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

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

Claudia Maria CHANU and Giovanni RASTELLI

C.M. Chanu and G. Rastelli

Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto
10, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received August 07, 2018, in final form February 14, 2019; Published online
February 23, 2019

We study twisted products $H=\alpha^{r}H_{r}$ 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-Stäckel 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 Stäckel separation
of variables. We classify the block-separable coordinates of $\mathbb{E}^{3}$.

Stäckel systems; partial separation of variables; position-dependent time
parametrisation

70H05; 37J15; 70H06

##  1 Introduction

In [28] Paul Stäckel started the study of (complete) separation of variables
in orthogonal coordinates for the Hamilton–Jacobi equation of natural
Hamiltonians with $N$ degrees of freedom. The characterization given by
Stäckel is both coordinate dependent – involving $N\times N$ Stäckel matrices
(see Theorem 2.1 below) – and invariant – involving $N$ quadratic first
integrals of the Hamiltonian. In the following years, the theory was widely
developed by Levi-Civita [21], Eisenhart [13, 14] and many others (see [18,
19, 24] for more complete references). We point out that complete separation
implies the completeness of the separated integral of the Hamilton–Jacobi
equation, because of the existence of $N$ independent constants of motion in
involution and, consequently, the Liouville integrability of the Hamiltonian
system.

Stäckel himself considered in [29] 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 the Hamiltonian. Partial separation of variables gained much less
interest than complete separation. This fact is certainly related with the
little use of partial separation in the search of solutions of the
Hamilton–Jacobi equation. Moreover, partial separation does not guarantee the
existence of complete integrals of the partially separated Hamilton–Jacobi
equation, neither Liouville integrability. Nevertheless, as pointed out by
[20], in this case the Jacobi method of inversion can, sometimes, produce
additional first integrals of the Hamiltonian. Recent papers develop partial
separation theory for Hamilton–Jacobi and Schrödinger equations, improving
somehow the results of Stäckel, by giving a more detailed characterization of
the metric coefficients in partially separable coordinates and by providing
further conditions for the separation of the quantum systems [17, 22]. Partial
separation of Hamilton–Jacobi and Helmholtz equations on four-dimensional
manifolds is briefly considered in [4]. A different approach to non-complete
additive separation is represented by non-regular separation which relies on
the existence of an additively separated solution on proper submanifolds only
[5, 6, 19].

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 in some case a simpler analysis of the original system. The separated
blocks of coordinates, considered together, form a $N$-dimensional coordinate
system on the base manifold of $H$ and in these coordinates the
$N$-dimensional metric tensor takes a block-diagonal form. This fact motivates
the name we choose for this kind of separation. We prefer to not use the
expression partial separation since it is already associated with
Hamilton–Jacobi theory, which we do not consider here. By shifting the focus
from Hamilton–Jacobi theory to the dynamics, we remove the obstruction
represented by the completeness of the integral of the Hamilton–Jacobi
equation, which is strictly connected with Stäckel theory of complete
separation of variables. Indeed, 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) generally
fails. On the contrary, our dynamical interpretation of block separation is
basically insensitive to complete or partial separation. 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 and Eisenhart equations, and of
more recent theorems about complete separation.

The main result of block-separation is the splitting of a $N$-dimensional
Hamiltonian system into lower-dimensional systems (the blocks). Thus, methods
of analysis disposable only for low-dimensional systems become available, such
as, for example, the topological classification of integrable Hamiltonian
systems [3].

In Section 2 we recall the basic theorems about Stäckel complete separation of
variables which we rewrite 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 block-like formulations of Levi-Civita, Eisenhart and other theorems,
and we provide an invariant characterization of block separation. The explicit
example we deal with is the four-body Calogero system. In Section 6, we
characterize, at least with necessary conditions, all the possible block-
separable coordinates of $\mathbb{E}^{3}$. Section 7 contains our final
considerations and comments.

##  2 Outline of Stäckel separation

We briefly recall the principal theorems regarding complete separation of the
Hamilton–Jacobi equation, see [18] and [1] for further details. The theory of
complete additive separation of the Hamilton–Jacobi equation begins with the
work of Stäckel [28, 29] about separability of the Hamilton–Jacobi equation in
orthogonal coordinates. The Einstein summation convention on equal indices is
understood, unless otherwise stated.

######  Theorem 2.1.

In a given orthogonal coordinate system $(q^{i})$, the Hamilton–Jacobi
equation

$\displaystyle H=\frac{1}{2}\left(g^{ii}\left(\frac{\partial W}{\partial
q^{i}}\right)^{2}+V(q)\right)=c_{1},$ (2.1)

admits a complete integral in the separated form,

$\displaystyle W=\sum_{i=1}^{N}W_{i}\big{(}q^{i},c_{1},c_{a}\big{)},\qquad
a=2,\dots,N,\qquad\det\left(\frac{\partial^{2}W}{\partial q^{i}\partial
c_{j}}\right)\neq 0,\qquad c_{j}=(c_{1},c_{a}),$

if and only if

  1. $1)$

there exists a $N\times N$ matrix $S$ which is invertible and such that each
element of its $j$-th row depends on $q^{j}$ only, such that the
$\big{(}g^{ii}\big{)}$ are a row of $S^{-1}$. The matrix $S$ is called Stäckel
matrix.

  2. $2)$

$V$ is a Stäckel multiplier, i.e., there exist $N$ functions
$v_{i}\big{(}q^{i}\big{)}$ such that

$\displaystyle V=v_{i}\big{(}q^{i}\big{)}g^{ii}.$

As a consequence of $(1)$ and $(2)$, there exist $N-1$ independent quadratic
first integrals $(K_{a})$ of $H$ $(a=2,\dots,N)$ such that
$(c_{i})=(c_{1},c_{a})$ are the constant values of $(H,K_{a})$, where
$(K_{a})=K_{2},\dots,K_{N}$.

A complete integral of (2.1) is then determined by the $N$ separated equations

$\displaystyle\left(\frac{{\rm d}W_{r}}{{\rm
d}q^{r}}\right)^{2}+v_{r}=2S^{i}_{r}c_{i},$

where all $S^{i}_{r}$ depend $(i=1,\dots,N)$ on the coordinate $q^{r}$ only.

We remark that completeness for the integral $W$ of the Hamilton–Jacobi
equation means that it depends on $N$ parameters $(c_{j})$, constants of
motion, such that

$\displaystyle\det\left(\frac{\partial W}{\partial q^{i}\partial
c_{j}}\right)\neq 0.$

Therefore, the $(c_{j})$ can be part of a new set of canonical coordinates in
which the Hamiltonian flow becomes trivially integrable.

Later, Levi-Civita [21] obtained necessary and sufficient conditions for the
complete separability of a generic Hamiltonian in a general coordinate system.
For natural Hamiltonians and orthogonal coordinates,
$H=\frac{1}{2}g^{ii}p_{i}^{2}+V(q)$, the Levi-Civita equations split into

$\displaystyle
g^{ii}g^{jj}\partial_{ij}g^{kk}-g^{ii}\partial_{i}g^{jj}\partial_{j}g^{kk}-g^{jj}\partial_{j}g^{ii}\partial_{i}g^{kk}=0,$
(2.2)

with $i\neq j$ not summed, $i,j,k=1,\dots,N$, for the components of the metric
tensor, and

$\displaystyle
g^{ii}g^{jj}\partial_{ij}V-g^{ii}\partial_{i}g^{jj}\partial_{j}V-g^{jj}\partial_{j}g^{ii}\partial_{i}V=0,$

with $i\neq j$ not summed, $i,j=1,\dots,N$, called Bertrand–Darboux equations
and whose solution (for $g^{ii}$ satisfying (2.2)) is $V$ in the form of a
Stäckel multiplier.

