# Classical $n$-body system in geometrical and volume variables. I. Three-body
case

A. M. Escobar-Ruiz<EMAIL_ADDRESS>Departamento de Física, Universidad
Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México, CDMX, 09340
México  R. Linares<EMAIL_ADDRESS>Departamento de Física, Universidad
Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México, CDMX, 09340
México  Alexander V Turbiner<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, UNAM, México DF 04510, Mexico  Willard Miller, Jr
<EMAIL_ADDRESS>School of Mathematics, University of Minnesota, Minneapolis
MN 55455, U.S.A.

###### Abstract

We consider the classical 3-body system with $d$ degrees of freedom $(d>1)$ at
zero total angular momentum. The study is restricted to potentials $V$ that
depend solely on relative (mutual) distances $r_{ij}=\mid{\bf r}_{i}-{\bf
r}_{j}\mid$ between bodies. Following the proposal by Lagrange, in the center-
of-mass frame we introduce the relative distances (complemented by angles) as
generalized coordinates and show that the kinetic energy does not depend on
$d$, confirming results by Murnaghan (1936) at $d=2$ and van Kampen-Wintner
(1937) at $d=3$, where it corresponds to a 3D solid body. Realizing
$\mathbb{Z}_{2}$-symmetry $(r\rightarrow-r)$ we introduce new variables
$\rho=r^{2}$, which allows us to make the tensor of inertia non-singular for
binary collisions. The 3 body positions form a triangle (of interaction) and
the kinetic energy is $\mathcal{S}_{3}$-permutationally invariant wrt
interchange of body positions and masses (as well as wrt interchange of edges
of the triangle and masses). For equal masses, we use lowest order symmetric
polynomial invariants of $\mathbb{Z}_{2}^{\otimes 3}\oplus\mathcal{S}_{3}$ to
define new generalized coordinates, they are called the geometrical variables.
Two of them of the lowest order (sum of squares of sides of triangle and
square of the area) are called volume variables. It is shown that for
potentials, which depend on geometrical variables only (i) and those which
depend on mass-dependent volume variables alone (ii), the Hamilton’s equations
of motion can be considered as being relatively simple. In the case (ii) all
trajectories are mass-independent!

We study three examples in some detail: (I) 3-body Newton gravity in $d=3$,
(II) 3-body choreography on the algebraic lemniscate by Fujiwara et al, and
(III) the (an)harmonic oscillator.

Classical mechanics, 3-body system, Symmetries

##  I Introduction

The Lagrangian for a classical system of two non-relativistic particles with
masses $m_{1}$ and $m_{2}$, each of them with $d$ degrees of freedom, has the
form

${\cal L}_{2}\ =\ \frac{1}{2}m_{1}\,{\dot{{\bf r}}_{1}}^{2}\ +\
\frac{1}{2}m_{2}\,{\dot{{\bf r}}_{2}}^{2}\ -\ V(\mid{\bf r}_{1}-{\bf
r}_{2}\mid)\ ,$ (1)

where ${\bf r}_{i}\in\mathbb{R}^{d}$ denotes the position vector of the $i$th
particle, $\dot{{\bf r}}_{i}\equiv\frac{d}{dt}{\bf r}_{i}$ its velocity and
$V$ is a translational invariant interaction potential with rotational
symmetry. Solving this well known system is one of the basic problems in
classical mechanics: it appears in many textbooks. After separation of the
$d$-dimensional center-of-mass (cms) motion, the $2d$-dimensional problem is
reduced to a $d$-dimensional one in the _space of relative motion_. In the
particular case of _zero total angular momentum_ the number of degrees of
freedom in the Lagrangian (1) is reduced to one. Thus, the relative motion is
described by the Lagrangian

$\tilde{\cal L}_{2}\ =\ \frac{1}{2}\mu\,{\dot{r}}^{2}\ -\ V(r)\ ,$ (2)

where $\mu=\frac{m_{1}\,m_{2}}{m_{1}+m_{2}}$ is the reduced mass of the
system. The _dynamical variable_ $r\equiv\mid{\bf r}_{1}-{\bf r}_{2}\mid$ has
an elementary _geometrical interpretation_ : it is the length of the interval
which connects two particles. We call this the _interval of interaction_.

In the case of three bodies with $d$-degrees of freedom, $d\geq 1$, the number
of relative distances is equal to the number of edges of the triangle which is
formed by taking the bodies positions as vertices. This triangle defines the
natural geometrical structure in which the interaction of the bodies occurs.
We call it the triangle of interaction. In 1936, using the 3 mutual distances
$r_{ij}$ (edges of the triangle of interaction) as generalized coordinates
($r-$representation), Murnaghan Murnaghan  introduced a canonical
transformation to reduce the planar problem ($d=2$) from the configuration
space $\mathbb{R}^{6}$ to $\mathbb{R}^{3}$. It implies that the original
configuration space $\mathbb{R}^{6}$ is decomposed into the product
$\mathbb{R}^{2}\times\mathbb{R}_{\rm rel}^{3}\times\mathcal{S}^{1}$ where the
first factor corresponds to the planar cms motion whilst the last one
corresponds to the cyclic (angular) motion of the system around the center-of-
mass. Therefore, the problem is formulated in the three-dimensional space
$\mathbb{R}_{\rm rel}^{3}\subset\mathbb{R}^{3}$ of the relative motion. The
natural extension of this reduction to the case of $3$-degrees of freedom
($d=3$) was then elaborated in Ref.Kampen .

It follows from Ref.Murnaghan  and Ref.Kampen  that at zero total angular
momentum, for both two and three degrees of freedom ($d=2,3$) the space of
relative motion is the same $\mathbb{R}_{\rm rel}^{3}$ and the corresponding
reduced Hamiltonians defined in a six-dimensional phase space $\mathbb{R}_{\rm
rel}^{3}\times\mathbb{R}^{3}$ coincide. It will be shown in this paper that it
remains true for any number of degrees of freedom, $d>1$. Let us denote this
reduced Hamiltonian as ${\cal H}_{0}$. It has the form of the kinetic energy
of the 3D solid body with a certain tensor of inertia plus external potential.
Following identification of the coefficients of the tensor of inertia in
${\cal H}_{0}$ as the entries of a contravariant metric (cometric), the
emerging Hamiltonian describes a three-dimensional particle moving in a curved
space with cometric $g^{\mu\,\nu}(r_{ij})$. Remarkably, the components of
$g^{\mu\,\nu}(r_{ij})$ are rational functions of $r_{ij}$ as well as its
determinant111This result was known by Murnaghan Murnaghan ..

The triangle of interaction is characterized by three edges $r_{ij}$
(intervals of interaction). It is easy to see that the free Hamiltonian ${\cal
H}_{0}$, i.e. when the potential $V$ is absent, is $\mathbb{Z}_{2}^{\otimes
3}$-invariant under the reflections $r_{ij}\rightarrow-r_{ij}$ Murnaghan . It
seems natural to make symmetry reduction introducing
$\mathbb{Z}_{2}$-invariant coordinates $\rho_{ij}=r_{ij}^{2}$ with their
corresponding canonical momentum (it will be called $\rho-$representation).
Lagrange in Ref.Lagrange  already paid attention to the importance of $r-$ and
$\rho$-variables, but never used them as dynamical coordinates (generalized
coordinates). The corresponding symmetric reduced Hamiltonian was never
constructed by him explicitly. It was done much later by MurnaghanMurnaghan
in $r-$variables only. Up to our present knowledge nobody used the $\rho$\- or
volume variables as dynamical variables (generalized coordinates). It is worth
mentioning that Lamaître, in Ref.Lemaitre , derived a reduced Hamiltonian
using certain radial variables but those variables suffer lacking the property
of $\mathcal{S}_{3}$ permutation invariance. In $\rho$-coordinates, we will
see that the free Hamiltonian ${\cal H}_{0}$ is a second degree polynomial in
momentum variables with linear $\rho$-dependent coefficients on the phase
space TME3-d . In the case of equal masses the free Hamiltonian ${\cal H}_{0}$
has extra permutational symmetry $\mathcal{S}_{3}$ with respect to
permutations of the intervals of interaction $r_{ij}$ (or their squares
$\rho_{ij}$). It suggests the introduction of new
$\mathcal{S}_{3}$-permutationally invariant coordinates in $\rho$-space

