side of the latter identity using (11) for the brackets {H si, H sj } and {H si, H sj } and (12) for the bracket {H si, H sj } we obtain ( H si / α p ) ( H sj / α q ) { H sp, H sq } = 0, p,q=1 whence using (2) we readily find that for all p, q = 1,...,k we have { H sp, H sq } = 0. However, Hsp are independent of α q for all q = 1,...,k, so { H si, H sj } = { H si, H sj } = 0, and the result follows. As we have already proved i), to prove ii) we only need to show that if {H si, H j } = 0 for all i = 1,..., k and all j = 1,...,r then { H si, H j } = 0 for all i = 1,..., k and all j = 1,..., r such that j s p for all p = 1,..., k. As H i, i = 1,...,r, are independent of α p for all p = 1,...,k by construction, we have { H si, H j } = { H si, H j }. Moreover, as j s p for all p = 1,..., k by assumption, by virtue of (4) the relation { H si, H j } = 0 is equivalent to { H si, H j } = 0. In turn, using (11) we can rewrite the Poisson bracket { H si, H j } as follows: { H si, H j } = { H si, H j } As { H sp, H si } = 0 by i), we see that ( H j / α p ) { H sp, H si } { H si, H j } = { H si, H j }. Now, in analogy with the proof of i), consider the identity 0 = { α p, H j } = {H sp, H j }. 5 p=1

6 Using (11) and our assumptions yields 0 = {H sp, H j } = ( H sp / α i ) { H si, H j }. i=1 Finally, using (2) we conclude that { H si, H j } = 0, (13) whence { H si, H j } = 0, and the result follows. Part iii) is proved in analogy with ii). Namely, in view of i) and ii) we only need to prove that the conditions {H i, H j } = 0, i, j = 1,..., r imply { H i, H j } = 0 for all i, j = 1,...,r such that i s p and j s p for all p = 1,..., k. For i s p and j s p for all p = 1,...,k we have { H i, H j } = {H i, H j } Using (12) and (13) we find that {H i, H j } = 0, and the result follows. Note that the computations in the above proof bear considerable resemblance with those in the theory of Hamiltonian systems with second-class constraints, see e.g. the classical work by Dirac [12]. From Proposition 1 it is immediate that the transformation defined by (3) and (4) preserves (super)integrability. Namely, under the assumptions of Proposition 1, iii) let dim M = 2n, rank P = 2n, and r = n. Then the dynamical system associated with any of H i is Liouville integrable, as it has n commuting functionally independent integrals, H j, j = 1,..., n, in involution. But by Proposition 1, iii) the dynamical system associated with any of H i enjoys the same property, the required involutive integrals of motion now being H i, i = 1,...,n. Likewise, under the assumptions of Proposition 1, ii) let dim M = 2n, rankp = 2n, and r > n. Then the Hamiltonian H sj is superintegrable for any j {1,...,k} as it has r > n integrals H i, i = 1,...,r, and by Proposition 1, ii) H sj is superintegrable for any j {1,...,k} as well, the integrals now being H i, i = 1,...,r. The multiparametric generalized Stäckel transform can be thought of as a very powerful tool for solving the Hamilton Jacobi equations (and hence the equations of motion) for Hamiltonian dynamical systems. Indeed, if we can solve the stationary Hamilton Jacobi equations for the transformed Hamiltonians H i, then we can do this for the original Hamiltonians H i as well, and vice versa, see Proposition 4 below for further details. Moreover, the separation of variables in the Hamilton Jacobi equation is nothing but a particular case of the generalized Stäckel transform. Indeed, suppose that M = R 2n, P is a canonical Poisson structure on M, and λ i, µ i, i = 1,...,n, are the Darboux coordinates for P, i.e., we have {λ i, µ j } = δ ij, {λ i, λ j } = 0, {µ i, µ j } = 0, i, j = 1,...,n. Let λ = (λ 1,...,λ n ) and µ = (µ 1,...,µ n ). Let r = k = n, s i = i, i = 1,..., n, and let H i = ϕ i (λ i, µ i, α 1,...,α n ), i = 1,...,n, 6