In [13], 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 $\mathbb{E}^{3}$. He determined eleven types of orthogonal
separable coordinate systems, which are described, for example, in [24]. The
fundamental Eisenhart equations

$\displaystyle\partial_{i}\lambda_{j}=(\lambda_{i}-\lambda_{j})\partial_{i}\ln\big{|}g^{jj}\big{|},\qquad
i,j=1,\dots,N$

characterize the eigenvalues of a Killing tensor: let $K$ be a 2-tensor with
eigenvalues $(\lambda_{i})$ and eigenvectors $(\partial_{i})$, then $K$ is a
Killing tensor if and only if the Eisenhart equations are satisfied. The
integrability conditions of the Eisenhart equations for a Killing tensor with
simple eigenvalues coincide with the Levi-Civita equations (2.2).

The coordinate systems can be geometrically understood as foliations of
hypersurfaces called coordinate webs. The 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 (up to possible
singular sets of zero measure) the basis of coordinate vectors [1]. We recall
that a symmetric Killing $2$-tensor $K$ is defined by the equivalent equations

$\displaystyle[g,K]=0,\qquad\nabla_{(i}K_{jk)}=0,$

where $[\cdot,\cdot]$ is the Schouten bracket and $\nabla$ is the covariant
derivative with respect to the metric $g$.

######  Theorem 2.2.

The Hamilton–Jacobi equation of a natural Hamiltonian with scalar potential
$V$ is completely separable in an orthogonal coordinate web if and only if
there exists a characteristic Killing $2$-tensor $K$ whose eigenvectors are
normal to the foliations of the web and such that ${\rm d}(K{\rm d}V)=0$.

The last condition is equivalent to say that $V$ is a Stäckel multiplier in
the orthogonal coordinates associated with $K$. In the formula of Theorem 2.2
$K$ is considered as a linear operator mapping one-forms into one-forms.

Necessary and sufficient conditions for a Killing tensor to be characteristic
are given in [10].

######  Theorem 2.3 (Tonolo–Schouten–Nijenhuis [26, 27, 30]).

A $2$-tensor $K$ with real distinct eigenvalues has normal eigenvectors if and
only if the following conditions are satisfied

$\displaystyle N^{l}_{[ij}g_{k]l}=0,\qquad N^{l}_{[ij}K_{k]l}=0,\qquad
N^{l}_{[ij}K_{k]m}K^{m}_{l}=0,$

where $N^{i}_{jk}$ are the components of the Nijenhuis tensor of $K^{i}_{j}$
defined by

$\displaystyle N^{i}_{jk}=K^{i}_{l}K_{[j,k]}+K^{l}_{[j}K^{i}_{k],l}.$

An equivalent formulation of Theorem 2.2 involves $N$ independent quadratic
first integrals, therefore, $N$ Killing 2-tensors, instead of a single
characteristic Killing tensor [1].

######  Theorem 2.4.

The natural Hamiltonian $H$ is separable in some orthogonal coordinates
$\big{(}q^{i}\big{)}$, if and only if

  1. $1)$

there exist other $N-1$ independent quadratic in the momenta functions $K_{a}$
such that

$\displaystyle\\{H,K_{a}\\}=0,$

  2. $2)$

the Killing two-tensors $(k_{a})$ associated with $(K_{a})$ are simultaneously
diagonalized with pointwise independent eigenvalues and have common normally
integrable eigevectors.

It follows that $\\{K_{a},K_{b}\\}=0$.

The original formulation of the theorem requires the reality of the
eigenvalues, however, this request is unnecessary if one accepts also complex
separable coordinates [11].

##  3 Twisted products of Hamiltonians

Let

$\displaystyle M=\times_{r=1}^{n}M_{r},$

be the product of $n$ Riemannian or pseudo-Riemannian manifolds
$(M_{r},g_{r})$ of dimension $n_{r}$, so that dim$(M)=n_{1}+\dots+n_{n}=N$,
and let $\alpha^{r}$ be $n$ non zero functions on $M$. The manifold $M$ with
metric tensor

$\displaystyle G=\alpha^{1}g_{1}+\cdots+\alpha^{n}g_{n},$

is a Riemannian or pseudo-Riemannian manifold called twisted product manifold
of the $(M_{r})$ with twist functions $(\alpha^{r})$ [23]. In the case when
$\alpha^{1}=1$ and $\alpha^{2},\dots,\alpha^{n}$ 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

$\displaystyle
H_{r}=\frac{1}{2}\big{(}g_{r}^{r_{i}r_{j}}p_{r_{i}}p_{r_{j}}+V_{r}\big{(}q^{r_{i}}\big{)}\big{)},$

where $\big{(}q^{r_{i}},p_{r_{i}}\big{)}$, $i=1,\dots,n_{r}$, are canonical
coordinates on $T^{*}M_{r}$, we construct twisted product

$\displaystyle H=\alpha^{r}H_{r}=\alpha^{1}H_{1}+\dots+\alpha^{n}H_{n},$

of the $H_{r}$ with twist functions $\alpha^{r}\in\mathcal{F}(M)$.

Then, $H$ is a natural Hamiltonian on $T^{*}M$ with metric $G$ and potential
$V=\alpha^{r}V_{r}$.

The manifold $M$ is naturally endowed with block-diagonal coordinates
$(q^{r_{i}})$ such that the components of $G$ are in the form

$\displaystyle G^{r_{i}r_{j}}=\alpha^{r}g^{r_{i}r_{j}}_{r},\qquad
G^{r_{i}s_{j}}=0,\qquad s\neq r,$

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 $\tau_{r}$ the Hamiltonian
parameter of $H_{r}$. From Hamilton’s equations we get

$\displaystyle\frac{{\rm d}q^{r_{i}}}{{\rm d}t}=\frac{\partial H}{\partial
p_{r_{i}}}=\alpha^{r}\frac{\partial H_{r}}{\partial
p_{r_{i}}}=\alpha^{r}\frac{{\rm d}q^{r_{i}}}{{\rm d}\tau_{r}},$

and

$\displaystyle\frac{{\rm d}p_{r_{i}}}{{\rm d}t}=-\frac{\partial H}{\partial
q^{r_{i}}}=-\alpha^{r}\frac{\partial H_{r}}{\partial
q^{r_{i}}}-H_{s}\frac{\partial\alpha^{s}}{\partial
q^{r_{i}}}=\alpha^{r}\frac{{\rm d}p_{r_{i}}}{{\rm
d}\tau_{r}}-H_{s}\frac{\partial\alpha^{s}}{\partial q^{r_{i}}}.$

Therefore, the relation between the Hamiltonian vector fields $X_{H}$ of $H$
and $X_{r}$ of $H_{r}$ is

$\displaystyle
X_{H}=\bar{X}_{1}+\dots+\bar{X}_{N}-H_{s}\frac{\partial\alpha^{s}}{\partial
q^{r_{i}}}\frac{\partial}{\partial p_{r_{i}}},$

where

$\displaystyle\bar{X}_{r}=\alpha^{r}X_{r},\qquad r\;\text{not summed},$

is the rescaled Hamiltonian vector field of $H_{r}$.

##  4 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
$N$-dimensional Hamiltonian system determined by $H$ and the dynamics of the
$N$ one-dimensional Hamiltonians $H_{r}$, so that $n_{1}=\dots=n_{N}=1$,
determined by the separated equations of $H$. Separation of variables for the
Hamilton–Jacobi equation of $H$ will not be of primary interest in what
follows. See Section 2 for definitions of Stäckel matrix, Stäckel multiplier
and separated equations.