$\displaystyle\sigma_{1}\ =\ \rho_{12}\ +\ \rho_{23}\ +\ \rho_{31}\ ,$ (3)
$\displaystyle\sigma_{2}\ =\ \rho_{12}\,\rho_{23}\ +\ \rho_{12}\,\rho_{31}\ +\
\rho_{23}\,\rho_{31}\ ,$ $\displaystyle\sigma_{3}\ =\
\rho_{12}\,\rho_{23}\,\rho_{31}\ .$

These coordinates, as well as ones of the original 3-body system, are
invariant under the permutations of the bodies. In variables (3), the free
Hamiltonian ${\cal H}_{0}$ remains a polynomial in phase space
$(\sigma,p_{\sigma})$, in agreement with Ref.TME3-d . Following the Cayley-
Menger formula CayleyMenger , instead of $\sigma_{2}$ one can introduce the
_square_ of the area of the triangle of interaction,
$S=\frac{1}{16}(4\,\sigma_{2}-\sigma_{1}^{2})$. Making a canonical
transformation one can show that the free Hamiltonian ${\cal H}_{0}$ in
coordinates $(P\equiv\frac{1}{2}\sigma_{1},\,S,\,T\equiv\sigma_{3})$ remains a
polynomial. The variables $(P,S)$ are called the volume variables. The key
task of this article is to derive the explicit form of the free Hamiltonian
${\cal H}_{0}$ that answers the question: _What is the form of the reduced
Hamiltonian in variables $(P,S,T)$?_

The goal of the present study is to write, for the 3-body system with zero
total angular momentum and arbitrary $d>1$, the reduced Hamiltonian ${\cal
H}_{0}$ in the $\rho-$representation and also in the $(P,S,T)$-representation.
The motivation of the present paper is three-fold. Our first aim is to
demonstrate that the choice of the intervals of interaction _squared_ ,
$\rho_{ij}=r^{2}_{ij}$, as dynamical variables leads to a deep connection
between the geometrical characteristics of the triangle of interaction and the
dynamics of the system. Accordingly, it will be shown that for arbitrary
masses the corresponding Hamiltonian in $\rho$ variables describes a particle
in a curved space with a certain (essentially non-flat) metric
$g^{\mu\,\nu}(\rho)$. Unlike the $r-$representation, now the components of
$g^{\mu\,\nu}(\rho)$ are first degree polynomials in $\rho$-variables and its
determinant $\text{Det}[g^{\mu\,\nu}(\rho)]$ is proportional to the square of
the area of the triangle of interaction.

Secondly, just as for the case of three identical particles
$m_{1}=m_{2}=m_{3}=1$ we will show that the set of dynamical coordinates
$(P,S,T)$ exhibit outstanding properties. In particular,

  * •

The Det$\big{[}g^{\mu\,\nu}(P,S,T)\big{]}$ is, in fact, the discriminant of
the fourth degree polynomial equation that defines the physically relevant
3-body Newtonian gravity potential at $d=3$ in terms of the aforementioned
variables.

  * •

Remarkably, for planar 3-body choreographic motion on an algebraic lemniscate
by Jacob Bernoulli, studied by Fujiwara et al.Fujiwara , two of the variables
$P,T$ become _particular_ constants of motion ATJC ; Turbiner:2013p  and the
trajectory is an elliptic curve.

Thirdly, we investigate how the volume variables $P,S$ are modified in the
case of arbitrary masses. Assuming that the potential $V$ depends on above-
introduced volume variables $(P,S)$, we will show that for the original
Hamiltonian there exist trajectories which depend on these two volume
variables alone.

The main long term goal for doing this work is to explore the possible
practical advantages of representations for which the involved generalized
coordinates posses the following two properties: (i) they encode the
symmetries of the free reduced Hamiltonian ${\cal H}_{0}$, and (ii) ${\cal
H}_{0}$ is a polynomial function in these variables. The reason to consider
such variables comes not only from esthetics (being esthetics a sufficient
reason, though). Upon the standard quantization procedure the associated
Hamiltonian operator can be constructed from the classical one. Then, due to
the property (ii) the kinetic energy term will be an algebraic operator, i.e.
the coefficients in front of the derivatives are polynomial functions.
Algebraic operators has been intimately related with the existence of exactly
and quasi-exactly-solvable quantum models. Hence, we think that a study of
classical systems within the aforementioned representations makes sense and it
is worthwhile to pursue.

Also, in the general $n$-body case in $d-$dimensions ($n\geq d-1$) the
dynamics of the system can be thought of as a type of ”breathing” _polytope of
interaction_. At fixed $n$ there exist ($n-1$) _volume-variables_ made out of
geometric elements with different dimensionality (edges, faces, cells and so
on) of this polytope of interaction, each variable being
$\mathcal{S}_{n}$-permutationally symmetric. Therefore, another rationale of
the present consideration is to shed light on the links between the theory of
regular polytopes, fundamental symmetric polynomials and the dynamics of an
$n$-body system.

The structure of the article is organized very simply. In Section II, we first
review the symmetric reduction of the planar 3-body system. Separating the cms
motion and using the conserved total angular-momentum $p_{\Omega}$, the
problem is reduced to one of three degrees of freedom in which the coordinates
are the lengths $r_{12},r_{23},r_{31}$ of the sides of the triangle of
interaction. It is shown that the corresponding reduced Hamiltonian ${\cal
H}_{0}$ describes a $3$-dimensional particle moving in a curved space. Next,
we introduce the variables $\rho_{ij}=r^{2}_{ij}$ ($\rho$-representation) for
which ${\cal H}_{0}$ becomes a polynomial at $p_{\Omega}=0$. In Section III we
prove that for zero total angular momentum and arbitrary dimension $d>1$ the
trajectories are described by exactly the same Hamiltonian ${\cal H}_{0}$. The
$3$-body chain of harmonic oscillators is briefly revisited to illustrate this
representation. The case of three identical particles $m_{1}=m_{2}=m_{3}=1$ is
studied in Section IV. For ${\cal H}_{0}$, the novel set of variables
$(P,S,T)$ is introduced and the corresponding equations of motion are
presented explicitly. The $3$-body Newtonian gravity potential in $d=3$ and
planar choreographic trajectories on algebraic lemniscate ($d=2$) are used to
exemplify the properties of this representation. Finally, in Section V in the
case of arbitrary masses with a certain class of potentials a further
_reduction_ of the problem to one of two degrees of freedom is accomplished
using modified, mass-dependent volume variables as dynamical coordinates. For
conclusions and future outlook see Section VI.

##  II Three-body system: planar case ($d=2$)

We consider a planar ($d=2$) classical system of three interacting point-like
particles with masses $m_{1},m_{2}$ and $m_{3}$, respectively. The Lagrangian
is of the form,

${\cal L}\ =\ {\cal T}\ -\ V(r_{12},\,r_{23},\,r_{31})\ ,$ (4)

where the kinetic energy is given by

${\cal T}\ =\ \frac{1}{2}m_{1}\,{\dot{{\bf r}}_{1}}^{2}\ +\
\frac{1}{2}m_{2}\,{\dot{{\bf r}}_{2}}^{2}\ +\ \frac{1}{2}m_{3}\,{\dot{{\bf
r}}_{3}}^{2}\ ,$ (5)

here ${\bf r}_{i}\in\mathbb{R}^{2}$ is the vector position of the $i$th body,
$\dot{{\bf r}_{i}}\equiv\frac{d}{dt}{\bf r}_{i}$ and the potential $V$ depends
on the relative distances

$r_{ij}\ \equiv\ \mid{\bf r}_{i}-{\bf r}_{j}\mid\ ,$

between particles only. This implies that ${\cal L}$ is rotationally
symmetric. The configuration space is six-dimensional $\mathbb{R}^{2}({\bf
r}_{1})\times\mathbb{R}^{2}({\bf r}_{2})\times\mathbb{R}^{2}({\bf r}_{3})$.

