Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians

We study twisted products $H=\alpha^rH_r$ of natural autonomous Hamiltonians $H_r$, each one depending on a separate set, called here separate $r$-block, of variables. We show that, when the twist functions $\alpha^r$ are a row of the inverse of a block-St\"ackel matrix, the dynamics of $H$ reduces to the dynamics of the $H_r$, modified by a scalar potential depending only on variables of the corresponding $r$-block. It is a kind of partial separation of variables. We characterize this block-separation in an invariant way by writing in block-form classical results of St\"ackel separation of variables. We classify the block-separable coordinates of $\mathbb E^3$.


Introduction
Paul Stäckel opened in [23] the theory of complete separation of variables for the Hamilton-Jacobi equation of natural Hamiltonians H with N degrees of freedom, written in orthogonal coordinates. The characterization, both coordinate dependent, involving N ×N Stäckel matrices (invertible matrices with each row depending on one different variable) and invariant, involving N quadratic first integrals of the Hamiltonian. Remark that complete separation implies the completeness of the separated integral of the Hamilton-Jacobi equation, thanks to the existence of then N constants of motion in involution. The theory greatly developed in the following years, thanks to the works of Levi-Civita [16], Eisenhart [11,12] and many others (see [15,21] for more complete references).
Stäckel himself considered in [24] the case of partial separation of variables and obtained a sufficient characterization of it in terms of Stäckel matrices of reduced dimension and of a corresponding number of quadratic first integrals of H. Differently from complete separation, partial separation of variables gained much less interest, certainly due to its scarce utility for finding solutions of the Hamilton-Jacobi equation. Moreover, partial separation does not guarantee the existence of complete integrals of the partially separated Hamilton-Jacobi equation. However, as pointed out by [17], in this case the Jacobi method of inversion can, sometimes, produce additional first integrals of H. Recent articles developing partial separation theory for Hamilton-Jacobi and Schrödinger equations develop somehow the results of Stäckel, giving a more detailed characterization of the metric coefficients in partially separable coordinates and further conditions for the separation of the quantum systems [18,14]. Another approach to non-complete additive separation is represented by non-regular separation wich relies on the existence of a complete additively separated solution on proper submanifolds only [5,3].
In our study, we shift for the first time the interest from the Hamilton-Jacobi equation to the dynamics of the system. We observe that the partial separation introduced by Stäckel, as well as the complete separation, establishes a dynamical relationship between H and the (partially) separated equations when these are considered as Hamiltonians on submanifolds of the original phase space. Namely, we find that the projection of the orbits of H on these submanifolds, spanned by the separated blocks of coordinates, coincides with the orbits of the separated Hamiltonians on the same submanifolds. The only difference is a position-dependent rescaling of the corresponding Hamiltonian parameters. As a consequence, the dynamics of H can be decomposed into a number of lower-dimensional Hamiltonian systems, allowing a simpler analysis of the original system. The separated blocks of coordinates, considered together as a N dimensional coordinate system on the base manifold of H, correspond to a block-diagonal form of the N -dimensional metric tensor, from this the name we choose for this kind of separation. We prefer do not use the expression partial separation since it is already associated with Hamilton-Jacobi theory, which we do not consider here. Indeed, by shifting the focus from Hamilton-Jacobi theory to the dynamics, we remove the discontinuity represented by the completeness of the integral of the Hamilton-Jacobi equation, which is strictly connected with Stäckel theory of complete separation of variables. Partial separation does not imply the existence of a complete integral, so that the Jacobi method (the construction of a canonical transformation to a trivially integrable Hamiltonian) is not in general applicable. On the contrary, our dynamical interpretation of block separation is basically insensitive to complete or partial separation, the only difference being the dimension of the separated blocks of coordinates. We find useful and natural to relate block-separation with the structure of twisted product that the Hamiltonian assumes when the separation is possible. This allows us to state our results in a form very close to analogous results in classical Stäckel separation, analogy missing in all the other studies about partial separation. Namely, we can characterize block-separation by introducing "block" versions of celebrated Levi-Civita, Eisenhart and more recent theorems about complete separation.
In Section 2 we recall the basic theorems about Stäckel complete separation of variables which we restate in block-separable form. In Section 3 we define twisted products of Hamiltonians and state some relevant properties of them. In Section 4 we give a dynamical interpretation of Stäckel separation, providing examples of the related properties of time-scaling. In Section 5 we introduce block-separation and our main results about its characterization, with the blocklike formulations of Levi-Civita, Eisenhart and other theorems, and we provide an invariant characterization of block separation. As example we consider the four-body Calogero system. In Section 6, we characterize, at least with necessary conditions, all the possible block-separable coordinates of E 3 . Section 7 contains our final considerations and comments.