Let be $H_{r}=\frac{1}{2}(p_{r}^{2}+V_{r})$ and assume that $\alpha^{r}$ are a
row (say the first one) of the inverse of a Stäckel matrix $S$ for given
coordinates $(q^{r})$. Then, the twisted product $H=\alpha^{r}H_{r}$ admits
separation of variables and we have the separated equations

$\displaystyle H_{r}=S^{i}_{r}c_{i},$ (4.1)

where $c_{i}$ are $N$ constants, corresponding to the $N$ constants of motion
$K_{i}$ of $H=K_{1}$ and

$\displaystyle K_{a}=\big{(}S^{-1}\big{)}_{a}^{r}H_{r},\qquad a=2,\dots,N.$

The Hamilton–Jacobi complete separated integral
$W=W_{1}\big{(}q^{1},c_{i}\big{)}+\dots+W_{N}\big{(}q^{N},c_{i}\big{)}$ is
given by integration of

$\displaystyle\left(\frac{{\rm d}W_{r}}{{\rm
d}q^{r}}\right)^{2}+V_{r}=2S_{r}^{i}c_{i}.$

The Hamilton’s equations of $H$, in time $t$, are

$\displaystyle\dot{q}^{r}=\alpha^{r}p_{r},$
$\displaystyle\dot{p}_{r}=-\partial_{r}\alpha^{i}H_{i}-\alpha^{i}\partial_{r}H_{i}=-\partial_{r}\alpha^{i}S_{i}^{j}c_{j}-\frac{1}{2}\alpha^{r}\frac{\rm
d}{{\rm d}q^{r}}V_{r},$ (4.2)

where we use the separated equations (4.1) to replace $H_{r}$ along the
integral curves. Since $\alpha^{r}=\big{(}S^{-1}\big{)}^{r}_{1}$ is a row of
the inverse of $S$, we have

$\displaystyle\partial_{r}\alpha^{r}S_{i}^{j}=\partial_{r}\big{(}\alpha^{i}S_{i}^{j}\big{)}-\alpha^{i}\partial_{r}S^{j}_{i}=\partial_{r}\big{(}\delta^{j}_{1}\big{)}-\alpha^{r}\frac{{\rm
d}}{{\rm d}q^{r}}S^{j}_{r}=-\alpha^{r}\frac{{\rm d}S^{j}_{r}}{{\rm d}q^{r}},$
(4.3)

and the same for all other elements of the rows of $S^{-1}$. Then, we can
write

$\displaystyle\dot{p}_{r}=\alpha^{r}\frac{{\rm d}}{{\rm
d}q^{r}}\left(c_{j}S^{j}_{r}-\frac{1}{2}V_{r}\right).$ (4.4)

Let $\gamma_{P}$ be the integral curve of $X_{H}$ containing a point $P\in
T^{*}M$. We consider the values $c_{i}=K_{i}(P)$ and we introduce the
Hamiltonians

$\displaystyle\tilde{H}_{r}=H_{r}-c_{j}S^{j}_{r},$

with Hamiltonian parameters $\tilde{\tau}_{r}$. We can write the equations of
Hamilton for $\tilde{H}_{r}$ as

$\displaystyle\frac{{\rm d}}{{\rm
d}\tilde{\tau}_{r}}q^{r}=p_{r}=\partial^{r}\tilde{H}_{r},$ (4.5)
$\displaystyle\frac{\rm d}{{\rm d}\tilde{\tau}_{r}}p_{r}=\frac{{\rm d}}{{\rm
d}q^{r}}\left(c_{j}S^{j}_{r}-\frac{1}{2}V_{r}\right)=-\partial_{r}\tilde{H}_{r}.$
(4.6)

Therefore,

######  Proposition 4.1.

For each orbit of $H$, the $N$ Hamiltonian vector fields $X_{\tilde{H}_{r}}$
of the $\tilde{H}_{r}$ are proportional to the components with respect to
$(\partial_{r},\partial^{r})$ of the Hamiltonian vector field $X_{H}$ of $H$,
where $\alpha^{r}$ are the proportionality functions.

###### Proof.

Let be

$\displaystyle X_{\tilde{H}_{r}}=\frac{\partial\tilde{H}_{r}}{\partial
p_{r}}\partial_{r}-\frac{\partial\tilde{H}_{r}}{\partial q^{r}}\partial^{r},$

and

$\displaystyle(X_{H})_{r}=\dot{q}^{r}\partial_{r}+\dot{p}_{r}\partial^{r}.$

Due to (4.5) and (4.6) we can write (4.2) and (4.4) as

$\displaystyle\dot{q}^{r}=\alpha^{r}\partial^{r}\tilde{H}_{r},\qquad\dot{p}_{r}=-\alpha^{r}\partial_{r}\tilde{H}_{r},$

and it follows immediately

$\displaystyle(X_{H})_{r}=\alpha^{r}X_{\tilde{H}_{r}}.$ ∎

######  Remark 4.2.

After Proposition 4.1 we can put

$\displaystyle\alpha^{r}=\frac{{\rm d}\tilde{\tau}_{r}}{{\rm d}t},$

and consider the twist functions as determining position-dependent time-
scalings between the Hamiltonian parameters $t$ and $\tilde{\tau}_{r}$.

From Proposition 4.1 follows the important result

######  Proposition 4.3.

The projection of each orbit of $H$ on each coordinate manifold
$\big{(}q^{r},p_{r}\big{)}$ coincides with the orbit of
$\tilde{H}_{r}=H_{r}-c_{j}S^{j}_{r}$.

######  Remark 4.4.

The Lagrange equations of the dynamics of $\tilde{H}_{r}$, expressed in times
$\tilde{\tau}_{r}$ are

$\displaystyle\frac{{\rm d}^{2}q^{r}}{{\rm d}\tilde{\tau}_{r}^{2}}=\frac{{\rm
d}}{{\rm d}q^{r}}\left(c_{j}S^{j}_{r}-\frac{1}{2}V_{r}\right).$

######  Remark 4.5.

Observe that

$\displaystyle\tilde{H}=\alpha^{r}\tilde{H}_{r}=H-c_{1},\qquad\tilde{K}_{j}=\big{(}S^{-1}\big{)}^{r}_{j}\tilde{H}_{r}=K_{j}-c_{j},\qquad
j\neq 1,$

i.e., the Stäckel systems associated with $H$ and $\tilde{H}$ coincide up to
additive constants. To the constants $(c_{i})$ for $(H,K_{a})$ correspond the
costants $(\tilde{c}_{i}=0)$ for $\big{(}\tilde{H},\tilde{K}_{a}\big{)}$.

######  Example 4.6.

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

$\displaystyle H_{i}=\frac{1}{2}\big{(}p_{i}^{2}-\cos q^{i}\big{)},\qquad
i=1,2,\qquad H_{3}=\frac{1}{2}p_{3}^{2},$

corresponding to two pendula and a purely inertial term, coupled together by
the first row of the inverse of the $3\times 3$ matrix

$\displaystyle S=\left(\begin{matrix}2&&1+q^{1}&&2\big{(}q^{1}\big{)}^{2}+2\\\
3&&q^{2}&&\big{(}q^{2}\big{)}^{3}+2\\\
4&&q^{3}&&\big{(}q^{3}\big{)}^{2}+1\end{matrix}\right),$

which is a Stäckel matrix in a neighborhood of the origin, since the Taylor
expansion up to the second order terms of its determinant $\Delta$ around
$(0,0,0)$ is $\Delta=5+5q^{1}-6q^{2}+2q^{3}$. The elements of the matrix
$S^{-1}$ are therefore quite complicated rational functions that we do not
need to compute explicitly but make the coupling of the $H_{r}$ suitable to
enhance the effect of the time-scaling. We take as $(\alpha^{r})$ the first
row of $S^{-1}$ and consider

$\displaystyle H=\alpha^{r}H_{r}=c_{1}.$

The quadratic first integrals of $H$ are determined by the remaining rows of
$\big{(}S^{-1}\big{)}$

$\displaystyle K_{a}=\big{(}S^{-1}\big{)}_{a}^{r}H_{r}=c_{a},\qquad a=2,3.$

We already know from the previous section that, despite the complicated
expression of the coupling terms $(\alpha^{r})$, the relation among the
dynamics of $H$ and of the separated Hamiltonians
$\tilde{H}_{r}=H_{r}-c_{a}S^{a}_{r}$ reduces to a simple position-dependent
time scaling.