7 so for i j we have H i / λ j = H i / µ j = 0. Further assume that (2) holds and define H i = H i (λ, µ, α 1,..., α n ), i = 1,...,n, by means of (3). It is immediate that {H i, H j } = 0 for all i, j = 1,...,n and therefore by Proposition 1, i) we have { H i, H j } = 0, i, j = 1,...,n. (14) By the implicit function theorem the condition (2) ensures that the system of the stationary Hamilton Jacobi equations for H i, H i (λ, S/ λ, α 1,..., α n ) = Ẽi, i = 1,..., n, (15) is equivalent (see (5)) to the system of the stationary Hamilton Jacobi equations for H i with α i replaced by Ẽi: ϕ i (λ i, S/ λ i, Ẽ1,..., Ẽn) = α i, i = 1,...,n. (16) The system (16) is in the separated form, i.e., it is in fact a system of ordinary differential equations for S, and we have a separated complete integral for (16), and therefore for (15), of the form S = n S j (λ j, Ẽ1,...,Ẽn, α j, β j ), (17) where β j are arbitrary constants and for each i the function S i is a general solution of the ordinary differential equation ϕ i (λ i, ds i /dλ i, Ẽ1,...,Ẽn) = α i. Of course, the existence of the separated complete integral (17) for (15) means that λ i, µ i, i = 1,..., n, are separation coordinates for H j, j = 1,..., n. The relations (3) in our case are nothing but the separation relations for H j in the sense of [27], and (14) reproduces the well-known result on commutativity of separable Hamiltonians. Thus, the generalized Stäckel transform indeed includes the separation of variables as a particular case. 2 Reciprocal transformations for the equations of motion Recall that the equations of motion associated with a Hamiltonian H and a Poisson structure P on M read (see e.g. [3]) dx b /dt H = (X H ) b, b = 1,..., dim M, (18) where x b are local coordinates on M, X H = PdH is the Hamiltonian vector field associated with H, and t H is the corresponding evolution parameter (time). Here and below the differentials are computed under the assumption that the parameters are considered to be constant, i.e., if H = H(x, α 1,...,α k ) then in the local coordinates x b on M we have dh = dim M b=1 H x bdxb. 7

8 Suppose that {H si, H sj } = 0 for all i, j = 1,...,k,and consider simultaneously the equations of motion (18) for the Hamiltonians H si with the times t si and for H si with the times t si : dx b /dt si = (X Hsi ) b, b = 1,...,dim M, i = 1,...,k, (19) dx b /d t si = (X Hsi ) b, b = 1,...,dim M, i = 1,...,k. (20) In analogy with [15] consider a reciprocal transformation (see e.g. [23, 25, 26] for general information on such transformations) relating the times t si and t sj : d t si = ( ) Hsj dt sj, i = 1,..., k. (21) α i Proposition 2 Under the assumptions of Proposition 1, i), consider the equations of motion (19) for H si, i = 1,...,k, restricted onto the common level surface N α = {x M H si (x, α 1,...,α k ) = α i, i = 1,...,k} of H si. Then the transformation (21) is well defined on these restricted equations of motion and sends them into the equations of motion (20) for H si, i = 1,..., k, restricted onto the common level surface Ñ α = {x M H si (x, α 1,..., α k ) = α i, i = 1,...,k} of Hsi. Proof. First of all show that (21) is well-defined, that is, we have 2 t si t sp t sq = by virtue of equations (19) restricted onto N α. Using (21) we find that (22) boils down to ( ) Hsp α i = t sq N α 2 t si t sq t sp, p, q = 1,..., k, (22) ( ) Hsq α i In turn, using (19) we readily find that (23) takes the form { ( Hsp ) } { ( Hsq ) }, H sq =, H sp, α i α i N α N α t sp, p, q = 1,..., k. (23) N α and the latter equality can be proved by taking the partial derivative of the relation {H sp, H sq } = 0 with respect to α i. Next, Eq.(21) yields d/dt si = ( ) Hsi d/d t sj, α j i = 1,...,k. 8

9 Taking into account (19) and (20) we conclude that we have to prove that ( ( Hsi ) ) X Hsi N α = X α j Hsj N α, i = 1,...,k, (24) N α where N α denotes restriction onto N α. As X H = PdH for any smooth function H on M, Eq.(24) boils down to ( ( ( ) )) Hsi P dh si + d α H sj = 0, i = 1,...,k. (25) j N α On the other hand, taking the differential of (3) we obtain (dh si ) + ( ) Hsi (d α H sj ) = 0, i = 1,..., k. (26) j As H sj are independent of α p for all p = 1,...,k we have (d H sj ) = d H sj, so (26) yields ( ) Hsi d α H sj = (dh si ), j and (24) takes the form ( P ( dhsi (dh si ) )) N α = 0, i = 1,...,k. In the local coordinates x b on M we have ( ( ( ( ) ( ( )) dim M H si Hsi P dhsi (dh si ) = P N α b=1 x b x b ( ( ) ) = dim M H si Hsi ( ) Pdx b, i = 1,..., k. x b x b N α N α b=1 ) dx b )) N α (27) By virtue of (3) and (5) N α and Ñα represent the same submanifold of M, whence ( ( ) ) ( H si Hsi ( ) ) H si Hsi ( ) ( ) Hsi Hsi = = = 0. x b x b x b x b x N α Ñα b Ñ α x b Ñ α We used here an easy identity ( ( Hsi ) ) x b = Ñα ( ) Hsi x b Ñ α. Thus, the left-hand side of (27), and therefore that of (25), vanishes, and the result follows. Now assume that all H i are in involution: {H i, H j } = 0, i, j = 1,...,r. Then by Proposition 1 so are H i, i.e., { H i, H j } = 0, i, j = 1,...,r, 9