Outline of Stäckel separation
We recall shortly the principal theorems regarding complete separation of the Hamilton-Jacobi equation see [15] and [1] for further details. The theory of complete additive separation of the Hamilton-Jacobi equation begins with the work of Stäckel [23,24] Theorem 1. The necessary and sufficient conditions such that the Hamilton-Jacobi equation admits a complete integral via separation of variables, As a consequence of i and ii, there exist N − 1 independent quadratic first integrals (K a ) of H such that (c a ) are the constant values of (H, K a ).
A complete integral of the (1) is then determined by the N separated equations dW r dq r Remark that completeness for the integral W of the Hamilton-Jacobi equation means that it depends from N parameters (c a ), constants of motion, such that Therefore, the (c a ) can be part of a new set of canonical coordinates in which the Hamiltonian flow becomes trivially integrable. Later, Levi-Civita [16] obtained necessary and sufficient conditions for the complete separability of given coordinate systems. For natural Hamiltonians and orthogonal coordinates, with i = j not summed, i, j, l = 1, . . . , N , for the components of the metric tensor, and with i = j not summed, i, j = 1, . . . , N , called Bertrand-Darboux equations and whose solution is V in the form of a Stäckel multiplier.
In [11], Eisenhart provided a geometrical characterization of complete separation of variables in orthogonal coordinates introducing Killing tensors and, later, determined all the possible orthogonal separable coordinate systems of E 3 . The systems are 11 and they are described, for example, in [21]. In Eisenhart's characterization are fundamental the Eisenhart equations expressing necessary and sufficient conditions for the 2-tensor tensor k of simple eigenvalues (λ i ) and (∂ i ) as base of its eigenvectors is a Killing tensor. The integrability conditions of Eisenhart equations coincide with the Levi-Civita equations (3).
After Eisenhart, the coordinate systems can be geometrically understood as foliations of hypersurfaces called coordinate webs. The geodesic separability of a coordinate web can be characterized by a single characteristic Killing tensor, i.e. a symmetric Killing 2-tensor with pointwise simple eigenvalues and normally integrable eigenvectors which determine in each point of the space (apart possible singular sets of zero measure) the basis of coordinate vectors [1]. Recall that a symmetric Killing 2-tensor k is defined by the equivalent equations where [, ] are the Schouten brackets and ∇ is the covariant derivative with respect to the metric g. Necessary and sufficient conditions for a Killing tensor to be characteristic are given in [8] Theorem 3. Tonolo-Schouten-Nijenhuis. A 2-tensor K with real distinct eigenvalues has normal eigenvectors iff the following conditions are satisfied where N i jk are the components of the Nijenhuis tensor of K i j defined by An equivalent formulation of Theorem 2 involves N independent quadratic first integrals, therefore N Killing 2-tensors, instead of a single characteristic Killing tensor [1]  ii) the Killing two-tensors (k a ) associated with (K a ) are simultaneously diagonalized with pointwise simple eigenvalues and have common normally integrable eigevectors.
It follows that {K a , K b } = 0.
The original formulation of the Theorem requires reality of the eigenvalues, however, this request is non necessary if one allows complex separable coordinates [9].

Twisted products of Hamiltonians
Let M = × n r=1 M r , be the product of n Riemannian or pseudo-Riemannian manifolds (M r , g r ) of dimension n r , so that dim(M ) = n 1 + . . . + n n = N , and let α r be n non zero functions on M . The manifold M with metric tensor is a Riemannian or pseudo-Riemannian manifold called twisted product manifold of the (M r ) with twist functions (α r ) [19]. In the case when α 1 = 1 and α 2 , . . . , α n are functions on M 1 , the manifold M is called warped product. We extend to functions on T * M and T * M r the concept of twisted and warped products in a natural way. In particular, for each r we consider natural Hamiltonians where (q ri , p ri ), i = 1, . . . , n r , are canonical coordinates on T * M r , we construct twisted product of the H r with twist functions α r ∈ F(M ). Then, H is a natural Hamiltonian on T * M with metric G and potential The manifold M is naturally endowed with block-diagonal coordinates (q ri ) such that the components of G are in the form we call these coordinates twisted coordinates.
We have now n + 1 Hamiltonians, each one with its own Hamiltonian parameter. We call t the Hamiltonian parameter of H and τ r the Hamiltonian parameter of H r . From Hamilton's equations we get and dp ri dt = − ∂H ∂q ri = −α r ∂H r ∂q ri − H s ∂α s ∂q ri = α r dp ri dτ r − H s ∂α s ∂q ri . Therefore, the relation between the Hamiltonian vector fields X H of H and X r of H r is whereX r = α r X r , r not summed, is the rescaled Hamiltonian vector field of H r .