We plot the numerical evaluation of the systems of Hamiltonian $H$ and

$\displaystyle\tilde{H}_{1}=H_{1}-2c_{1}-\big{(}1+q^{1}\big{)}c_{2}-2\big{(}\big{(}q^{1}\big{)}^{2}+1\big{)}c_{3}$

respectively, and project the orbits on $\big{(}q^{1},p_{1}\big{)}$,
obtaining, for the initial conditions $p_{1}=0$, $p_{2}=0$, $p_{3}=0$,
$q^{1}=0.2$, $q^{2}=-0.2$, $q^{3}=0$ and consequently $c_{1}=H=0.09494666248$,
$c_{2}=0.0916913483$, $c_{3}=-0.3797866499$, the graphs in Fig. 1,

Figure 1: Projections of the orbits of $H$ and $\tilde{H}_{1}$ on
$\big{(}p_{1},q^{1}\big{)}$.

where we see that the orbits on $\big{(}q^{1},p_{1}\big{)}$ of the two systems
coincide. But, if we include the dependence on the different Hamiltonian
parameters (denoted in both the graphs as $t$), we get Fig. 2

Figure 2: Projections of the orbits of $H$ and $\tilde{H}_{1}$ on
$\big{(}t,p_{1},q^{1}\big{)}$ and $\big{(}\tau_{1},p_{1},q^{1}\big{)}$.

and we can see how the dependence on the Hamiltonian parameters can be
extremely different in the two cases.

######  Example 4.7.

_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

$\displaystyle
H_{i}=\frac{1}{2}\big{(}p_{i}^{2}+\omega_{i}^{2}\big{(}q^{i}\big{)}^{2}\big{)},\qquad
i=1,\dots,n$

be a finite set of harmonic oscillators. Let

$\displaystyle H=\alpha^{i}H_{i},\qquad\alpha^{i}\in\mathbb{R}^{+},$

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

$\displaystyle\frac{{\rm d}q^{i}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle\alpha^{i}\frac{{\rm d}q^{i}}{{\rm d}\tau_{i}}=\alpha^{i}p_{i},$
$\displaystyle\frac{{\rm d}p_{i}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle-\alpha^{i}\frac{{\rm d}p_{i}}{{\rm
d}\tau_{i}}=-\alpha^{i}\omega_{i}^{2}q^{i}.$

The general solution of these equations is

$\displaystyle
q^{i}(t)=c_{1}^{i}\sin\big{(}\alpha^{i}\omega^{i}t\big{)}+c_{2}^{i}\cos\big{(}\alpha^{i}\omega^{i}t\big{)},\qquad
c^{i}_{j}\in\mathbb{R}.$

We see that, for example, the choice $\alpha^{i}=k/\omega_{i}$, 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 real positive
frequency for different choices of $k$ for each $i$). Namely, the rescaling is
in this case

$\displaystyle
t=\sum_{i=1}^{n}\frac{1}{\alpha^{i}}\tau_{i}+t_{0},\qquad\text{or}\qquad\tau_{i}=\alpha^{i}t+\tau_{i}^{0}.$

The frequency of each oscillator $H_{i}$ with respect to its own Hamiltonian
parameter $\tau_{i}$ remains clearly $\omega_{i}$.

##  5 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 $n\leq N$, consider for
each integer $r=1,\dots,n$ the integers $n_{r}$ such that

$\displaystyle N=n_{1}+\dots+n_{n}.$

The coordinate system is therefore composed of $n$ blocks, and for each $r\leq
n$ we have an $r$-block of coordinates that we denote as
$\big{(}q^{r_{1}},\dots,q^{r_{n_{r}}}\big{)}$. We call $M_{r}$ the manifold
spanned by the $r$-block of coordinates. Consider $T^{*}M$ with the conjugate
$r$-block momenta $(p_{r_{1}},\dots,p_{r_{n_{r}}})$.

Let us consider

$\displaystyle
H_{r}=\frac{1}{2}g^{r_{i}r_{j}}_{r}p_{r_{i}}p_{r_{j}}+V_{r}\big{(}q^{r_{k}}\big{)},$

and the $n$ block-separated equations

$\displaystyle H_{r}=S_{r}^{a}c_{a},\qquad a=1,\dots,n,$

where we assume that $g_{r}^{r_{i}r_{j}}$, $S_{r}^{a}$ and $V_{r}$ are
functions of coordinates of the $r$-block only and $c_{a}$ are constants. If
we assume that the $n\times n$ matrix $(S_{r}^{a})$ is invertible, and we call
it block-Stäckel matrix, then we can write the $n$ equations

$\displaystyle\big{(}S^{-1}\big{)}_{a}^{r}H_{r}=c_{a}.$ (5.1)

We denote $\alpha^{r}=\big{(}S^{-1}\big{)}_{1}^{r}$ and call

$\displaystyle H=\alpha^{r}H_{r},\qquad
K_{a}=\big{(}S^{-1}\big{)}_{a}^{r}H_{r},\qquad a=2,\dots,n.$

Hence, $H$ is in the form of twisted product and it is a natural Hamiltonian
whose metric tensor $G$ is block-diagonalized, with components

$\displaystyle G^{r_{i}r_{j}}=\alpha^{r}g^{r_{i}r_{j}}_{r},\qquad
G^{r_{i}s_{j}}=0,\qquad s\neq r,$

and whose scalar potential has the form

$\displaystyle V=\alpha^{r}V_{r},$

while the scalar potentials in $K_{a}$ are

$\displaystyle W_{a}=\big{(}S^{-1}\big{)}_{a}^{r}V_{r}.$ (5.2)

A necessary condition for the procedure of above is that the (5.1) are indeed
constants of motion of $H$.

######  Proposition 5.1.

The $n$ functions $(H,K_{a})$ are all independent, quadratic in the momenta
and pairwise in Poisson involution.

###### Proof.

Since $S$ is a block-Stäckel matrix, in analogy to (4.3) we have

$\displaystyle\partial_{r_{i}}\alpha^{s}_{a}S_{s}^{j}=\partial_{r_{i}}\big{(}\alpha^{s}_{a}S_{s}^{j}\big{)}-\alpha^{s}_{a}\partial_{r_{i}}S^{j}_{s}=\partial_{r_{i}}\big{(}\delta^{j}_{1}\big{)}-\alpha^{r}_{a}\partial_{r_{i}}S^{j}_{r}=-\alpha^{r}_{a}\partial_{r_{i}}S^{j}_{r},$

where $r$ is not summed and $\alpha^{r}_{a}=\big{(}S^{-1}\big{)}^{r}_{a}$.
Then, from the definition of Poisson bracket and of block-separated
coordinates, we get the statement. ∎

For the dynamics of $H$, an analogue of Proposition 4.3 holds.

######  Proposition 5.2.

The dynamics of $H$ coincides in each $r$-block with the dynamics of

$\displaystyle\tilde{H}_{r}=H_{r}-c_{a}S^{a}_{r},$

up to a reparametrization of the Hamiltonian parameter given by
$\alpha^{r}={\rm d}\tilde{\tau}_{r}/{\rm d}t$. So, the projections of the
orbits of $H$ on each $T^{*}M_{r}$ coincide with the orbits of
$\tilde{H}_{r}$.

###### Proof.

The proof follows the same reasoning of the proof of Proposition 4.3. It
follows that, denoting with $(X_{H})_{r}$ the $r$-block component of the
Hamiltonian vector field $X_{H}$ of $H$,

$\displaystyle(X_{H})_{r}=\dot{q}^{r_{i}}\partial_{r_{i}}+\dot{p}_{r_{i}}\partial^{r_{i}},$

we have

$\displaystyle(X_{H})_{r}=\alpha^{r}X_{\tilde{H}_{r}},$

where $X_{\tilde{H}_{r}}$ is the Hamiltonian vector field of $\tilde{H}_{r}$.
Therefore, if $(X_{H})_{r}$ is tangent to any submanifold $f\subseteq T^{*}M$,
then also $X_{\tilde{H}_{r}}$ 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 the
$\tilde{H}_{r}$, with the difference that the $\tilde{H}_{r}$ 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 Hamiltonians
$\tilde{H}_{r}$. The $\tilde{H}_{r}$ 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
[12] and generalized by Stäckel in [29]. He introduced the $n\times n$ matrix
$S$ and his results are analogue to our Proposition 5.1. Stäckel obtained
sufficient conditions for partial separation of the Hamilton–Jacobi equation
of natural Hamiltonians. His work has been extended more recently in [17] and
[22], 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 [29] and
[17], since these last results 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 [17] 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
Hamilton–Jacobi equation. This is understandable when one considers that
Hamilton–Jacobi theory is of not easy application, apart the simplest cases,
even when completely separated integrals of the Hamilton–Jacobi equations do
exist. Some applications of partially separated integrals of the
Hamilton–Jacobi equation, in order to generate new possible first integrals of
the Hamiltonian, are presented in [20] and [25]. Our approach based upon the
block-separated dynamics, instead of the partially separated Hamilton–Jacobi
equation, appears to be completely new and could be more suitable for
applications of the theory, particularly in the analysis of systems with many
degrees of freedom.