10 and we can consider two sets of simultaneous evolutions, dx b /dt i = (X Hi ) b, b = 1,..., dim M, i = 1,...,r, (28) dx b /d t i = (X Hi ) b, b = 1,..., dim M, i = 1,...,r, (29) and the extension of (21), r ( ) Hj d t si = dt j, i = 1,...,k, α i t q = t q, q = 1, 2,...,r, q s p for any p = 1,...,k. (30) In analogy with Proposition 2 we can prove the following result. Proposition 3 Under the assumptions of Proposition 1, iii), consider the equations of motion (28) for H i, i = 1,..., r, restricted onto N α. Then the transformation (30) is well defined on these restricted equations of motion and sends them into the equations of motion (29) for H i, i = 1,..., r, restricted onto Ñα. Note that the transformations from Propositions 2 and 3 do not change the dynamical variables x. In particular, under the assumptions of Proposition 2 for any given i from 1 to k the trajectories of the dynamical system associated with H si are identical to those of the dynamical system associated with H si, if we consider the trajectories as non-parametrized curves. In other words, the transformation (21) amounts to the reparametrization of the times associated with H sj for all j = 1,...,k. Notice, however, that the reparametrization in question is different for different trajectories, as one can readily infer from (21). 3 Canonical Poisson structure Let P be a canonical Poisson structure on M = R 2n. Then the Hamilton Jacobi equations for H i and H i have a common solution, cf. [9]. Namely, we have the following generalization of the results of [9] to the Hamiltonians that are not necessarily quadratic in the momenta: Proposition 4 Under the assumptions of Proposition 1, i) let M = R 2n, P be a canonical Poisson structure on M, and λ i, µ i, i = 1,...,n, be the Darboux coordinates for P, i.e., {λ i, µ j } = δ ij. Let λ = (λ 1,...,λ n ) and µ = (µ 1,...,µ n ). Let S = S(λ, α 1,...,α k, E s1,...,e sk, a 1,...,a n k ), where a i are arbitrary constants, be a complete integral of the stationary Hamilton Jacobi equation for the Hamiltonians H si = H si (λ, µ, α 1,...,α k ), H si (λ, S/ λ, α 1,...,α k ) = E si, i = 1,...,k. If we set E si = α i and α i = Ẽs i for all i = 1,..., k then S also is a complete integral of the stationary Hamilton Jacobi equation for the Hamiltonians H si = H si (λ, µ, α 1,... α k ), H si (λ, S/ λ, α 1,... α k ) = Ẽs i. 10