The kinetic energy (5) can be expressed in terms of center-of-mass and the
relative coordinates of the three bodies Murnaghan ,

$\displaystyle{\cal T}$ $\displaystyle=\ \frac{1}{2}M\,{\dot{\bf Y}}^{2}\ +\
\frac{1}{2}{\mu_{12}}\,{\dot{{\bf r}}_{12}}^{2}\ +\
\frac{1}{2}{\mu_{23}}\,{\dot{{\bf r}}_{23}}^{2}\ +\
\frac{1}{2}{\mu_{31}}\,{\dot{{\bf r}}_{31}}^{2}$ (6) $\displaystyle=\
\frac{1}{2}M\,{\dot{\bf Y}}^{2}\ +\
\frac{1}{2}{\mu_{12}}\,({\dot{r}}_{12}^{2}\ +\
{r}_{12}^{2}\,{\dot{\theta}}_{12}^{2})\ +\
\frac{1}{2}{\mu_{23}}\,({\dot{r}}_{23}^{2}\ +\
{r}_{23}^{2}\,{\dot{\theta}}_{23}^{2})\ +\
\frac{1}{2}{\mu_{31}}\,({\dot{r}}_{31}^{2}\ +\
{r}_{31}^{2}\,{\dot{\theta}}_{31}^{2})\ ,$

where $M=m_{1}+m_{2}+m_{3}$ is the total mass, ${\bf Y}=\frac{m_{1}\,{\bf
r}_{1}+m_{2}\,{\bf r}_{2}+m_{3}\,{\bf r}_{3}}{M}$ is the center-of-mass
vector, and $\mu_{ij}\equiv\frac{m_{i}\,m_{j}}{M}$ denotes a reduced-like
mass. In (6), we also introduce polar coordinates for relative vectors, i.e.
${\bf r}_{ij}\equiv(r_{ij},\,\theta_{ij})$ in the space of relative motion.

For future convenience, we select the center-of-mass system as the inertial
frame, ${\bf Y}=0,{\dot{\bf Y}}=0$. Eventually, in this inertial frame the
system is characterized by four degrees of freedom.

###  II.1 $r$-representation

Now, following Ref.Murnaghan  we consider the three _relative distances_
$r_{12},\,r_{23},\,r_{31},\,$ and the _angle_

$3\,\Omega\ =\ \theta_{12}\ +\ \theta_{23}\ +\ \theta_{31}\ ,$ (7)

as the four generalized coordinates of the Lagrangian (4). It turns out that
the kinetic energy (6) does not depend on $\Omega$ (7). Therefore, the angle
$\Omega$ is a cyclic variable and its canonical momentum

$\displaystyle p_{{}_{\Omega}}$ $\displaystyle\equiv\
\frac{\partial}{\partial\dot{\Omega}}\,{\cal L}\ =\
\frac{\partial}{\partial\dot{\Omega}}\,{\cal T}$ (8) $\displaystyle=\
\mu_{12}\,{r}_{12}^{2}\,{\dot{\theta}}_{12}^{2}\ +\
\mu_{23}\,{r}_{23}^{2}\,{\dot{\theta}}_{23}^{2}\ +\
\mu_{31}\,{r}_{31}^{2}\,{\dot{\theta}}_{31}^{2}\ ,$

is a constant of motion (saying differently, the first integral),
$\dot{p}_{{}_{\Omega}}=0$. From (8) it follows that $p_{{}_{\Omega}}$ is
nothing but the total angular momentum of the system about its center of mass.
The conserved quantity $p_{{}_{\Omega}}$ allows us to reduce the problem to
one of three degrees of freedom in which the corresponding coordinates are the
relative distances $r_{12},\,r_{23}$ and $r_{31}\,$ Murnaghan . To this end,
it is convenient to introduce the Routhian ${\cal R}$ defined
LandauLifshitzBook  by the Legendre transformation

${\cal R}(r_{ij},{\dot{r}}_{ij},p_{{}_{\Omega}})\ =\ {\cal L}\ -\
p_{{}_{\Omega}}\,{\dot{\Omega}}\ .$ (9)

It is a function of the radial variables $r_{ij},{\dot{r}}_{ij}$ and the
integral $p_{{}_{\Omega}}$ only.

####  II.1.1 Hamiltonian for the reduced three-dimensional problem

In this Section we switch, for convenience, from the Lagrangian formalism to
the Hamiltonian one. From (9) it follows that the reduced Hamiltonian of the
system is defined in a six-dimensional phase space where the dynamical
variables are the relative distances $r_{ij}$ and its conjugate momentum
variables

$p_{12}\ \equiv\ \frac{\partial\,{\cal R}}{\partial{\dot{r}}_{12}}\ ,\qquad
p_{23}\ \equiv\ \frac{\partial\,{\cal R}}{\partial{\dot{r}}_{23}}\ ,\qquad
p_{31}\ \equiv\ \frac{\partial\,{\cal R}}{\partial{\dot{r}}_{31}}\ .$

In particular, for zero total angular momentum222In atomic and subatomic
quantum systems the lowest energy state is usually one of zero total angular
momentum ($S-$states) $p_{{}_{\Omega}}=0$, the momentum variable $p_{12}$
reads

$\displaystyle p_{12}\ =\
\frac{1}{16\,I\,S^{2}_{\bigtriangleup}}\,\bigg{[}\,4\,r_{12}^{2}\,\big{[}\left(\mu_{12}\,\mu_{23}+\mu_{12}\,\mu_{31}+\mu_{23}\,\mu_{31}\right)r_{23}^{2}\,r_{31}^{2}\
+\ 4\,\mu_{12}^{2}\,S^{2}\,\big{]}\,{\dot{r}}_{12}\ -\ $ (10) $\displaystyle
r_{12}\,r_{23}\,\big{[}\,2\,\left(\mu_{12}+\mu_{23}\right)\,\mu_{31}\left(r_{12}^{2}+r_{23}^{2}-r_{31}^{2}\right)\,r_{31}^{2}+\mu_{12}\,\mu_{23}\,\left(\left(r_{12}^{2}-r_{23}^{2}\right){}^{2}-r_{31}^{4}\right)\big{]}\,{\dot{r}}_{23}\
-\ $ $\displaystyle
r_{12}\,r_{31}\,\big{[}\,2\,\left(\mu_{12}+\mu_{31}\right)\,\mu_{23}\left(r_{12}^{2}-r_{23}^{2}+r_{31}^{2}\right)\,r_{23}^{2}+\mu_{12}\,\mu_{31}\,\left(\left(r_{12}^{2}-r_{31}^{2}\right){}^{2}-r_{23}^{4}\right)\big{]}\,{\dot{r}}_{31}\bigg{]}\
,$

here

$I\ \equiv\ \mu_{12}\,r_{12}^{2}\ +\ \mu_{23}\,r_{23}^{2}\ +\
\mu_{31}\,r_{31}^{2}\ ,$ (11)

is the moment of inertia with respect to the center of mass, and

$S^{2}_{\bigtriangleup}\ \equiv\
\frac{1}{16}\big{(}\,2\,r_{12}^{2}\,r_{23}^{2}\ +\ 2\,r_{12}^{2}\,r_{31}^{2}\
+\ 2\,r_{23}^{2}\,r_{31}^{2}\ -\ r_{12}^{4}\ -\ r_{23}^{4}\ -\
r_{31}^{4}\,\big{)}\ ,$ (12)

is the _square_ of the area of the triangle formed by the three particles. It
was called the _triangle of interaction_ , see Fig. 1. By a cyclic arrangement
of the labels $(1,2,3)$ ( i.e. $(2,3,1)$ or $(3,1,2)$ ) in (10) we obtain the
other two momenta $p_{23}$ and $p_{31}$. The two geometrical quantities
$P_{m}\equiv\frac{m_{1}+m_{2}+m_{3}}{2\,m_{1}\,m_{2}\,m_{3}}\,I$ and
$S_{m}\equiv\frac{3\,m_{1}\,m_{2}\,m_{3}}{m_{1}+m_{2}+m_{3}}S^{2}_{\bigtriangleup}$
will play an important role in the present study. They will be called modified
volume variables.