Stäckel systems as twisted Hamiltonians
In this section we study Stäckel systems in their nature of twisted Hamiltonians. Our aim, is to enlighten the relations among the dynamics of the N Let be H r = 1 2 (p 2 r + V r ) and assume that α r are a row (say the first one) of the inverse of a Stäckel matrix S for given coordinates (q ri ). Then, the twisted product H = α r H r admits separation of variables and we have the separated equations where c i are N constants, corresponding to the N constants of motion K i of H = K 1 and K a = (S −1 ) r a H r , a = 2, . . . , N. The Hamilton-Jacobi complete separated integral W = W 1 (q 1 , c i )+. . .+W N (q N , c i ) is given by integration of The Hamilton's equations of H, in time t, arė where we use the separated equations (4) to replace H r along the integral curves.
and the same for all other elements of the rows of S −1 . Then, we can writė Let be γ P the integral curve of X H based in any point P ∈ T * M . We consider the values c i = K i (P ) and we introduce the Hamiltonians with Hamiltonian parametersτ r , we can write the equations of Hamilton forH r as Therefore, Proof. Let be Due to (9) and (10) we can write (6) and (8) aṡ and it follows immediately Remark 1. After Proposition 5 we can put and consider the twist functions as determining position-dependent time-scalings between the Hamiltonian parameters t andτ r .

From Proposition 5 follows the important result
Proposition 6. The projection of each orbit of H on the coordinate manifolds (q r , p r ) coincides with the orbit ofH r = H r − c j S j r .
Remark 2. The Lagrange equation of the dynamics ofH r , expressed in times τ r is Remark 3. The equations (14) are associated with the Weierstrass equations i.e. the Stäckel systems associated with H andH coincide up to additive constants. To the constants Example 1. Twisted product of pendula. In order to show the effect of the time-scaling described above, we consider the twisted product of the following three one-dimensional Hamiltonians corresponding to two pendula and a purely inertial term, coupled together by the first row of the inverse of the 3 × 3 matrix which is a Stäckel matrix in a neighborhood of the origin, since the Taylor expansion up to the second order terms of its determinant ∆ around (0, 0, 0) is ∆ = 5 + 5q 1 − 6q 2 + 2q 3 . The elements of the matrix S −1 are therefore quite complicated rational functions that we do not need to explicit but make the coupling of the H r suitable to enhance the effect of the time-scaling. We take as (α r ) the first row of S −1 and consider The quadratic first integrals of H are simply determined by the remaining rows of (S −1 ) K a = (S −1 ) r a H r = c a , a = 2, 3. We already know from the previous section, that, despite the complicated expression of the coupling terms (α r ), the relation among the dynamics of H and of the separated HamiltoniansH r = H r −c a S a r reduces to a simple positiondependent time scaling.
We plot the numerical evaluation of the systems of Hamiltonian H and respectively, and project the orbits on (q 1 , p 1 ), obtaining, for the initial condi-   where we see that the orbits on (q 1 , p 1 ) of the two systems coincide. But, if we include the dependence from the different Hamiltonian parameters (denoted in both graphs as t), we get Fig.2. and we can see how the dependence from the Hamiltonian parameters can be extremely variate in the two cases.
Example 2. Twisted product with constant coefficients of harmonic oscillators. Taking twisted products of Hamiltonians seems an interesting way to establish an interaction among Hamiltonian systems. An example, even if somehow trivial, is provided by the twisted product with constant coefficients of harmonic oscillators. Let . . , n be a finite set of harmonic oscillators. Let be their twisted product with constant twist functions. The H i are all constants of motion of H and there is actually no interaction among them. However, some effect of the twisted product is nevertheless evident. The Hamilton equations of H are The general solution of these equations is We see that, for example, the choice α i = k/ω i , for any real positive k, determines a time-scaling that gives to all the oscillators the same frequency k with respect to t (as well as any other different real positive frequency for different choices of k for each i). Namely, the rescaling is in this case The frequency of each oscillator H i with respect to its own Hamiltonian parameter τ i remains clearly ω i .