11 Moreover, if the assumptions of Proposition 1, iii) are satisfied as well, and r n, let S = S(λ, α 1,...,α k, E 1,..., E r, a 1,...,a n r ) (31) where a i are arbitrary constants, be a complete integral for the system of stationary Hamilton Jacobi equations H i (λ, S/ λ, α 1,...,α k ) = E i, i = 1,..., r. If we set α j = Ẽs j, E sj = α j, j = 1,...,k, and E i = Ẽi, i = 1,..., r, i s p for all p = 1,...,k, then S (31) is also a complete integral for the system H i (λ, S/ λ, α 1,..., α k ) = Ẽi, i = 1,..., r. As for the equations of motion, in addition to general Propositions 2 and 3, a somewhat more explicit result can be obtained by straightforward computation. Corollary 1 Under the assumptions of Proposition 3 let M = R 2n, P be a canonical Poisson structure on M, and λ i, µ i, i = 1,..., n be the Darboux coordinates for P, that is, we have {λ i, µ j } = δ ij, {λ i, λ j } = 0, {µ i, µ j } = 0, i, j = 1,..., n. Let λ = (λ 1,...,λ n ) and µ = (µ 1,...,µ n ). Suppose that r = n, 2 H i / α j µ = 0 for all i = 1,...,n and all j = 1,...,k, and that λ j, j = 1,..., n, can be chosen as local coordinates on the Lagrangian submanifold N E = {(λ, µ) M H i (λ, µ, α 1,...,α k ) = E i, i = 1,...,n} (in other words, the system H i (λ, µ, α 1,...,α k ) = E i, i = 1,...,n, can be solved for µ), and that we have α j = Ẽs j, E sj = α j, j = 1,..., k, and E i = Ẽi, i = 1,..., n, i s p for all p = 1,...,k. (32) Then the reciprocal transformation (30) turns the system dλ/dt i = ( H i / µ) NE, i = 1,...,n, (33) into dλ/d t i = ( H i / µ) ÑẼ, i = 1,...,n, (34) where ÑẼ = {(λ, µ) M H i (λ, µ, α 1,..., α k ) = Ẽi, i = 1,..., n}. Recall that N E and NẼ in fact represent the same Lagrangian submanifold of M, cf. proof of Proposition 1. For instance, if we have k = 1, and let α 1 α, s 1 = s, and H i = 1 2 (µ, G i(λ)µ) + V i (λ) + αw i (λ), i = 1,...,n, (35) where (, ) stands for the standard scalar product in R n and G i (λ) are n n matrices, then the system (33) reads dλ/dt i = G i (λ)m, (36) where µ = M(λ, α, E 1,...,E n ) is a general solution of the system H i (α, λ, µ) = E i, i = 1,...,n. 11

12 If we eliminate M from (36) then we obtain the dispersionless Killing systems (cf. [8]) λ ti = G i (G s ) 1 λ ts, i = 1, 2,..., s 1, s + 1,...,n, (37) and the reciprocal transformation (30), which in our case reads n d t s = W i (λ)dt i, t i = t i, i s, i=1 turns (37) into λ t i = G i ( G s ) 1 λ t s, i = 1, 2,..., s 1, s + 1,...,n, (38) where the contravariant metrics G s = G s /W s and G i = G i W i G s /W s, i = 1, 2,...,s 1, s+1,...,n, are related to the Hamiltonians H i = 1 2 (µ, G i (λ)µ) + Ṽi(λ) + α W i (λ), i = 1,...,n, (39) which are Stäckel-equivalent to H i, i = 1,...,n. 4 Solving the reduced equations of motion We can now apply Proposition 4 in order to obtain the solutions of equations of motion (33) and (34) as follows: Corollary 2 Under the assumptions of Corollary 1, suppose that S = S(λ, α 1,...,α k, E 1,...,E n ) (40) is a complete integral for the system of stationary Hamilton Jacobi equations H i (λ, S/ λ, α 1,...,α k ) = E i, i = 1,...,n. Then a general solution of (33) for i = d can be written in implicit form as S/ E j = δ jd t d + b j, j = 1,..., n, (41) where b j are arbitrary constants, and by virtue of (32) a general solution of (34) for i = d can be written in implicit form as S/ Ẽj = δ jd t d + b j, j = 1,..., n. (42) Comparing (41) and (42) and using (32) we readily see that, in perfect agreement with (30), t i = t i for i s 1,...,s k, but t sj = S/ E sj b sj = S/ α j b sj while t sj = S/ Ẽs j b sj = S/ α j b sj. Thus, the above approach does not yield an explicit formula expressing t sj as functions of λ, µ, and t si. In order to find a complete integral (40) we can use separation of variables as follows (see e.g. [27, 7] and references therein; cf. also the discussion in Section 1). Under the assumptions of Corollary 2 suppose that λ i, µ i, i = 1,..., n, are separation coordinates for the Hamiltonians H i, i = 1,..., n, that is, the system of equations H i (λ, µ, α 1,..., α k ) = E i, i = 1,...,n, is equivalent to the following one: ϕ i (λ i, µ i, α 1,...,α k, E 1,...,E n ) = 0, i = 1,...,n, (43) 12