It is worth mentioning that in Ref.Hsiang  the virtual motions of the
_triangle of interaction_ , through which the triangle changes its kinematic
invariants such as size, shape and orientation (position) and velocity are the
starting point for a different systematic geometric-based approach to the
three-body problem. However, unlike us, in Ref.Hsiang  $\mathcal{S}_{n}$
permutationally-invariant variables are not considered as the fundamental
dynamical generalized coordinates.

Figure 1: Triangle of interaction in $d=3$: the individual coordinate vectors
${\bf r}_{i}$ mark positions of vertices of the triangle with sides $r_{ij}$.
The center-of-mass (the barycenter of the triangle) is marked by a (blue)
bubble.

Eventually, we arrive to the reduced Hamiltonian Murnaghan

$\displaystyle{\cal H}_{r}$ $\displaystyle=\ p_{12}\,{{\dot{r}}_{12}}\ +\
p_{31}\,{{\dot{r}}_{31}}\ +\ p_{23}\,{{\dot{r}}_{23}}\ -\ {\cal R}$ (13)
$\displaystyle=\ \frac{1}{2}\bigg{[}\,\frac{p_{12}^{2}}{m_{12}}\ +\
\frac{p_{23}^{2}}{m_{23}}\ +\ \frac{p_{31}^{2}}{m_{31}}\ +\
\frac{r_{12}^{2}+r_{31}^{2}-r_{23}^{2}}{m_{1}\,r_{12}\,r_{31}}\,p_{12}\,p_{31}\
+\
\frac{r_{12}^{2}+r_{23}^{2}-r_{31}^{2}}{m_{2}\,r_{12}\,r_{23}}\,p_{12}\,p_{23}\
+\ $
$\displaystyle\qquad\frac{r_{23}^{2}+r_{31}^{2}-r_{12}^{2}}{m_{3}\,r_{23}\,r_{31}}\,p_{23}\,p_{31}\,\bigg{]}\
+\
\frac{2}{3}\,p_{{}_{\Omega}}\,S_{\bigtriangleup}\bigg{[}\,p_{12}\,\frac{m_{1}\,r_{31}^{2}-m_{2}\,r_{23}^{2}}{m_{1}\,m_{2}\,r_{12}\,r_{23}^{2}\,r_{31}^{2}}\
+\
p_{23}\,\frac{m_{2}\,r_{12}^{2}-m_{3}\,r_{31}^{2}}{m_{2}\,m_{3}\,r_{23}\,r_{12}^{2}\,r_{31}^{2}}$
$\displaystyle\quad\ +\
p_{31}\,\frac{m_{3}\,r_{23}^{2}-m_{1}\,r_{12}^{2}}{m_{1}\,m_{3}\,r_{31}\,r_{23}^{2}\,r_{31}^{2}}\,\bigg{]}\
+\ V_{\text{eff}}\quad,\qquad m_{ij}\equiv\frac{m_{i}\,m_{j}}{m_{i}+m_{j}}\ ,$

where

$V_{\text{eff}}\ =\ V\ +\
\frac{p^{2}_{{}_{\Omega}}}{9}\bigg{[}\frac{1}{m_{12}\,r_{12}^{2}}\,+\,\frac{1}{m_{23}\,r_{23}^{2}}\,+\,\frac{1}{m_{31}\,r_{31}^{2}}\,-\,\frac{r_{12}^{2}}{2\,m_{3}\,r_{23}^{2}\,r_{31}^{2}}\,-\,\frac{r_{23}^{2}}{2\,m_{1}\,r_{12}^{2}\,r_{31}^{2}}\,-\,\frac{r_{31}^{2}}{2\,m_{2}\,r_{12}^{2}\,r_{23}^{2}}\bigg{]}\
,$

is an effective potential. The angular momentum $p_{{}_{\Omega}}$ is a
Liouville integral, it Poisson-commutes with the Hamiltonian (13),
$\\{p_{{}_{\Omega}},\,{\cal H}_{r}\\}=0$. At zero angular momentum
$p_{{}_{\Omega}}=0$ the second term in kinetic energy vanishes as well as the
second term in the effective potential $V_{\rm eff}$. Explicitly, at
$p_{{}_{\Omega}}=0$ the above Hamiltonian (13) becomes

$\displaystyle{H}_{r}$ $\displaystyle=\
\frac{1}{2}\bigg{[}\,\frac{p_{12}^{2}}{m_{12}}\ +\ \frac{p_{23}^{2}}{m_{23}}\
+\ \frac{p_{31}^{2}}{m_{31}}\ +\
\frac{r_{12}^{2}+r_{31}^{2}-r_{23}^{2}}{m_{1}\,r_{12}\,r_{31}}\,p_{12}\,p_{31}\
+\
\frac{r_{12}^{2}+r_{23}^{2}-r_{31}^{2}}{m_{2}\,r_{12}\,r_{23}}\,p_{12}\,p_{23}\
+\ $ (14)
$\displaystyle\qquad\frac{r_{23}^{2}+r_{31}^{2}-r_{12}^{2}}{m_{3}\,r_{23}\,r_{31}}\,p_{23}\,p_{31}\,\bigg{]}\
+\ V(r_{12},\,r_{23},\,r_{31})\ .$

The way how the reduced Hamiltonian (14) is written we call the
$r-$representation.

####  II.1.2 The metric $g^{\mu\nu}(r)$

The associated cometric for (13) defined by coefficients in front of the
quadratic terms in variables $(p_{12},\,p_{31},\,p_{23})$ is given by

$g^{\mu\nu}(r)\ =\left|\begin{array}[]{ccc}\frac{1}{2\,m_{12}}&\
\frac{r_{12}^{2}+r_{31}^{2}-r_{23}^{2}}{4\,m_{1}\,r_{12}\,r_{31}}&\
\frac{r_{12}^{2}+r_{23}^{2}-r_{31}^{2}}{4\,m_{2}\,r_{12}\,r_{23}}\\\ &&\\\
\frac{r_{12}^{2}+r_{31}^{2}-r_{23}^{2}}{4\,m_{1}\,r_{12}\,r_{31}}&\
\frac{1}{2\,m_{31}}&\
\frac{r_{31}^{2}+r_{23}^{2}-r_{12}^{2}}{4\,m_{3}\,r_{23}\,r_{31}}\\\ &&\\\
\frac{r_{12}^{2}+r_{23}^{2}-r_{31}^{2}}{4\,m_{2}\,r_{12}\,r_{23}}&\
\frac{r_{31}^{2}+r_{23}^{2}-r_{12}^{2}}{4\,m_{3}\,r_{23}\,r_{31}}&\frac{1}{2\,m_{23}}\end{array}\right|\
.$ (15)

Its determinant $\mathcal{D}_{m}=\text{Det}[g^{\mu\nu}(r)]$ possesses the
remarkable factorization property

$\mathcal{D}_{m}\ =\
\frac{{(m_{1}+m_{2}+m_{3})}^{2}}{2\,m_{1}^{2}\,m_{2}^{2}\,m_{3}^{2}}\,\frac{I\,\
S^{2}_{\triangle}}{r_{12}^{2}\,r_{23}^{2}\,r_{31}^{2}}\ ,$

thus, it is proportional to the moment of inertia $I$ (11) and the area
(squared) $S^{2}_{\bigtriangleup}$ (12) of the triangle of interaction.
Remark. The above determinant $\mathcal{D}_{m}$ is a rational function in
$r$-variables. It effectively depends on the three coordinates $I$,
$S^{2}_{\bigtriangleup}$ and $T\equiv r_{12}^{2}\,r_{23}^{2}\,r_{31}^{2}$.
Note that it vanishes, $\mathcal{D}_{m}=0$, iff the area of the triangle of
interaction is equal to zero or a triple-body collision occurs. It is singular
at the two-body collision point.