###  5.1 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 $\big{(}q^{i}\big{)}$ are twisted coordinates
$\big{(}q^{r_{i}}\big{)}$, then for $G$ we have

$\displaystyle G_{r_{i}r_{j}}=\frac{1}{\alpha^{r}}g_{r_{i}r_{j}}^{r},\qquad r\
\text{not summed},$

so that

$\displaystyle G_{r_{k}a}G^{as_{j}}=\delta_{r_{k}}^{s_{j}}.$

Moreover, if we assume that the coordinates $\big{(}q^{r_{i}}\big{)}$ are
block-separated

$\displaystyle
k^{r_{i}r_{j}}_{a}=\big{(}S^{-1}\big{)}^{r}_{a}g^{r_{i}r_{j}}_{r},$

where $k^{bc}_{a}$ are the components of the 2-tensor associated with $K_{a}$.
Then,

$\displaystyle(k_{a})^{r_{i}}_{r_{j}}=\frac{1}{\alpha^{r}}\big{(}S^{-1}\big{)}^{r}_{a}\delta^{r_{i}}_{r_{j}},$

and we can consider the functions

$\displaystyle\lambda_{a}^{r}=\frac{1}{\alpha^{r}}\big{(}S^{-1}\big{)}^{r}_{a},\qquad
r\;\text{not summed},$

as the analogue of eigenvalues of Killing tensors $k_{a}$ in Stäckel theory.
They are indeed the eigenvalues with respect to $G$ of the Killing tensors
$(k_{a})$ associated with the first integrals $(K_{a})$.

######  Proposition 5.3.

In block-separated coordinates, we have that

$\displaystyle\\{H,K_{a}\\}=0,$

if and only if the block-Eisenhart equations

$\displaystyle\partial_{r_{k}}\lambda^{s}_{a}=\big{(}\lambda_{a}^{r}-\lambda_{a}^{s}\big{)}\partial_{r_{k}}\ln|\alpha^{s}|,$
(5.3)

hold, with $r$, $s=1,\dots,n$, for all $r_{i}$, $s_{j}$ in the respective
separated blocks, and (5.2) hold.

###### Proof.

By expanding

$\displaystyle\\{H,K_{a}\\}=0,$

in block-separable coordinates, collecting homogeneous terms in the momenta
and dividing by $\alpha^{r}\alpha^{s}$, from the higher order terms in the
momenta we have,

$\displaystyle\big{(}\lambda^{s}_{a}-\lambda^{r}_{a}\big{)}\partial_{r_{k}}g^{s_{i}s_{j}}+g^{s_{i}s_{j}}\big{[}\partial_{r_{k}}\lambda^{s}_{a}-\big{(}\lambda_{a}^{r}-\lambda_{a}^{s}\big{)}\partial_{r_{k}}\ln|\alpha^{s}|\big{]}=0,$
(5.4)

for all $r$, $s$, $r_{k}$, $s_{i}$, $s_{j}$ in the respective blocks. If $r=s$
the equations become

$\displaystyle g^{s_{i}s_{j}}\partial_{s_{k}}\lambda^{s}_{a}=0,$

and, if $r\neq s$, then $\partial_{r_{k}}g^{s_{i}s_{j}}=0$. Hence, (5.4) is
equivalent to

$\displaystyle
g^{s_{i}s_{j}}\big{[}\partial_{r_{k}}\lambda^{s}_{a}-\big{(}\lambda_{a}^{r}-\lambda_{a}^{s}\big{)}\partial_{r_{k}}\ln|\alpha^{s}|\big{]}=0.$

If $g^{s_{i}s_{j}}=0$, the equations are identically satisfied, otherwise, we
have (5.3). Since not all the $g^{s_{i}s_{j}}$ are zero, we have the
statement. The first-order terms in the momenta vanish if and only if (5.2)
hold. ∎

By definition of the $\lambda_{a}^{s}$ and of the $\alpha^{s}$, we have the
equations

$\displaystyle S^{s}_{r}\alpha^{r}=\delta^{s}_{1},\qquad
S^{s}_{r}\lambda_{a}^{r}\alpha^{r}=\delta^{s}_{a}.$

We observe that, after putting $\alpha^{r}=g^{rr}$, 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 5.4.

The block-Eisenhart equations (5.3) 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 considered
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
$\big{(}q^{1},\dots,q^{N}\big{)}$ are block-separated for the Hamiltonian $H$
if and only if the block-Levi-Civita equations

$\displaystyle\alpha^{r}\alpha^{s}\partial_{r_{i}s_{j}}\alpha^{m}-\alpha^{r}\partial_{r_{i}}\alpha^{s}\partial_{s_{j}}\alpha^{m}-\alpha^{s}\partial_{s_{j}}\alpha^{r}\partial_{r_{i}}\alpha^{m}=0,$

are satisfied, where the coordinates $r_{i}$, $s_{j}$ are in different blocks
and $m=1,\dots,n$. The scalar potential $V$ is already in the form of a block-
Stäckel multiplier, thanks to the form of $H$, and satisfies

$\displaystyle\alpha^{r}\alpha^{s}\partial_{r_{i}s_{j}}V-\alpha^{r}\partial_{r_{i}}\alpha^{s}\partial_{s_{j}}V-\alpha^{s}\partial_{s_{j}}\alpha^{r}\partial_{r_{i}}V=0.$

See (2.2) for a comparison.

###  5.2 Invariant characterization

As in Stäckel theory, we can use the previous results for an invariant
characterization of block-separation. Therefore, we have the analogue of the
Eisenhart–Kalnins–Miller–Benenti theorem [1].

######  Proposition 5.5.

The twisted Hamiltonian $H=\alpha^{r}H_{r}$, $r=1,\dots,n$, is block-separated
in twisted coordinates $\big{(}q^{r_{i}}\big{)}$ if and only if

  1. $1)$

there exist other $n-1$ independent quadratic in the momenta functions
$K_{a}=\alpha_{a}^{r}H_{r}$ such that

$\displaystyle\\{H,K_{a}\\}=0,$

  2. $2)$

the Killing two-tensors $(k_{a})$ are simultaneously block-diagonalized and
have common normally integrable eigenspaces

Moreover, it follows that $\\{K_{a},K_{b}\\}=0$.

######  Remark 5.6.

The proof of the previous statement relies on the assumption that each $H_{r}$
depends only on coordinates in $T^{*}M_{r}$. Otherwise, if the metric tensor
of $H$ is only block-diagonal in $\big{(}q^{r_{i}}\big{)}$, conditions 1) and
2) are only necessary.

Consequently, the strategy for finding block-separated coordinates of a given
$N$-dimensional natural Hamiltonian $H$ is the following

  * •

Find a number $n\leq N$ 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
$H=\alpha^{r}H_{r}$ for some functions $H_{r}$, the $\alpha^{r}$ being
determined by the block-Stäckel matrix determined by $(k_{a})$.

  * •

The last step consists in checking that each $H_{r}$ depends only on
coordinates in $T^{*}M_{r}$.

This is indeed the procedure applied in Example 5.9.