Block-separation
The results of the previous section can be generalized as follows, leading to a kind of partial separation of variables that we call block-separation. Let M be a N -dimensional manifold. Let us consider a partition of a coordinate system on M organized as follows. For n ≤ N , consider for each integer r = 1, . . . , n the integers n r such that N = n 1 + . . . + n n .
The coordinate system is therefore composed of n blocks, and for each r ≤ n we have an r-block of coordinates that we denote as (q r1 , . . . , q rn r ). We call M r the manifold spanned by the r-block of coordinates. Consider T * M with the conjugate r-block momenta (p r1 , . . . , p rn r ).
Let us consider and the n block-separated equations H r = S a r c a , a = 1, . . . , n, where we assume that g rirj r , S a r and V r are functions of coordinates of the rblock only and c a are constants. If we assume that the n × n matrix (S a r ) is invertible, and we call it block-Stäckel matrix, then we can write the n equations We denote α r = (S −1 ) r 1 and call H = α r H r , K a = (S −1 ) r a H r , a = 2, . . . , n.
Hence, H is in the form of twisted product and it is a natural Hamiltonian whose metric tensor G is block-diagonalized, with components G rirj = α r g rirj r , G risj = 0, s = r, and whose scalar potential has the form while the scalar potentials in K a are A necessary condition for the above procedure is that the (17)  Proof. We have, since S is a block-Stäckel matrix, the analogous of (7) where r is not summed, and the same for the other elements Proof. The proof follows the same reasoning of the proof of Proposition 6. It follows that, denoting with (X H ) r the r-block component of the Hamiltonian vector field X H of H, (X H ) r =q ri ∂ ri +ṗ ri ∂ ri , where XH r is the Hamiltonian vector field ofH r . Therefore, if (X H ) r is tangent to any submanifold f ⊆ T * M , then also XH r is, and vice-versa. Hence, just as for the Stäckel systems, the dynamics of H is determined in each r-block, up to reparametrizations of the Hamiltonian parameter, by the dynamics of theH r , with the difference that theH r are no longer one-dimensional.
In this way, the time-independent dynamics of H can be exactly decomposed into the n lower-dimensional separated dynamics of HamiltoniansH r . TheH r share with the H r , factors of the twisted product H, the same inertial terms, while the scalar potential is modified by the addition of the term −c a S a r . Partial separation of Hamilton-Jacobi equation was introduced by di Pirro in [10] and generalized by Stäckel in [24]. He introduced the n × n matrix S and his results are analogous to our Proposition 7. Stäckel obtained sufficient conditions for partial separation of the H-J equation of natural Hamiltonians. His work has been continued more recently in [14] and [18], including the study of partial separation of the Schrödinger equation, obtaining again sufficient conditions for partial separation, and a more detailed form of the components of the metric tensor in partially separable coordinates. We remark that, by introducing twisted products, our characterization of block-separation provides necessary and sufficient conditions for it, in analogy with Stäckel theory of complete separation. We do not make here a strict comparison between our results and those of [24] and [14], since these last are strictly related to Hamilton-Jacobi theory and there is no consideration of the dynamical relations among the N -dimensional Hamiltonian and the separated Hamiltonians, which is our main interest. The detailed characterization of the partially separable metric's components in [14] should eventually coincide with a similar characterization of block-separable metrics, we do not consider here the distinction between linear and quadratic in the momenta first integrals (from linear first integrals one can always obtain quadratic ones). It is remarkable that in the last century very few works have been devoted to partial separation of H-J equation. This is understandable when one considers that Hamilton-Jacobi theory is of not easy application, apart the simplest cases, already when completely separated integrals of the H-J equations do exist. Some applications of partially separated integrals of the H-J equation, in order to generate new possible first integrals of the Hamiltonian, are presented in [17] and [22]. Our approach based upon the block-separated dynamics, instead of the partially separated Hamilton-Jacobi equation, appears to be completely new and can be more suitable for applications of the theory, particularly in the analysis of systems with many degrees of freedom.