###  II.2 $\rho$-representation

The cometric (15) as well as the original kinetic energy (6) are invariant
formally under reflections
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$,

$r_{12}\,\rightarrow\,-r_{12}\ ,\qquad r_{23}\,\rightarrow\,-r_{23}\ ,\qquad
r_{31}\,\rightarrow\,-r_{31}\ ,$

and w.r.t. $\mathcal{S}_{3}$-group action (permutations of the bodies). If we
introduce new variables,

$\rho_{12}\,=\,r_{12}^{2}\ ,\qquad\rho_{23}\,=\,r_{23}^{2}\
,\qquad\rho_{31}\,=\,r_{31}^{2}\ ,$ (16)

with the corresponding canonical momenta

$P_{12}\,=\,\frac{1}{2\,r_{12}}\,p_{12}\ ,\qquad
P_{23}\,=\,\frac{1}{2\,r_{23}}\,p_{23}\ ,\qquad
P_{31}\,=\,\frac{1}{2\,r_{31}}\,p_{31}\ ,$ (17)

we immediately arrive at the ${\mathbb{Z}}_{2}$-symmetry reduced Hamiltonian

$\displaystyle{\cal H}_{\rho}$ $\displaystyle=\
2\,\bigg{[}\,\frac{\rho_{12}\,P_{12}^{2}}{m_{12}}\ +\
\frac{\rho_{23}\,\,P_{23}^{2}}{m_{23}}\ +\
\frac{\rho_{31}\,\,P_{31}^{2}}{m_{31}}\ +\
\frac{\rho_{12}+\rho_{31}-\rho_{23}}{m_{1}}\,P_{12}\,P_{31}\ +\
\frac{\rho_{12}+\rho_{23}-\rho_{31}}{m_{2}}\,P_{12}\,P_{23}$ (18)
$\displaystyle+\
\frac{\rho_{23}+\rho_{31}-\rho_{12}}{m_{3}}\,P_{23}\,P_{31}\,\bigg{]}\ +\
\frac{4}{3}\,p_{{}_{\Omega}}\,S_{\bigtriangleup}\bigg{[}\,P_{12}\,\frac{m_{1}\,\rho_{31}-m_{2}\,\rho_{23}}{m_{1}\,m_{2}\,\rho_{23}\,\rho_{31}}\
+\
P_{23}\,\frac{m_{2}\,\rho_{12}-m_{3}\,\rho_{31}}{m_{2}\,m_{3}\,\rho_{12}\,\rho_{31}}$
$\displaystyle+\
P_{31}\,\frac{m_{3}\,\rho_{23}-m_{1}\,\rho_{12}}{m_{1}\,m_{3}\,\rho_{23}\,\rho_{31}}\,\bigg{]}\
+\ V_{\text{eff}}\ ,$

where

$V_{\text{eff}}\ =\ V\ +\
\frac{p^{2}_{{}_{\Omega}}}{9}\bigg{[}\frac{1}{m_{12}\,\rho_{12}}\,+\,\frac{1}{m_{23}\,\rho_{23}}\,+\,\frac{1}{m_{31}\,\rho_{31}}\,-\,\frac{\rho_{12}}{2\,m_{3}\,\rho_{23}\,\rho_{31}}\,-\,\frac{\rho_{23}}{2\,m_{1}\,\rho_{12}\,\rho_{31}}\,-\,\frac{\rho_{31}}{2\,m_{2}\,\rho_{12}\,\rho_{23}}\bigg{]}\
.$

The Hamiltonian (18) is written in what we call the $\rho-$representation (cf.
(13)).

####  II.2.1 The metric $g^{\mu\nu}(\rho)$

The associated contravariant metric for (18) defined by coefficients in front
of the quadratic terms in momentum variables $(P_{12},\,P_{31},\,P_{23})$

$g^{\mu\nu}(\rho)\ =\left|\begin{array}[]{ccc}2\,\frac{\rho_{12}}{m_{12}}&\
\frac{\rho_{12}+\rho_{31}-\rho_{23}}{m_{1}}&\
\frac{\rho_{12}+\rho_{23}-\rho_{31}}{m_{2}}\\\ &&\\\
\frac{\rho_{12}+\rho_{31}-\rho_{23}}{m_{1}}&\ 2\,\frac{\rho_{31}}{m_{31}}&\
\frac{\rho_{31}+\rho_{23}-\rho_{12}}{m_{3}}\\\ &&\\\
\frac{\rho_{12}+\rho_{23}-\rho_{31}}{m_{2}}&\
\frac{\rho_{31}+\rho_{23}-\rho_{12}}{m_{3}}&2\,\frac{\rho_{23}}{m_{23}}\end{array}\right|\
,$ (19)

is linear in $\rho$-coordinates, cf.(15). Its determinant is

$D_{m}\ =\ \text{Det}\big{[}g^{\mu\nu}({\rho})\big{]}\ =\
2\,\frac{m_{1}+m_{2}+m_{3}}{m_{1}^{2}\,m_{2}^{2}\,m_{3}^{2}}\times$
$\left(m_{1}m_{2}\rho_{12}+m_{1}m_{3}\rho_{13}+m_{2}m_{3}\rho_{23}\right)\left(2\rho_{12}\rho_{13}+2\rho_{12}\rho_{23}+2\rho_{13}\rho_{23}-\rho_{12}^{2}-\rho_{13}^{2}-\rho_{23}^{2}\right)\
,$ (20)

and it is positive definite. It is worth noting a remarkable factorization
property of the determinant

$D_{m}\ =\
32\,\frac{{(m_{1}+m_{2}+m_{3})}^{2}}{m_{1}^{2}\,m_{2}^{2}\,m_{3}^{2}}\,I\
S^{2}_{\triangle}\ =\
\frac{64}{3}\,\frac{{(m_{1}+m_{2}+m_{3})}^{2}}{m_{1}^{2}\,m_{2}^{2}\,m_{3}^{2}}\,P_{m}\
S_{m}\ .$

Hence, $D_{m}$ is a polynomial function of the $\rho$-variables and it depends
on two coordinates alone, namely the product of $I$ and ${S}_{\triangle}^{2}$.

The three dimensional $\rho$-space with metric (19) is not flat.