We recall (see [15]) that in a Riemannian manifold any symmetric 2-tensor is
pointwise diagonalizable, that is there exist $n$ vector fields $E_{i}$,
pointwise orthogonal eigenvectors of $k$, such that
$k=\sum_{i}\lambda^{i}E_{i}\otimes E_{i}$. The $\lambda^{i}$ are the roots of
the characteristic equation $\det(k-\lambda g)=0$ and the geometric
multiplicity of each eigenvalue $\lambda^{i}$ coincides with its algebraic
multiplicity.

In our case, the algebraic multiplicity of each $\lambda^{r}_{a}$ is $n_{r}$
at least (for some $a$, we can have $\lambda_{a}^{r}=\lambda_{a}^{s}$, and the
algebraic multiplicity is in this case $n_{r}+n_{s}$).

Proposition 5.5 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 $T^{\kappa}_{\lambda}$ 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 $E_{N-n_{r}}$ such that
$\big{\langle}X_{a},E^{b}\big{\rangle}=0$, $\forall\,E^{b}\in E_{N-n_{r}}$. We
assume that $E_{N-n_{r}}$ is a regular distribution of constant rank
$N-n_{r}$.

The necessary and sufficient condition for the integrability of the
distributions $E_{N-n_{r}}$ is given by the Haantjes theorem [16].

######  Theorem 5.7.

Let $T^{\lambda}_{\kappa}$ 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 $E_{N-n_{r}}$ determined
by these vectors are integrable if and only if

$\displaystyle
H^{\kappa}_{\nu\sigma}T^{\nu}_{\mu}T^{\sigma}_{\lambda}-2H^{\sigma}_{\nu[\lambda}T^{\nu}_{\mu]}T^{\kappa}_{\sigma}+H^{\nu}_{\mu\lambda}T^{\kappa}_{\sigma}T^{\sigma}_{\nu}=0,$

where

$\displaystyle
H^{\kappa}_{\mu\lambda}=2T^{\nu}_{[\mu}\partial_{|\nu|}T^{\kappa}_{\lambda]}-2T^{\kappa}_{\nu}\partial_{[\mu}T^{\nu}_{\lambda]},$

is the so-called Haantjes tensor of $T$.

Therefore, in analogy with the characterization of the Stäckel separable
coordinate systems, we have

######  Proposition 5.8.

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 ${\rm d}(T{\rm d}V)=0$. 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 analogue result about characteristic Killing tensors of Stäckel systems is
given in [10] 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 [18]. Many common computer-algebra softwares include specific
commands for the determination of Killing tensors of Riemannian manifolds.

######  Example 5.9.

_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

$\displaystyle V=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}(x_{i}-x_{j})^{-2}.$

The Hamiltonian is therefore

$\displaystyle H_{(N)}=\frac{1}{2}\sum_{i=1}^{N}p_{i}^{2}+V,$

in Cartesian coordinates $\big{(}x^{i}\big{)}$ and it is known to be maximally
superintegrable for any $N$ and multiseparable for $N<4$ [2, 31].

In constant curvature manifolds, quadratic first integrals
$K_{a}=\frac{1}{2}k^{ij}_{a}p_{i}p_{j}+W_{a}$ of natural Hamiltonians can be
determined in a systematic way. Indeed, see for example [2], $K_{a}$ is a
first integral of $H$ if and only if the functions $k^{ij}_{a}$ are the
components of a symmetric Killing 2-tensor $k_{a}$ and ${\rm d}W_{a}=k_{a}{\rm
d}V$. It follows that a necessary condition on $k_{a}$ for $K_{a}$ to be a
first integral of $H$ is ${\rm d}(k_{a}{\rm d}V)=0$. As in any constant
curvature manifold, the generic Killing 2-tensor of $\mathbb{E}^{4}$ is a
linear combination (depending on 50 real parameters) of symmetric product of
pairs of Killing vectors. By imposing the condition ${\rm d}(k_{a}{\rm d}V)=0$
to the elements of this space, one finds that $H_{(4)}$ 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

$\displaystyle K_{a}=\frac{1}{2}k_{a}^{ij}p_{i}p_{j}+W_{a},$

where $W_{a}$ are suitable functions that we will make explicit later on, and

$\displaystyle k_{1}^{ii}=\sum_{j,k}x^{j}x^{k},\qquad j,k=1,\dots,4,\qquad
j<k,\qquad j,k\neq i,$ $\displaystyle
k_{1}^{rs}=\frac{1}{2}\left(\big{(}x^{l}\big{)}^{2}+\big{(}x^{m}\big{)}^{2}+x^{r}x^{s}-\sum_{j<k}x^{j}x^{k}\right),$

where $j,k=1,\dots,4$, $l$, $m$, $r$, $s$ all different,

$\displaystyle k_{2}^{ii}=\sum_{j\neq i}\big{(}x^{j}\big{)}^{2},\qquad
k_{2}^{ij}=-x^{i}x^{j},\qquad i\neq j.$

The eigenvalues of $k_{1}$ are

$\displaystyle\big{\\{}0,\big{(}x^{1}\big{)}^{2}+\big{(}x^{2}\big{)}^{2}+\big{(}x^{3}\big{)}^{2}+\big{(}x^{4}\big{)}^{2}\big{\\}}$

with multiplicity 1, and

$\displaystyle\sum_{i,j}x^{i}x^{j}-\frac{1}{2}\big{(}\big{(}x^{1}\big{)}^{2}+\big{(}x^{2}\big{)}^{2}+\big{(}x^{3}\big{)}^{2}+\big{(}x^{4}\big{)}^{2}\big{)},\qquad
i,j=1,\dots,4,\qquad i<j,$

with multiplicity 2.

The eigenvalues of $k_{2}$ are $0$, of multiplicity $1$, and

$\displaystyle\big{(}x^{1}\big{)}^{2}+\big{(}x^{2}\big{)}^{2}+\big{(}x^{3}\big{)}^{2}+\big{(}x^{4}\big{)}^{2},$

of multiplicity 3. Since the tensors of components $k_{1}^{ij}$ and
$k_{2}^{ij}$ commute as linear operators and the metric is positive definite,
they can be diagonalized simultaneously in some coordinate system.

By using some properties of the eigenvalues of Killing tensors [9], one finds
that these coordinates $(r,\phi_{1},\phi_{2},\phi_{3})$ are spherical and
determined by the consecutive transformations [7]

$\displaystyle z^{1}=2^{-1/2}\big{(}x^{1}-x^{2}\big{)},$ $\displaystyle
z^{2}=6^{-1/2}\big{(}x^{1}+x^{2}-2x^{3}\big{)},$ $\displaystyle
z^{3}=12^{-1/2}\big{(}x^{1}+x^{2}+x^{3}-3x^{4}\big{)},$ $\displaystyle
z^{4}=2^{-1}\big{(}x^{1}+x^{2}+x^{3}+x^{4}\big{)},$

and

$\displaystyle z^{4}=r\cos\phi_{1},$ $\displaystyle
z^{3}=r\sin\phi_{1}\cos\phi_{2},$ $\displaystyle
z^{2}=r\sin\phi_{1}\sin\phi_{2}\cos\phi_{3},$ $\displaystyle
z^{1}=r\sin\phi_{1}\sin\phi_{2}\sin\phi_{3}.$

The scalar potential $V$ of $H_{(4)}$ becomes in these coordinates

$\displaystyle V=\frac{1}{r^{2}\sin^{2}\phi_{1}}f(\phi_{2},\phi_{3}),$

where $f(\phi_{2},\phi_{3})$ is a rather complicated rational function of
trigonometric functions of $\phi_{1}$, $\phi_{2}$.