Block-Eisenhart and block-Levi-Civita equations
We can see, with some surprise, that the characterisation of block-separation includes tools developed for Stäckel complete separation. Indeed, we can formulate classical results by Eisenhart and Levi-Civita in block form. If we assume that (q i ) are twisted coordinates (q ri ), then for G we have G rirj = 1 α r g r rirj , r not summed, so that G r k a G asj = δ sj r k . Moreover, assuming (q ri ) are block-separated hold, with r, s = 1, . . . , n, for all r i , s j in the respective separated blocks, and (19) hold.
Proof. By expanding {H, K a } = 0, in block-separable coordinates and collecting homogeneous terms in the momenta, dividing by α r α s , from the higher order terms in the momenta, we have, for all r, s, r k , s i , s j in the respective blocks. If r = s the equations become and, if r = s, then ∂ r k g sisj = 0. Hence, (23) is equivalent to If g sisj = 0, the equations are identically satisfied, otherwise, we have the (22).
Since not all the g sisj are zero, we have the statement. The first-order terms in the momenta vanish if and only if (19) hold.
By definition of the λ s a and of the α s , we have the equations S s r α r = δ s 1 , S s r λ r a α r = δ s a .
We observe that, after putting α r = g rr , the previous equations are identical to the relations typical of Stäckel systems. In the same way, the block-Eisenhart equations become the standard Eisenhart equations.

Proposition 10. The block-Eisenhart equations (22) hold if and only if (S r a ) is a block-Stäckel matrix.
As for the Stäckel systems, the block-Levi-Civita equations can be interpreted as the integrability conditions of the block-Eisenhart equations. The derivation is essentially the same as in [1].
It is therefore straightforward to see that the block-diagonalized coordinates (q 1 , . . . , q N ) are block-separated for the Hamiltonian H if and only if the block-Levi-Civita equations are satisfied, where the coordinates r i , s j are in different blocks and m = 1, . . . , n. The scalar potential V is already in the form of a block-Stäckel multiplier thanks to the form of H and satisfies See (3) for a comparison.

Invariant characterization
As in Stäckel theory, we can use the previous results for an invariant characterization of block-separation. Therefore, we have the analogous of the Eisenhart-Kalnins-Miller-Benenti theorem [1], Consequently, the strategy for finding block-separated coordinates of a given N -dimensional natural Hamiltonian H is the following • Find a number n ≤ N of independent quadratic first integrals (K a ) of H in involution among themselves, whose associated Killing tensors (k a ) admit common block-diagonalized normally integrable eigenspaces. The number n corresponds to the number of blocks. The dimension of the common eigenspaces equals the dimension of each block.
• At this point, we have block-diagonalized coordinates and we can write H = α r H r for some functions H r , the α r being determined by the block-Stäckel matrix determined by the (k a ).
• The last step consists in checking that each H r depends only from coordinates in T * M r . This is indeed the procedure applied in Example 3. The algebraic multiplicity of each λ r a is n r at least (for some a, we can have λ r a = λ s a , and the algebraic multiplicity is in this case n r + n s ). Proposition 11 provides an invariant characterization of block-separable coordinates in terms of what we can call block-Killing-Stäckel algebras generated by the (k a ). See [1,2] for a definition of Killing-Stäckel algebras.
If we assume that for N -dimensional 2-tensors T κ λ to each eigenvalue of algebraic multiplicity n r it corresponds a space of n r linearly independent covariant eigenvectors {X a }, we can consider the (N − n r )-dimensional space of vectors E N −nr such that < X a , E b >= 0, ∀E b ∈ E N −nr . We assume that E N −nr is a regular distribution of constant rank N − n r .
The necessary and sufficient condition for the integrability of the distributions E N −nr is given by the Haantjes theorem [13] Theorem 12. Let T λ κ be a tensor such that to each root with multiplicity n r of the characteristic equation belongs a set of n r linearly independent covariant eigenvectors. Then the E N −nr determined by these vectors are integrable if and only if , is the so-called Haantjes tensor of T .
Therefore, in analogy with the characterization of the Stäckel separable coordinate systems, we have Proposition 13. A natural Hamiltonian admits block-separable coordinates only if its metric tensor admits a symmetric Killing 2-tensor T satisfying the Haantjes theorem and its scalar potential V satisfies dT dV = 0. We call T the characteristic tensor of the block-separable coordinates.
The coordinates are therefore divided into n blocks, where n is the number of the pointwise different eigenvalues of T , the dimension of each block equals the multiplicity of the corresponding eigenspace.
An analogous result about characteristic Killing tensors of Stäckel systems is given in [8] making use of theorems due to Tonolo, Schouten and Nijenhuis (see Section 2). The main difference is due to the fact that, in that case, the eigenvalues of the tensors are simple.
The characterization of block-separation via Killing 2-tensors is extremely powerful in view of applications. For example, in any Riemannian manifold of constant curvature, all Killing tensors, of any order, are linear combinations with constant coefficients of symmetric products of Killing vectors, i.e. isometries [15]. Many common computer-algebra softwares include specific commands for the determination of Killing tensors of Riemannian manifolds.
Example 3. The four-body Calogero system. The N -body Calogero system is the Hamiltonian system of N points of unitary mass on a line, subject to the interaction The Hamiltonian is therefore in Cartesian coordinates (x i ) and it is known to be maximally superintegrable for any N and multiseparable for N < 4 [25,2]. Indeed, H (4) admits, other than the Hamiltonian, only two quadratic independent first integrals in involution, and not the three necessary for standard Stäckel separation. The two quadratic first integrals of H (4) can be chosen as follows where W a are suitable functions that we will make explicit later on, and x j x k , j, k = 1 . . . 4, j < k, j, k = i, where j, k = 1 . . . 4, l, m, r, s all different, of multiplicity 2. The eigenvalues of K 2 are 0 of multiplicity 1 and of multiplicity 3. Since the tensors of components K ij 1 and K ij 2 commute as linear operators and the metric is definite positive, they can be diagonalized simultaneously in some coordinate system. By using some properties of the eigenvalues of Killing tensors [4], one finds that these coordinates (r, φ 1 , φ 2 , φ 3 ) are spherical and determined by the consecutive transformations [6] V = Therefore, the block-separated Hamiltonians arẽ The dynamics of the original Hamiltonian H is therefore decomposed into three separated blocks, corresponding to the two dynamics of Hamiltonians H 1 ,H 2 , with one degree of freedom, and the two-degrees of freedom dynamics generated byH 3 .