###  II.3 Reduced Hamiltonian at zero angular momentum: $\rho$-representation

For the special case of vanishing angular momentum $p_{{}_{\Omega}}=0$, the
Hamiltonian (18) reduces to

$\displaystyle{\cal H}_{0}$ $\displaystyle=\
2\,\bigg{[}\,\frac{\rho_{12}\,P_{12}^{2}}{m_{12}}\ +\
\frac{\rho_{23}\,\,P_{23}^{2}}{m_{23}}\ +\
\frac{\rho_{31}\,\,P_{31}^{2}}{m_{31}}\ +\
\frac{\rho_{12}+\rho_{31}-\rho_{23}}{m_{1}}\,P_{12}\,P_{31}\ +\ $ (21)
$\displaystyle\qquad\quad\frac{\rho_{12}+\rho_{23}-\rho_{31}}{m_{2}}\,P_{12}\,P_{23}\
+\ \frac{\rho_{23}+\rho_{31}-\rho_{12}}{m_{3}}\,P_{23}\,P_{31}\,\bigg{]}\ +\
V(\rho_{12},\,\rho_{23},\,\rho_{31}\,)\ .$

In this case, the coefficients in the kinetic energy are linear in the
$\rho$-coordinates. This classical Hamiltonian is in complete agreement with
the result obtained through the procedure of _dequantization_ from the quantum
one, see Ref.TME3-d . In the next section the above result will be generalized
to the non planar case $d>2$. Hereafter, the form (21) of the reduced
Hamiltonian will be used throughout the text. It is the central object to
explore in the present consideration.

##  III Three-body system: non-planar case $d>2$

###  III.1 $\rho$-representation

The extension of the symmetry reduction originally proposed by Murnaghan for
planar case $d=2$ to the non-planar case $d=3$ (from a 18-dimensional phase
space to 8-dimensional one) is presented in Eq.$(55_{2})$ in Ref.Kampen . In
this case, the reduced Hamiltonian depends on eight dynamical coordinates,
namely, the three mutual distances ($r_{12},\,r_{23}$, $r_{31}\,$), an angular
variable $\omega$ and their associated canonical conjugate momenta. However,
at zero angular momentum the $\omega-$dependent terms disappear and the system
is described by the same reduced Hamiltonian (21). In the
$\rho$-representation it can be proved that this result is valid in any
dimension $d>1$.

###### Theorem 1

The $3-$body system in $d$-dimensions ($d\geq 2$) with potential of the form
$V=V(\rho_{12},\,\rho_{23},\,\rho_{31}\,)$ and zero angular momentum is
described by a six-dimensional reduced Hamiltonian ${\cal H}_{0}$ (in
$\rho$-representation)

${\cal H}_{0}\ =\ {\cal T}_{0}\ +\ V(\rho_{12},\,\rho_{23},\,\rho_{31}\,)\ ,$
(22)

where the kinetic energy ${\cal T}_{0}$ is polynomial in coordinates
$\rho_{ij}\equiv r_{ij}^{2}$ and their conjugate momenta $P_{ij}$. Explicitly,

$\displaystyle{\cal T}_{0}\ =$ $\displaystyle\
2\,\bigg{[}\,\frac{\rho_{12}\,P_{12}^{2}}{m_{12}}\ +\
\frac{\rho_{23}\,\,P_{23}^{2}}{m_{23}}\ +\
\frac{\rho_{31}\,\,P_{31}^{2}}{m_{31}}\ +\
\frac{\rho_{12}+\rho_{31}-\rho_{23}}{m_{1}}\,P_{12}\,P_{31}\ +\ $ (23)
$\displaystyle\qquad\quad\frac{\rho_{12}+\rho_{23}-\rho_{31}}{m_{2}}\,P_{12}\,P_{23}\
+\ \frac{\rho_{23}+\rho_{31}-\rho_{12}}{m_{3}}\,P_{23}\,P_{31}\,\bigg{]}\ .$

Proof: Take free 3-body Hamiltonian (kinetic energy)

${\cal T}\ =\ \frac{1}{2\,m_{1}}\,{\bf p}^{2}_{1}\ +\ \frac{1}{2\,m_{2}}\,{\bf
p}^{2}_{2}\ +\ \frac{1}{2\,m_{3}}\,{\bf p}^{2}_{3}\ .$ (24)

(${\bf p}_{i}\,=\,m_{i}\,\dot{\bf r}_{i}$). Let us make a canonical
transformation (change of variables in the phase space) from $({\bf
r}_{i},{\bf p}_{i})$ variables to the $d-$dimensional center-of-mass variable,
the three coordinates (squares of mutual distances) $\rho_{ij}$, a suitable
set of $(2d-3)$ angular variables and their corresponding canonical momenta.
The center-of mass-variables can be separated out completely. The
$\rho$-variables are given by

$\rho_{\ell k}\ =\ \sum_{s=1}^{d}(x_{\ell,s}-x_{k,s})^{2},\quad
1\leq\ell<k\leq 3\ .$

Thus,

$p_{x_{\ell,s}}\ =\ 2\,\sum_{h\neq\ell,}(x_{\ell,s}-x_{h,s})\,P_{\rho_{\ell
h}}\ +\ \cdots\ ,$

where the non-explicit terms are proportional to the momentum variables
associated with the remaining angular coordinates. At zero total angular
momentum, such terms vanish. Hence, we can compute the coefficient of
$P_{\rho_{\ell k}}^{2}$ in (24) for $\ell<k$. It is

$2\,\bigg{(}\frac{1}{m_{\ell}}\ +\
\frac{1}{m_{k}}\bigg{)}\sum_{s=1}^{d}(\,x_{\ell,s}-x_{k,s}\,)^{2}\ =\
2\,\frac{m_{\ell}+m_{k}}{m_{\ell}m_{k}}\rho_{\ell k}\ .$

Similarly the coefficient of $P_{\rho_{\ell k}}\,P_{\rho_{\ell{k^{\prime}}}}$
for $k<k^{\prime}$ is

$\frac{4}{m_{\ell}}\sum_{s=1}^{d}(x_{\ell,s}-x_{k,s})(x_{\ell,s}-x_{{k^{\prime}},s})=\frac{4}{m_{\ell}}({\bf
r}_{\ell}-{\bf r}_{k})\cdot({\bf r}_{\ell}-{\bf
r}_{k^{\prime}})=\frac{2}{m_{\ell}}(\rho_{\ell{k}}+\rho_{\ell{k^{\prime}}}-\rho_{k{k^{\prime}}})\
,$

where the last equality follows from the law of cosines. The coefficient of
$P_{\rho_{\ell k}}P_{\rho_{{\ell^{\prime}}{k^{\prime}}}}$ for
$k,{k^{\prime}},\ell,{\ell^{\prime}}$ all pairwise distinct is $0$. At zero
center-of-mass momentum (at rest frame) we arrive at ${\cal T}_{0}$. $\Box$

Hence, for any dimension $d>1$ all trajectories with zero angular momentum are
governed by the Hamiltonian (21). This Hamiltonian also describes a three-
dimensional particle moving in a curved space subject to a potential $V$.

###  III.2 Three-body closed chain of interactive harmonic oscillators

As an application of the presented formalism, we consider the case of 3-body
oscillator with quadratic potentials of interaction which depend on relative
distances, $|{\bf r}_{i}-{\bf r}_{j}|$, only 333For quantum counterpart of the
is problem, see 3bodychain .. Needless to say that the two-body harmonic
oscillator can be reduced to a one-dimensional radial Jacobi oscillator, see
e.g. 3bodychain . In the 3-body case such a reduction is not possible in
general.