13 which is nothing but the set of the separation relations on the Lagrangian submanifold N E. On the other hand, under the identification (32) the system (43) is equivalent to H i (λ, µ, α 1,..., α k ) = Ẽi, i = 1,..., n. (44) Thus, the Stäckel-equivalent n-tuples of Hamiltonians share the separation relations (43) provided (32) holds. Consider the system of stationary Hamilton Jacobi equations for H i H i (λ, S/ λ, α 1,...,α k ) = E i, i = 1,...,n. (45) By the above, (45) is equivalent to the system ϕ i (λ i, S/ λ i, α 1,...,α k, E 1,..., E n ) = 0, i = 1,..., n. (46) Suppose that (43) can be solved for µ i, i = 1,..., n: µ l = M l (λ l, α 1,...,α k, E 1,..., E n ), l = 1,...,n. Then there exists a separated complete integral of (46), and hence of (45), of the form (cf. e.g. [7]) n S = M l (λ l, α 1,...,α k, E 1,...,E n )dλ l, (47) l=1 and general solutions for (33) and (34) can be found using the method of Corollary 2. In this case the formulas (41) take the form n ( M i (λ i, α 1,...,α k, E 1,...,E n )/ E j )dλ i = δ jd t d + b j, j = 1,...,n, (48) i=1 and expressing λ i as functions of t d from (48) is nothing but a variant of the Jacobi inversion problem. In particular, for d = s i we have n t si + b si = S/ Ẽs i = S/ α i = ( M l (λ l, α 1,...,α k, E 1,..., E n )/ α i )dλ l, i = 1,..., k. l=1 5 Generalized Stäckel transform and deformations of separation curves Under the assumptions of Corollary 1, suppose that λ i, µ i, i = 1,...,n, are separation coordinates for the n-tuple of commuting Hamiltonians H i, i = 1,...,n. Then the Lagrangian submanifold N E is defined by n separation relations (43). Further assume that all functions ϕ i are identical, ϕ i = ϕ(λ i, µ i, α 1,...,α k, E 1,...,E n ), i = 1,..., n. (49) Then relations (43) mean that the points (λ i, µ i ), i = 1,..., n, belong to the separation curve [27, 7] ϕ(λ, µ, α 1,...,α k, E 1,...,E n ) = 0. (50) 13

14 If the relations ϕ(λ i, µ i, α 1,..., α k, H 1,...,H n ) = 0, i = 1,..., n, uniquely determine the Hamiltonians H i for i = 1,...,n, then for the sake of brevity we shall say that H i for i = 1,..., n have the separation curve ϕ(λ, µ, α 1,...,α k, H 1,...,H n ) = 0. (51) Fixing values of all Hamiltonians H i = E i, i = 1,..., n, picks a particular Lagrangian submanifold from the Lagrangian foliation. It is also clear that the Stäckel-equivalent n-tuples of the Hamiltonians H i, i = 1,..., n, and H i, i = 1,..., n, share the separation curve (51) provided (3) and (5) hold. In the rest of this section we shall deal with a special class of separation curves of the form (cf. e.g. [7] and references therein) n H j λ β j = ψ(λ, µ), (52) where β j are arbitrary pairwise distinct non-negative integers, β 1 > β 2 > > β n. In fact one always can impose the normalization β n = 0 by diving the left- and right-hand side of (52) by λ βn if necessary, but we shall not impose this normalization in the present paper. For a given n, each class of systems (52) is labelled by a sequence (β 1,...,β n ) while a particular system from a class is given by a particular choice of ψ(λ, µ). In particular, the choice ψ(λ, µ) = 1 2 f(λ)µ2 +γ(λ) yields the well-known classical Stäckel systems. All these systems admit the separation of variables in the same coordinates (λ i, µ i ) by construction. We shall refer to the class with the separation curve n H j λ n j = ψ(λ, µ) (53) as to the seed class. Note that if ψ(λ, µ) = 1 2 f(λ)µ2 + γ(λ) we obtain precisely the Benenti class of Stäckel systems [1, 2]. The seed class is a rather general one: it includes the majority of known integrable systems with natural Hamiltonians [7]. It turns out that, roughly speaking, the n-tuple of Hamiltonians having the general separation curve (52) can be related via a suitably chosen generalized multiparameter Stäckel transform to an n-tuple of Hamiltonians having the separation curve (53) from the seed class. The exact picture is a bit more involved, as in fact we need to consider the deformations of the curves in question. Define first an operator R f k that acts as follows: R f k (F) = F + fλk (λ k /k!)( k F/ λ k ) λ=0. For instance, we have Let ( s ) a j λ j = fλ k + R f k j=0 s a j λ j. j=0,j k F 0 = n H j λ n j and F0 = n H j λ n j. 14