Therefore, the Hamiltonian becomes

$\displaystyle H_{(4)}=\alpha^{1}H_{1}+\alpha^{2}H_{2}+\alpha^{3}H_{3},$

with

$\displaystyle\alpha^{1}=1,\qquad\alpha^{2}=\frac{1}{r^{2}},\qquad\alpha^{3}=\frac{1}{r^{2}\sin^{2}\phi_{1}},$
$\displaystyle H_{1}=\frac{1}{2}p_{r}^{2}+V_{1},\qquad
H_{2}=\frac{1}{2}p_{\phi_{1}}^{2}+V_{2},\qquad
H_{3}=\frac{1}{2}\left(p_{\phi_{2}}^{2}+\frac{1}{\sin^{2}\phi_{2}}p_{\phi_{3}}^{2}\right)+V_{3},$

where

$\displaystyle V=\alpha^{i}V_{i},$

with $V_{1}=0$, $V_{2}=0$, $V_{3}=f(\phi_{2},\phi_{3})$. Moreover,

$\displaystyle
K_{1}=H_{2}+\frac{1-2\sin^{2}\phi_{1}}{\sin^{2}\phi_{1}}H_{3},\qquad
K_{2}=H_{2}+\frac{1}{\sin^{2}\phi_{1}}H_{3}.$

So that the inverse of the block-Stäckel matrix is

$\displaystyle
S^{-1}=\left(\begin{matrix}1&&\dfrac{1}{r^{2}}&&\dfrac{1}{r^{2}\sin^{2}\phi_{1}}\vspace{1mm}\\\
0&&1&&\dfrac{1-2\sin^{2}\phi_{1}}{\sin^{2}\phi_{1}}\vspace{1mm}\\\
0&&1&&\dfrac{1}{\sin^{2}\phi_{1}}\end{matrix}\right),$

and the block-Stäckel matrix

$\displaystyle S=\left(\begin{matrix}1&&0&&-\dfrac{1}{r^{2}}\vspace{1mm}\\\
0&&\dfrac{1}{2\sin^{2}\phi_{1}}&&\dfrac{2\sin^{2}\phi_{1}-1}{2\sin^{2}\phi_{1}}\vspace{1mm}\\\
0&&-\dfrac{1}{2}&&\dfrac{1}{2}\end{matrix}\right).$

Therefore, the block-separated Hamiltonians are

$\displaystyle\tilde{H}_{1}=H_{1}-c_{1}+\frac{1}{r^{2}}c_{3},$
$\displaystyle\tilde{H}_{2}=H_{2}-\frac{1}{2\sin^{2}\phi_{1}}c_{2}-\frac{2\sin^{2}\phi_{1}-1}{2\sin^{2}\phi_{1}}c_{3},$
$\displaystyle\tilde{H}_{3}=H_{3}+\frac{1}{2}c_{2}-\frac{1}{2}c_{3}.$

The dynamics of the original Hamiltonian $H$ is therefore decomposed into
three separated blocks, corresponding to the two dynamics of Hamiltonians
$\tilde{H}_{1}$, $\tilde{H}_{2}$, with one degree of freedom, and the two-
degrees of freedom dynamics generated by $\tilde{H}_{3}$.

######  Example 5.10.

_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 $2\times 2$ Stäckel matrix.
Indeed, from block-Eisenhart equations we have

$\displaystyle X_{1}\lambda_{1}=0,\qquad X_{2_{i}}\lambda_{2}=0,\qquad
i=1,\dots,N-1,$

where $X_{1}$ is the eigenvector corresponding to the eigenvalue $\lambda_{1}$
of multiplicity one and $X_{2_{i}}$ the eigenvectors of $\lambda_{2}$ of
multiplicity $N-1$. Hence, provided $\lambda_{1}$ is not a constant, we have
that the submanifolds $\lambda_{1}={\rm const}$ are orthogonal to the
eigenvector $X_{1}$, which is therefore normally integrable. We can put in
this case $X_{1}=\partial_{1}$ and the block separation is essentially
determined by the equations

$\displaystyle\lambda_{1}\big{(}q^{2},\dots,q^{N}\big{)}={\rm const},$

moreover $\lambda_{2}(q^{1})$.

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 [9] we show that Stäckel coordinates can be completely determined by 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 $\boldsymbol{\mathbb{E}^{3}}$

In dimension three, only two types of block-separable coordinates can exist,
plus the trivial case of a single three-dimensional block. So, or each block
is one-dimensional, and the coordinates are standard separable orthogonal
coordinates, or one block is one-dimensional and the other one is two-
dimensional. In this last case, by denoting the separable coordinates as
$(u,v,w)$, the geodesic Hamiltonian is

$\displaystyle H=\alpha^{1}H_{1}+\alpha^{2}H_{2},$

with

$\displaystyle H_{1}=g_{1}(u)p_{u}^{2},$

and, since any 2-dimensional Riemannian manifold is locally conformally flat,

$\displaystyle H_{2}=g_{2}(v,w)\big{(}p_{v}^{2}+p_{w}^{2}\big{)},$

where the choice of local Cartesian coordinates on the manifolds $u={\rm
const}$ 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

$\displaystyle S=\left(\begin{matrix}S_{1}^{1}(u)&&S_{1}^{2}(u)\\\
S_{2}^{1}(v,w)&&S_{2}^{2}(v,w)\end{matrix}\right).$

Since

$\displaystyle\alpha^{1}=\big{(}S^{-1}\big{)}_{1}^{1},\qquad\alpha^{2}=\big{(}S^{-1}\big{)}_{1}^{2},$

the contravariant metric $G$ associated with $H$ has components

$\displaystyle G^{uu}=\frac{S_{2}^{2}(v,w)}{\Delta},\qquad
G^{vv}=G^{ww}=\frac{-S_{1}^{2}(u)g}{\Delta},$

with $\Delta=S_{1}^{1}(u)S_{2}^{2}(v,w)-S_{1}^{2}(u)S_{2}^{1}(v,w)$.

We restrict ourselves to the space $\mathbb{E}^{3}$ by imposing that the
Riemann tensor of the metric $G$ is identically zero. We have

$\displaystyle
R_{2132}=-\frac{3\big{(}\partial_{u}S_{1}^{2}S_{1}^{1}-\partial_{u}S_{1}^{1}S_{1}^{2}\big{)}\big{(}\partial_{w}S_{2}^{2}S_{2}^{1}-\partial_{w}S_{2}^{2}S_{2}^{2}\big{)}}{4\Delta
S_{1}^{2}g},$

and

$\displaystyle
R_{3123}=-\frac{3\big{(}\partial_{u}S_{1}^{2}S_{1}^{1}-\partial_{u}S_{1}^{1}S_{1}^{2}\big{)}\big{(}\partial_{v}S_{2}^{2}S_{2}^{1}-\partial_{v}S_{2}^{2}S_{2}^{2}\big{)}}{4\Delta
S_{1}^{2}g}.$

Hence, two cases are possible

  1. i)

$\big{(}\partial_{u}S_{1}^{2}S_{1}^{1}-\partial_{u}S_{1}^{1}S_{1}^{2}\big{)}=0$,

  2. ii)

$\big{(}\partial_{w}S_{2}^{2}S_{2}^{1}-\partial_{w}S_{2}^{2}S_{2}^{2}\big{)}=0$
and $(\partial_{v}S_{2}^{2}S_{2}^{1}-\partial_{v}S_{2}^{2}S_{2}^{2})=0$.

### Case i

We have

$\displaystyle S_{1}^{1}=aS_{1}^{2},$

where $a$ is a constant. We observe that the components of $G$ become

$\displaystyle
G^{uu}=\frac{S_{2}^{2}}{S_{1}^{2}\big{(}aS_{2}^{2}-S_{2}^{1}\big{)}},\qquad
G^{vv}=G^{ww}=-\frac{g}{aS_{2}^{2}-S_{2}^{1}},$

that is, we can write without restrictions

$\displaystyle G^{uu}=h(u)^{-2}f(v,w)^{-2},\qquad G^{vv}=G^{ww}=l(v,w)^{-2},$

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 as an ignorable coordinate, therefore, associated with a Killing
vector $\partial_{u}$.