The potential of a three-body closed chain of harmonic oscillators takes the
form

$V^{(\text{har})}\ =\ 2\,\omega^{2}\bigg{[}\nu_{12}\,\rho_{12}\ +\
\nu_{13}\,\rho_{13}\ +\ \nu_{23}\,\rho_{23}\bigg{]}\ ,$ (25)

where $\omega>0$ is frequency and $\nu_{12},\nu_{13},\nu_{23}\geq 0$ define
spring constants. At zero total angular momentum in center-of-mass inertia
frame, this system is naturally described by the Hamiltonian

$\displaystyle{\cal H}_{0}^{\text{har}}\ =$ $\displaystyle\
2\,\bigg{[}\,\frac{\rho_{12}\,P_{12}^{2}}{m_{12}}\ +\
\frac{\rho_{23}\,\,P_{23}^{2}}{m_{23}}\ +\
\frac{\rho_{31}\,\,P_{31}^{2}}{m_{31}}\ +\
\frac{\rho_{12}+\rho_{31}-\rho_{23}}{m_{1}}\,P_{12}\,P_{31}\ +\
\frac{\rho_{12}+\rho_{23}-\rho_{31}}{m_{2}}\,P_{12}\,P_{23}$ (26)
$\displaystyle\qquad\quad\ +\
\frac{\rho_{23}+\rho_{31}-\rho_{12}}{m_{3}}\,P_{23}\,P_{31}\,\bigg{]}\ +\
2\,\omega^{2}\bigg{[}\nu_{12}\,\rho_{12}\ +\ \nu_{13}\,\rho_{13}\ +\
\nu_{23}\,\rho_{23}\bigg{]}\ ,$

cf.(21). Unlike the case of the standard three-dimensional harmonic oscillator
the Hamiltonian (26) is not superintegrable and it does not admit separation
of variables. However, in the special case

$m_{2}\,\nu_{13}=m_{3}\,\nu_{12}\ ,\qquad m_{1}\,\nu_{23}=m_{2}\,\nu_{13}\
,\qquad m_{3}\,\nu_{12}=m_{1}\,\nu_{23}\ ,$ (27)

(any two relations imply that the third relation should hold), the system (26)
admits 5 functionally independent constants of the motion, so it becomes
_maximally_ superintegrable 3bodychain . If one of the three constraints in
(27) is fulfilled only the system becomes _minimally_ superintegrable.

##  IV $(P,S,T)$-representation (geometrical representation)

In this Section, we consider the case of three equal masses
$m_{1}=m_{2}=m_{3}=1$. By using a canonical transformation, the geometrical
properties of the triangle of interaction are translated to a certain
dynamical variables (see below).

###  IV.1 Reduced Hamiltonian

Based on the $\mathbb{Z}_{2}^{\otimes 3}\oplus\mathcal{S}_{3}$ symmetry of the
free Hamiltonian ${\cal H}_{0}$ (21), let us introduce its invariants as new
variables

$(r_{12},\,r_{23},\,r_{31})\quad\rightarrow\quad(\rho_{12},\,\rho_{23},\,\rho_{31})\quad\rightarrow\quad(P,\,S,\,T)\
,$

where

$\displaystyle P\ \equiv\ \frac{3}{2}\,I\ =\ \frac{1}{2}(\,\rho_{12}\ +\
\rho_{23}\ +\ \rho_{31}\,)\ ,$ (28) $\displaystyle S\ \equiv\
S^{2}_{\bigtriangleup}\ =\ \frac{1}{16}\big{(}\,2\,\rho_{12}\,\rho_{23}\ +\
2\,\rho_{12}\,\rho_{31}\ +\ 2\,\rho_{23}\,\rho_{31}\ -\ \rho_{12}^{2}\ -\
\rho_{23}^{2}\ -\ \rho_{31}^{2}\,\big{)}\ ,$ $\displaystyle T\ \equiv\
\rho_{12}\,\rho_{23}\,\rho_{31}\ ,$

cf.(3), and the three associated canonical conjugate momentum $P_{P}$, $P_{S}$
and $P_{T}$, respectively.

In (28), the moment of inertia $I\propto P$ and the area (squared)
$S^{2}_{\bigtriangleup}=S$ of the triangle of interaction are translated to
two new dynamical variables explicitly. The quantities $P$ and $S$ were called
_volume variables_. It is worth mentioning that each of the variables
$(P,S,T)$ (28) is characterized by an _accidental_ permutation symmetry
$\mathcal{S}_{3}$ in $\rho-$coordinates in addition to the symmetry
$\mathcal{S}_{3}$ of interchange of any pair of bodies positions.

In new variables the Hamiltonian ${\cal H}_{0}$ (21) takes the form

$\displaystyle{\cal H}_{\rm geo}$ $\displaystyle=\ 3\,P\,P_{P}^{2}\ +\
P\,S\,P_{S}^{2}\ +\ {T}\,\bigg{[}48\,S\ +\ 4\,P^{2}\,\bigg{]}\,P_{T}^{2}$ (29)
$\displaystyle\ +\ 18\,{T}\,P_{P}\,P_{T}\ +\ 12\,S\,P_{P}\,P_{S}\ +\
\bigg{[}\,32\,S^{2}\ +\ 8\,S\,P^{2}\bigg{]}\,P_{T}\,P_{S}\ +\ V(P,S,T)\ .$

It corresponds to 3D solid body motion in an external potential $V$. This
Hamiltonian ${\cal H}_{\rm geo}$ can be interpreted as the way to describe a
three-dimensional particle in a curved space, see below. Tensor of inertia can
be identified with cometric in this case. The associated cometric for (29) is
given by

$g^{\mu\nu}(P,\,S,\,{T})\ =\left|\begin{array}[]{ccc}3\,P&\ 6\,S&\ 9\,{T}\\\
&&\\\ 6\,S&\ S\,P&\ 4S\,(4\,S\ +\ P^{2})\\\ &&\\\ 9\,{T}&\ 4S(4S\ +\ P^{2})&\
4\,(12\,S\ +\ P^{2})\,{T}\end{array}\right|\ .$ (30)

Its components are polynomials in the volume variables. Its determinant admits
factorization to three factors,

$D_{\rm geo}\ \equiv\ \text{Det}\big{[}\,g^{\mu\nu}(P,S,T)\,\big{]}\ =\
3\,P\,S\,\left(4\,P\,{T}\,\left(36\,S\ +\ P^{2}\right)\ -\ 16\,S\,\left(4\,S\
+\ P^{2}\right)^{2}\ -\ 27\,{T}^{2}\right)\ .$

Now, let us study under what conditions the determinant vanishes, $D_{\rm
geo}=0$, so that the metric (tensor of inertia) becomes degenerate. In this
case, the transformation (28) is singular (not-invertible). There are three
possibilities. The first possibility is when the first factor vanishes, $P=0$.
We discard it, since it corresponds to the situation when the configuration
space shrinks to a point (we call it triple collision point). The second
possibility is

$S=0\ ,$ (31)

it corresponds to a collinear three-body configuration, or, equivalently, when
three bodies are on the line, the configuration space becomes two-dimensional,
thus, it shrinks to plane, while the third possibility leads to,

$4\,P\,{T}\,\left(36\,S\ +\ P^{2}\right)\ -\ 16\,S\,\left(4\,S\ +\
P^{2}\right)^{2}\ -\ 27\,{T}^{2}\ =\ 0\ ,$ (32)

in which l.h.s. is homogeneous polynomial in $\rho-$coordinates. From (32) it
follows that the determinant $D_{\rm geo}$ vanishes for any isosceles triangle
of interaction, in particular, for equilateral triangle, where

$12\,S\ =\ P^{2}\qquad,\qquad 27\,{T}\ =\ 8\,P^{3}\ .$ (33)

The space with metric (30) is not flat.