Example 4.
Killing tensor with an eigenvalue of multiplicity N −1. If H admits a single quadratic first integral, this one determines block-separable coordinates if it has exactly one eigenvalue of multiplicity one and another one of multiplicity N − 1, so that we have a 2 × 2 Stäckel matrix. Indeed, from block-Eisenhart equations we have where X 1 is the eigenvector corresponding to the eigenvalue λ 1 of multiplicity one and X 2i the eigenvectors of λ 2 of multiplicity N − 1. Hence, provided λ 1 is not a constant, we have that the submanifolds λ 1 = const. are orthogonal to the eigenvector X 1 , which is therefore normally integrable. We can put in this case X 1 = ∂ 1 and the block separation is essentially determined by the equations λ 1 (q 2 , . . . , q N ) = const. moreover λ 2 (q 1 ).
We find in this way another (partial) analogy with Stäckel separation, since in that case, the existence of a single Killing 2-tensor with distinct eigenvalues in dimension two is enough to determine Stäckel separable coordinates and the eigenvalues themselves, if not constants, generate the separable coordinates.
In [4] we show that Stäckel coordinates can be completely determined from the eigenvalues of the associated Killing two-tensors. Part of those results can be easily extended to block-separable systems. However, we leave the analysis of these questions for future researches.
6 Block-separable coordinates of E 3 In dimension three, only two types of block-separable coordinates can exist, apart the trivial case of a single three-dimensional block. Either each block is one-dimensional, and the coordinates are standard separable orthogonal coordinates, or one block is one-dimensional an the other is two-dimensional. In this last case, denoting the separable coordinates as (u, v, w), the geodesic Hamiltonian is with and, since any 2-dimensional Riemannian manifold is locally conformally flat, where the choice of local Cartesian coordinates on the manifolds u = const. is not restrictive. Since g 1 can always be set equal to 1 by a rescaling of u, we can assume g 1 = 1 and call g 2 simply g.
The corresponding general block-Stäckel matrix has the form Since . We restrict ourselves to the space E 3 by imposing that the Riemann tensor of the metric G is identically zero. We have Hence, two cases are possible