15 For any integer m define [7] the so-called basic separable potentials V (m) j by means of the relations that must hold for λ = λ i, i = 1,...,n. λ m + n V (m) j λ n j = 0 (54) Under the assumptions of Corollary 1, consider an n-tuple of commuting Hamiltonians of the form H i = H (0) i + where γ j, j = 1,..., k, are pairwise distinct integers. Suppose that the Hamiltonians (55) have the separation curve of the form α j V (γ j) i, (55) α j λ γ j + F 0 = ψ(λ, µ), (56) where γ j > n 1 for all j = 1,...,k and γ i γ j if i j for all i, j = 1,..., k. Now pick k n distinct numbers s i {1,..., n} and define the Hamiltonians H i by means of the following separation curve H sj λ γ j + R α 1 n s 1 R α k n s k ( F 0 ) = ψ(λ, µ). (57) This means that H i are the solutions of the system of linear algebraic equations obtained from (57) upon substituting λ i for λ and µ i for µ into (57) for i = 1,...,n. Proposition 5 Under the above assumptions the n-tuple of Hamiltonians H i, i = 1,...,n, is Stäckelequivalent to H i, i = 1,...,n. The n-parameter generalized Stäckel transform in question reads as follows: H si = det B i / det B, (58) where B = V (γ 1) s 1 V (γ k) s V (γ 1) s k V (γ k) s k is a k k matrix, and B i are obtained from B by replacing V (γ i) s j by H s (0) j α j for all j = 1,...,k; H i = H (0) i + H sj V (γ j) i, i = 1,...,r, i s j for j = 1,..., k, (59) where H si are given by (58). 15

16 Proof. First of all, note that the above formulas for H i indeed constitute the Stäckel transform, as Eq.(58) is readily seen to imply the relations of the type (3), namely H (0) s i + H sj V (γ j) s i = α i, i = 1,..., k, (60) cf. the discussion after (8). Now we only have to prove that the Hamiltonians H i defined by (58) and (59) have the separation curve (57). As we have already mentioned above, the Stäckel-equivalent n-tuples of separable commuting Hamiltonians share the separation relations provided (32) holds. Therefore, in order to prove our claim it suffices to show that the separation curves (56) and (57) can be identified by virtue of (60). Indeed, upon plugging the relations λ γ j = n p=1 V (γ j) p λ n p, j = 1,...,k, (61) that follow from (54) into (56), collecting the coefficients at the powers of λ, and taking into account (55), the separation curve (56) takes the form n H (0) j λ n j = ψ(λ, µ). (62) On the other hand, plugging (61) into (57) and proceeding in a similar fashion as above, we obtain ( n ) H sj V (γ j) p λ n p + R α 1 n s 1 R α k n s k ( F 0 ) = ψ(λ, µ). (63) p=1 By virtue of the relations (60), which can be further rewritten as H (0) s i = H sj V (γ j) i + α i, i = 1,..., k, along with (59), we find that the curves (63) and (62) are indeed identical, and hence so are the curves (57) and (56). Remark 1 In fact the above argument can be inverted, that is, we can obtain the relations (60) (and hence (58)) and (59) by requiring the curves (56) and (57) to coincide and comparing the coefficients at the powers of λ on the left-hand sides of these curves, or equivalently (by virtue of (54)), of (63) and (62). As a final remark, note that upon setting the parameters α i and α i to zero for all i = 1,...,k the formulas (58) and (59) indeed relate the Hamiltonians H i with the separation curve (53) and the Hamiltonians H i with the separation curve (52). In this case we essentially recover the formulas from [7] relating the Hamiltonians from the seed class and from the so-called r-hole deformation thereof (in our language, the deformed systems are precisely those having the separation curve (52)) up to a suitable renumeration of the Hamiltonians H i. 16