The unknown functions are now reduced to two, and we can consider the
remaining components of the Riemann tensor. The resulting equations are

$\displaystyle
l\left(\partial_{vv}l+\partial_{ww}l\right)-(\partial_{v}l)^{2}-(\partial_{w}l)^{2}=0,$
$\displaystyle
l\partial_{vv}f-\partial_{v}l\partial_{v}f+\partial_{w}l\partial_{w}f=0,$
$\displaystyle
l\partial_{ww}f+\partial_{v}l\partial_{v}f-\partial_{w}l\partial_{w}f=0,$
$\displaystyle
l\partial_{vw}f-\partial_{v}l\partial_{w}f-\partial_{w}l\partial_{v}f=0.$
(6.1)

######  Proposition 6.1.

The coordinate leaves $u={\rm const}$ are planes and $\partial_{u}$ is
proportional to a Killing vector.

###### Proof.

Equation (6.1) means that the Riemann tensor of the coordinate leaves $u={\rm
const}$ is zero. ∎

The block-Stäckel matrix is in this case

$\displaystyle S=\left(\begin{matrix}a&&1\\\
\left(1-\dfrac{a}{f^{2}}\right)gl^{2}&&-\dfrac{gl^{2}}{f^{2}}\end{matrix}\right),$

and the Hamiltonian $H=\alpha_{1}^{1}H_{1}+\alpha_{1}^{2}H_{2}$ and the first
integral $K=\alpha_{2}^{1}H_{1}+\alpha_{2}^{2}H_{2}$ are

$\displaystyle
H=\frac{1}{f^{2}}p_{u}^{2}+\frac{1}{l^{2}}\big{(}p_{v}^{2}+p_{w}^{2}\big{)},\qquad
K=\left(1-\frac{a}{f^{2}}\right)p_{u}^{2}-\frac{a}{l^{2}}\big{(}p_{v}^{2}+p_{w}^{2}\big{)}.$

Since $K=p_{u}^{2}-aH$, we have that the first integral associated with this
kind of block-separation is simply $p_{u}^{2}$. An expected result, because
the variable $u$ is ignorable.

We observe that the solution of the equation (6.1) is

$\displaystyle l=c_{2}{\rm e}^{\frac{c_{0}}{2}(w^{2}-v^{2})-c_{1}v-c_{3}w},$
(6.2)

where $(c_{i})$ are real constants. The substitution of (6.2) in the remaining
equations yields

$\displaystyle\partial_{vv}f+(c_{1}-c_{0}v)\partial_{v}f-(c_{3}+c_{0}w)\partial_{w}f=0,$
$\displaystyle\partial_{ww}f-(c_{1}-c_{0}v)\partial_{v}f+(c_{3}+c_{0}w)\partial_{w}f=0,$
$\displaystyle\partial_{vw}f+(c_{3}+c_{0}w)\partial_{v}f+(c_{1}-c_{0}v)\partial_{w}f=0.$
(6.3)

The sum of the first two equations implies that $f$ is a solution of the
Laplace equation. Then,

$\displaystyle f=f_{1}(v+{\rm i}w)+f_{2}(v-{\rm i}w).$

The substitution of this equation into (6.3) gives, after few manipulations,
the equivalent system

$\displaystyle f_{1}^{\prime\prime}+(c_{1}-{\rm i}c_{3}-c_{0}(v+{\rm
i}w))f_{1}^{\prime}=0,$ $\displaystyle f_{2}^{\prime\prime}+(c_{1}+{\rm
i}c_{3}-c_{0}(v-{\rm i}w))f_{2}^{\prime}=0.$

######  Example 6.2.

_Rotational and cylindrical coordinates._ Given in the Euclidean plane any
coordinate system with a symmetry axis, the coordinates of $\mathbb{E}^{3}$
obtained by rotating the plane around the symmetry axis and by 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 Case ii gives the equivalent condition

$\displaystyle S_{2}^{1}=aS_{2}^{2}.$

We have

$\displaystyle G^{uu}=\frac{1}{S_{1}^{1}-aS_{1}^{2}},\qquad
G^{vv}=G^{ww}=-\frac{S_{1}^{2}g}{S_{2}^{2}\big{(}S_{1}^{1}-aS_{1}^{2}\big{)}},$

and we can write without restrictions

$\displaystyle G^{uu}=h(u)^{2},\qquad G^{vv}=G^{ww}=l(u)^{2}f(v,w)^{2},$

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, $\partial_{u}$ must be parallel to a conformal Killing vector [8].

The equations arising from the imposition that the Riemann tensor is equal to
zero are now

$\displaystyle l^{\prime\prime}l-2(l^{\prime})^{2}=0,$ $\displaystyle
l^{4}\big{(}f(\partial_{vv}f+\partial_{ww}f)-(\partial_{v}f)^{2}-(\partial_{w}f)^{2}\big{)}-(l^{\prime})^{2}=0.$
(6.4)

The block-Stäckel matrix is in this case

$\displaystyle S=\left(\begin{matrix}1-al^{2}&&-l^{2}\\\
a\dfrac{g}{f^{2}}&&\dfrac{g}{f^{2}}\end{matrix}\right),$

and the Hamiltonian $H=\alpha_{1}^{1}H_{1}+\alpha_{1}^{2}H_{2}$ and first
integral $K=\alpha_{2}^{1}H_{1}+\alpha_{2}^{2}H_{2}$ are

$\displaystyle H=p_{u}^{2}+l^{2}f^{2}\big{(}p_{v}^{2}+p_{w}^{2}\big{)},\qquad
K=-ap_{u}^{2}+\big{(}1-al^{2}\big{)}f^{2}\big{(}p_{v}^{2}+p_{w}^{2}\big{)}.$

Since $K=f^{2}\big{(}p_{v}^{2}+p_{w}^{2}\big{)}-aH$, we have that the first
integral associated with this kind of block-separation is

$\displaystyle f^{2}\big{(}p_{v}^{2}+p_{w}^{2}\big{)},$

as expected, due to the warped form of the metric.

The solutions of (6.4) are

$\displaystyle l=-(c_{1}u+c_{2})^{-1},$

that, substituted in the second equation, give

$\displaystyle
f(\partial_{uu}f+\partial_{ww}f)-(\partial_{v}f)^{2}-(\partial_{w}f)^{2}-c_{1}^{2}=0.$
(6.5)

We remark that (6.5) means that the Ricci scalar of the submanifolds $u={\rm
const}$ is

$\displaystyle R=2l^{2}(u)c_{1}^{2},$

and its Riemann tensor has non null components

$\displaystyle R_{yzyz}=\frac{c_{1}^{2}}{f^{2}}.$

Therefore,

######  Proposition 6.3.

In Case ii, all the coordinate leaves orthogonal to $\partial_{u}$ are planes
$(c_{1}=0)$ or spheres. The vector $\partial_{u}$ is proportional to a
conformal Killing vector.

######  Example 6.4.

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 $c_{1}=0$,
$c_{2}\neq 0$, then the coordinates are cylindrical with the plane $(v,w)$ as
base.

######  Remark 6.5.

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
$\mathbb{E}^{3}$, they are Stäckel-separable, then block-separable in three
blocks, but they cannot be block-separable 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.

The classification of 4D block-separable coordinates, even in Euclidean
spaces, appears to be much less simple than in 3D spaces. We must leave the
classification of block-separable coordinates in 4D Euclidean spaces for other
works. Partial separation of Hamilton–Jacobi and Helmholtz equations on 4D
Riemannian manifolds is briefly considered in [4].

##  7 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 provide 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 block-separation in an invariant way, by adapting classical
results of complete separation theory for the Hamilton–Jacobi equation. A
deeper characterisation of the twisted form of the Hamiltonian is in progress.
Furthermore, we extend Eisenhart’s classification of completely separable
coordinate systems, the Stäckel systems, in $\mathbb{E}^{3}$ to block-
separable 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 lower-dimensional, block-separated
Hamiltonians can find many applications, even in the field of numerical
computations. We do not consider here the block-separation 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, for these equations,
it is less relevant the absence of completeness in partial separation and the
consequent impossibility of application of the Jacobi’s canonical
transformation. We will study block-separation of Schrödinger and related
equations in next papers.

### Acnowledgements

We are grateful to Raymond McLenaghan, Krishan Rajaratnam and Piergiulio
Tempesta for conversations useful to clarify some points of our research and
to the anonymous referees which gave a relevant contribution to the
improvement of this article.

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