Case i
We have S 1 1 = aS 2 1 , where a is a constant. We observe that the components of G become that is, we can write without restrictions with the obvious definitions of h, f and l.
A further rescaling of u allows to set h = 1 and u can always be considered an ignorable coordinate, therefore, associated with a Killing vector ∂ u .
The unknown functions are now reduced to two, and we can consider the remaining components of the Riemann tensor. The resulting equations are Proposition 14. The coordinate leaves u = const. are planes and ∂ u is proportional to a Killing vector.
Example 5. Rotational and cylindrical coordinates. Given in an Euclidean plane any coordinate system with a symmetry axis, the coordinates of E 3 obtained by rotating the plane around the symmetry axis and taking as third coordinate the angle of rotation are of this form. If f is constant and l −2 represents a metric in the Euclidean plane (v, w), then the cylindrical coordinate system with the plane (v, w) as base is of this form.

Case ii
A similar analysis for the case ii gives the equivalent condition We have , and we can write without restrictions with the obvious definitions of h, f and l.
Again, a rescaling of u allows to set h = 1, in this case, u is not necessarily ignorable, but the metric has the structure of a warped metric. Therefore, ∂ u must be parallel to a conformal Killing vector [7].
The equations arising from imposing Riemann tensor equal to zero are now The block-Stäckel matrix is in this case and the Hamiltonian H = α 1 1 H 1 + α 2 1 H 2 and first integral K = α 1 2 H 1 + α 2 2 H 2 are H = p 2 u + l 2 f 2 (p 2 v + p 2 w ), K = −ap 2 u + (1 − al 2 )f 2 (p 2 v + p 2 w ), since K = f 2 (p 2 v + p 2 w ) − aH, we have that the first integral associated with this kind of block-separation is f 2 (p 2 v + p 2 w ), as expected due to the warped form of the metric.
The solutions of (38) are that, substituted in the second equation, give We remark that (42) means that the Ricci scalar of the submanifolds u = const. is R = 2l 2 (u)c 2 1 , and its Riemann tensor has non null components Therefore, Proposition 15. In case ii, all the coordinate leaves orthogonal to ∂ u are planes (c 1 = 0) or spheres. The vector ∂ u is proportional to a conformal Killing vector.
Example 6. Spherical and cylindrical coordinates. An example of type ii of block separation is given by spherical-type coordinates, where u is the radius of the spheres and (v, w) are any coordinates on the sphere. If c 1 = 0, c 2 = 0, the coordinates are cylindrical with base the plane (v, w).

Remark 6.
From the previous remarks it follows that, if a coordinate system is block-separable in n > m blocks, then, not necessarily it is block separable in m blocks too. Indeed, if we consider the ellipsoidal coordinates in E 3 , they are Stäckel-separable, then block-separable in three blocks, but they cannot be blockseparable in 2 blocks, since these coordinates do not include planes or spheres. This fact is not surprising, since it is known that Stäckel separation in ellipsoidal coordinates cannot be achieved by successive separation of the single variables, and block-separation into two blocks in dimension three means exactly that one of the variables can be separated from the others.

Conclusions and future directions
By introducing the idea of twisted product of natural Hamiltonians and the analysis of the consequent relations among the Hamiltonian flows of the product and of its factors, we find a new, dynamical, interpretation of classical partial separation of variables of Hamilton-Jacobi equation, as well as of complete separation. We find that our block-separation, when possible, allows the reconstruction of the orbits of the product Hamiltonian from the orbits of the several lower-dimensional block-separated Hamiltonians. We characterize in an invariant way block-separation, by adapting classical results of complete separation theory for the Hamilton-Jacobi equation. We extend Eisenhart's classification of completely separable coordinate systems, the Stäckel systems, in E 3 to blockseparable coordinate systems, finding essentially coordinate-blocks of rotational, cylindrical, and spherical type. We are confident that the possibility of reducing the analysis of the dynamics of Hamiltonians with many degrees of freedom to the dynamics of its block-separated Hamiltonians can find many applications, even in the field of numerical computations. We do not consider here the blockseparation of Schrödinger's and other related equations of mathematical physics. Studies on partial separation of these equations are somehow more developed than those on partial separation of Hamilton-Jacobi equation, probably because the absence of completeness in partial separation and the consequent impossibility of application of the Jacobi's canonical transformation do not seem to be an obstacle in this case. We will study block-separation of Schrödinger and related equations in next papers.