17 6 Examples As a simple illustration of the above results, consider the Hamiltonian systems on a four-dimensional phase space M = R 4 with the coordinates (p 1, p 2, q 1, q 2 ) and canonical Poisson structure. For our first example let k = 1, r = 2, s 1 = 2, α 1 α and α 1 α. Consider the Hamiltonian H 1 = 1 2 p p2 2 + α(q2 1 q2 2 ) q 2 p 2 2α 2 q 2 1, which is Liouville integrable because it Poisson commutes with H 2 = q 1p 2 q 2 p 1 2αq 1 q 2 p 2. The above pair of commuting Hamiltonians was found by analogy with one of the models from [16]. The relation (3) in our case takes the form q 1 p 2 q 2 p 1 2 H 2 q 1 q 2 p 2 = α, whence and therefore by virtue of (4) we have H 2 = q 1p 2 q 2 p 1 αp 2 2q 1 q 2, H 1 = q2 1 + q2 2 2 αq 1 p 1 p 2 + α(q2 1 αq 1 + q2) 2 p 2 2q 1 q 2 2q 1 q By Proposition 1, ii) the relation {H 1, H 2 } = 0 implies { H 1, H 2 } = 0, so H 1 is Liouville integrable just like H 1. Interestingly enough, in this example the generalized Stäckel transform sends the Hamiltonian H 1 into a natural geodesic Hamiltonian H 1, but the metrics associated with H 1 is not flat and, moreover, has nonconstant scalar curvature unlike the metrics associated with H 1. By Proposition 2 the reciprocal transformation ( t 1 = t 1, d t 2 = 2q 1 p 1 + (q2 1 2 αq 1 + q2 2)p ) 2 dt 1 + 2q 1q 2 dt 2 q 2 p 2 takes the equations of motion for H 1 and H 2, with the respective evolution parameters t 1 and t 2, restricted onto the common level surface N α = {x R 4 H 1 (x, α 1, α 2 ) = α 1, H 2 (x, α 1, α 2 ) = α 2 } into the equations of motion for H 1 and H 2, with the respective evolution parameters t 1 and t 2, restricted onto the common level surface Ñα = {x R 4 H 1 (x, α 1, α 2 ) = α 1, H 2 (x, α 1, α 2 ) = α 2 }. It is easily seen that Ñα and N α represent the same submanifold of R 4. For the second example we set k = r = 2 and consider the (extended) Hénon Heiles system with the Hamiltonian which Poisson commutes with H 1 = 1 2 p ( 2 p2 2 α 1 q1 3 + q ) 1q2 2 α 2 q 1, 2 H 2 = 1 2 q 2p 1 p q 1p 2 2 α 1 17 ( ) q q2 1q2 2 q2 2 α

18 The separation curve for the system in question belongs to the seed class and reads α 1 λ 4 + α 2 λ 2 + H 1 λ + H 2 = λµ 2 /2. (64) The separation coordinates (λ i, µ i ), i = 1, 2, are related to p s and q s by the formulas q 1 = λ 1 + λ 2, q 2 = 2 λ 1 λ 2, p 1 = λ 1µ 1 + λ 2µ 2, p 2 = ( µ1 λ 1 λ 2 + µ ) 2. λ 1 λ 2 λ 2 λ 1 λ 1 λ 2 λ 2 λ 1 Let s 1 = 1, s 2 = 2, k = r = 2. Then (55) and (58) yield the following deformation of H 1 and H 2 : H 1 = 2 q 1 q 2 2 p q 3 2 H 2 = 4q2 1 + q2 2 2q 1 q (q q2 1 ) q 1 q 2 2 p 1 p 2 2(q q2 1 ) p 2 q 1 q q 1 q2 2 p 2 1 4(q q2 1 ) q 3 2 α 1 8(q q2 1 ) α q The corresponding separation curve reads (see Proposition 5) α α q2 4 2, p 1 p q q2 1 q2 2 + q4 2 p 2 q 1 q2 4 2 H 1 λ 4 + H 2 λ 2 + α 1 λ + α 2 = λµ 2 /2. (65) Using Proposition 2 and in analogy with the previous example we readily find that the reciprocal transformation (30) for the equations of motion restricted onto the appropriate Lagrangian manifolds in our case takes the form d t 1 = ( q q 1q Acknowledgments ) ( ) q 4 dt q2 1 q2 2 dt 2, d t 2 = q1 2 4 dt 1 + q2 2 4 dt 2. This research was supported in part by the Czech Grant Agency (GAČR) under grant No. 201/04/0538, by the Ministry of Education, Youth and Sports of the Czech Republic (MŠMT ČR) under grant MSM , by Silesian University in Opava under grant IGS 10/2007, and by the Polish State Committee For Scientific Research (KBN) under the KBN Research Grant No. 1 PO3B A.S. appreciates the warm hospitality of the Institute of Physics of the Adam Mickiewicz University, where the present work was completed. A.S. would also like to thank Prof. W. Miller, Jr. and Prof. S. Rauch-Wojciechowski for stimulating discussions. References [1] S. Benenti, Orthogonal separable dynamical systems, in: Differential geometry and its applications (Opava, 1992), Math. Publ., 1, Silesian University in Opava, Opava, 1993, pp ; available online at 18

19 [2] S. Benenti, Intrinsic characterization of the variable separation in the Hamilton- Jacobi equation, J. Math. Phys. 38, (1997); preprint version available at benenti/ricerca/icvs.pdf [3] M. B laszak, Multi-Hamiltonian theory of dynamical systems, Springer-Verlag, Berlin etc., [4] M. B laszak, On separability of bi-hamiltonian chain with degenerated Poisson structures, J. Math. Phys. 39, (1998). [5] M. B laszak and Wen-Xiu Ma, Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems, J. Geom. Phys. 47, (2003); preprint nlin.si/ (arxiv.org) [6] M. B laszak and A. Sergyeyev, Maximal superinegrability of Benenti systems, J. Phys. A: Math. Gen. 38, L1 L5 (2005); preprint nlin.si/ (arxiv.org) [7] M. B laszak, Separable systems with quadratic in momenta first integrals, J. Phys. A: Math. Gen. 38, (2005); preprint nlin.si/ (arxiv.org) [8] M. B laszak and K. Marciniak, From Stäckel systems to integrable hierarchies of PDE s: Benenti class of separation relations, J. Math. Phys. 47, (2006); preprint nlin.si/ (arxiv.org) [9] C.P. Boyer, E.G. Kalnins, and W.Miller, Jr., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal., 17, (1986); see also miller/stackel.pdf [10] A. Cannas da Silva, Lectures on symplectic geometry. Lecture Notes in Mathematics, Springer- Verlag, Berlin, [11] M. Crampin and W. Sarlet, A class of non-conservative Lagrangian systems on Riemannian manifolds, J. Math. Phys. 42, (2001). [12] P.A.M. Dirac, Lectures on Quantum Mechanics. Yeshiva Press, New York, [13] E. V. Ferapontov, Integration of weakly nonlinear hydrodynamic systems in Riemann invariants, Phys. Lett. A 158, (1991). [14] E. V. Ferapontov and A. P. Fordy, Separable Hamiltonians and integrable systems of hydrodynamic type, J. Geom. Phys. 21, (1997). [15] J. Hietarinta, B. Grammaticos and B. Dorizzi and A. Ramani, Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems, Phys. Rev. Lett. 53, (1984). [16] J. Hietarinta, New Integrable Hamiltonians with Transcendental Invariants, Phys. Rev. Lett. 52, (1984). [17] E.G. Kalnins, J.M. Kress, W.Miller, Jr., P. Winternitz, Superintegrable systems in Darboux spaces, J. Math. Phys. 44, (2003); preprint version at [18] E.G. Kalnins, J.M. Kress, W. Miller, Jr. Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46, (2005); preprint version at miller/supstructure2.pdf 19

20 [19] E.G. Kalnins, J.M. Kress, W. Miller, Jr. Second order superintegrable systems in conformally flat spaces. V. The classical 3D Stäckel transform and 3D classification theory, J. Math. Phys. 47, (2006). [20] E. Kalnins, Separation of variables for Riemannian spaces of constant curvature, John Wiley & Sons, New York, Online at miller/variableseparation.html [21] C. Lanczos, The Variational Principles of Mechanics, University of Toronto Press, Toronto, [22] W. Miller, Jr., Symmetry and separation of variables, Addison-Wesley, Reading, MA etc., 1977; available online at miller/separationofvariables.html [23] W. Oevel and C. Rogers, Gauge transformations and reciprocal links in (2+1) dimensions, Rev. Math. Phys. 5, (1993); preprint version available at walter/publications/gauges ps [24] S. Rauch-Wojciechowski, K. Marciniak and M. B laszak, Two Newton decompositions of stationary flows of KdV and Harry Dym hierarchies, Physica A 233, (1996); see also krzma/publications/ [25] C. Rogers and W.F. Shadwick, Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering Series, New York, Academic Press, [26] B.L. Roždestvenskiĭ, N.N. Janenko, Systems of quasilinear equations and their applicatons to gas dynamics, Translations of Mathematical Monographs, Vol. 55, Providence, RI, AMS, [27] E.K. Sklyanin, Separation of variables new trends, Progr. Theoret. Phys. Suppl. 118, (1995); preprint solv-int/ (arxiv.org) [28] A.V. Tsiganov, Canonical transformations of the extended phase space, Toda lattices and the Stäckel family of integrable systems, J. Phys. A: Math. Gen. 33, (2000); preprint solv-int/ (arxiv.org). [29] A.V. Tsiganov, The Maupertuis Principle and Canonical Transformations of the Extended Phase Space, J. Nonlin. Math. Phys. 8, (2001); preprint solv-int/ (arxiv.org). [30] A.P. Veselov, Time change in integrable systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh., no. 5, and 104 (1987), in Russian. 20