###  IV.2 Equations of motion

From the reduced Hamiltonian (29), we obtain the Newton equations of motion
for the variables $(P,S,T)$

$\displaystyle\ddot{P}\ =$ $\displaystyle\frac{3}{2\,D_{\rm
geo}}\bigg{[}\,-4\,S\,\dot{P}^{2}\,\left(4\,S\,\left(4S+P^{2}\right)^{2}-P\,T\left(12\,S+P^{2}\right)\right)\
+\ $ (34) $\displaystyle
3\,\dot{S}^{2}\,T\,\left(48\,S\,P+4\,P^{3}-27\,T\right)\ -\
3\,S\,\dot{T}^{2}\,\left(12\,S-P^{2}\right)\ -\
12\,S\,\dot{S}\,\dot{P}\,T\,\left(24\,S\,-2\,P^{2}\right)\ +\ $
$\displaystyle\dot{T}\left(6\,S\,\dot{P}\,\left(8\,S\left(4\,S+P^{2}\right)-3\,T\right)-12\,S\,\dot{S}\,\left(8\,S\,P+2\,P^{3}-9\,T\,\right)\right)\,\bigg{]}\
-\ $ $\displaystyle
6\,(3\,T\,\partial_{T}V\,+\,2\,S\,\partial_{S}V\,+\,P\,\partial_{P}V)\ ,$
$\displaystyle\ddot{S}\ =$ $\displaystyle\frac{3}{2\,D_{\rm
geo}\,P}\bigg{[}\dot{S}^{2}\,\left(4\,P\,T\left(-72\,S^{2}+30\,S\,P^{2}+P^{4}\right)-16\,S\,P^{2}\left(4\,S+P^{2}\right)^{2}+27\,T^{2}\,\left(6\,S-P^{2}\right)\right)\
+\ $ (35) $\displaystyle 6\,S^{2}\,\dot{T}^{2}\,\left(12\,S-\,P^{2}\right)\ +\
8\,S^{2}\,\dot{P}^{2}\,\left(4\,S\,\left(4\,S+P^{2}\right)^{2}-P\,T\left(12\,S+P^{2}\right)\right)\
+\ $ $\displaystyle
2\,S\,\dot{S}\,\dot{P}\,\left(4\,T\,\left(72\,S^{2}+30\,S\,P^{2}+P^{4}\right)-16\,S\,P\,\left(4\,S+P^{2}\right)^{2}-27\,P\,T^{2}\right)\
+\ $
$\displaystyle\dot{T}\,\left(24\,S^{2}\,\dot{S}\,\left(8\,S\,P+2\,P^{3}-9\,T\right)-12\,S^{2}\,\dot{P}\,\left(8\,S\left(4\,S+P^{2}\right)-3\,P\,T\,\right)\right)\,\bigg{]}$
$\displaystyle\ -\
2\,S\,(4\,\left(4\,S+P^{2}\right)\,\partial_{T}V\,+\,P\,\partial_{S}V\,+\,6\,\partial_{P}V)\
,$ $\displaystyle\ddot{T}\ =$ $\displaystyle\frac{3}{2\,D_{\rm
geo}\,P}\bigg{[}8\,S\,\dot{P}^{2}\,T\left(96\,S^{3}-48\,S^{2}\,P^{2}+14\,S\,P^{4}-3\,P^{3}\,T\right)\
+\ $ (36)
$\displaystyle\dot{S}^{2}\,T\,\left(16\,P^{2}\,\left(96\,S^{2}+12\,S\,P^{2}+P^{4}\right)-144\,P\,T\,\left(6\,S+P^{2}\right)+243\,T^{2}\right)\
+\ $ $\displaystyle
S\,\dot{T}^{2}\,\left(-192\,S^{2}\,P+12\,S\,\left(8\,P^{3}+9\,T\right)+4\,P^{5}-45\,P^{2}\,T\right)\
-\ $ $\displaystyle
8\,S\,\dot{S}\,\dot{P}\,T\,\left(192\,S^{2}\,P-12\,S\,\left(8\,P^{3}+9\,T\right)-4\,P^{5}+45\,P^{2}\,T\right)$
$\displaystyle\ +\
\dot{T}\bigg{(}\,16\,S\,\dot{P}\,\left(32\,S^{3}\,P-4\,S^{2}\,\left(4\,P^{3}+9\,T\right)-3\,S\,P^{2}\,\left(2\,P^{3}-5\,T\right)+P^{4}\,T\right)\
-\ $ $\displaystyle
4\,S\,\dot{S}\,\left(-6\,P\,T\,\left(36\,S+13\,P^{2}\right)+8\,P^{2}\,\left(4\,S+P^{2}\right)\left(12\,S+P^{2}\right)+81\,T^{2}\right)\,\bigg{)}\,\bigg{]}$
$\displaystyle\ -\
2\,(\,4\,T\,\left(12\,S+P^{2}\right)\partial_{T}V\,+\,4\,S\,\left(4\,S+P^{2}\right)\partial_{S}V\,+\,9\,T\,\partial_{P}V\,)\
.$

A further remark is in order. From the Hamiltonian (29) it follows that the
time evolution of $P_{T}$ is given by the Hamilton’s equation

$\dot{P}_{T}\ =\ -2\,P_{T}\,\big{[}\,9\,P_{P}\ +\
2\,P_{T}\,(12\,S+P^{2})\big{]}\ -\ \partial_{T}V\ .$ (37)

Note when potential $V$ depends on the volume variables $P$ and $S$ only, (37)
is reduced to

$\dot{P}_{T}\ =\ -2\,P_{T}\,\big{[}\,9\,P_{P}\ +\
2\,P_{T}\,(12\,S+P^{2})\big{]}\ ,$ (38)

which implies the existence of trajectories with $P_{T}=0$. From the
Hamilton’s equations we obtain

$P_{T}\ =\ \frac{3\,S}{2\,D_{\rm
geo}}\bigg{[}{\dot{P}}\left(8S\left(P^{2}+4S\right)-3PT\right)+{\dot{S}}\left(18T-4P\left(P^{2}+4S\right)\right)+{\dot{T}}\left(P^{2}-12S\right)\,\bigg{]}\
.$ (39)

Therefore, one can construct a _reduced_ Hamiltonian on the phase space
$(P,S,P_{P},P_{S})$. This construction will be elaborated in the next Section.

Now, within the ($P,S,T$)-representation we consider two particular three-body
examples: (i) the $\mathbb{R}^{3}$ Newtonian gravity and (ii) the planar,
$\mathbb{R}^{2}$ choreographic trajectories on the algebraic lemniscate by
Jacob Bernoulli (1694).

####  IV.2.1 Three-body Newtonian gravity potential

First, let us consider the three-body Newton problem in $\mathbb{R}^{3}$
($d=3$). The potential in (29) reads

$V\ \equiv\ V_{\gamma}\ =\ -\gamma\,\bigg{(}\frac{1}{r_{12}}\ +\
\frac{1}{r_{23}}\ +\ \frac{1}{r_{31}}\bigg{)}\ =\
-\gamma\,\bigg{(}\frac{1}{\sqrt{\rho_{12}}}\ +\ \frac{1}{\sqrt{\rho_{23}}}\ +\
\frac{1}{\sqrt{\rho_{31}}}\bigg{)}\ ,$ (40)

where $\gamma$ is the gravitational constant. In terms of the variables
($P,S,T$), see (28), one can show that $V_{\gamma}$ is one of the roots of the
fourth order algebraic equation

$T^{2}\,V_{\gamma}^{4}\ -\ 2\,\gamma^{2}\,(4\,S+P^{2})\,T\,V_{\gamma}^{2}\
{+}\ 8\,\gamma^{3}\,{T}^{\frac{3}{2}}\,V_{\gamma}\ +\
\gamma^{4}\,\bigg{(}{(4\,S+P^{2})}^{2}\ -\ 8\,{T}\,P\bigg{)}\ =\ 0\ .$ (41)

This equation is invariant with respect to simultaneous change
$(\gamma\rightarrow-\gamma)$ and $(V_{\gamma}\rightarrow-V_{\gamma})$ 444For
the case of $\mathbb{R}^{2}$ ($d=2$) Newtonian gravity the 3-body potential is
of logarithmic type, $V_{\gamma}\leavevmode\nobreak\ =\leavevmode\nobreak\
\gamma\log r_{12}r_{13}r_{23}$. Analogue of the equation (41) is very simple,
$V_{\gamma}=\frac{\gamma}{2}\,\log T$. It does not contain dependencies on
$P,S$.

Interestingly, the remaining three roots correspond to three different 3-body
Coulomb potentials for 3 unit charges of different signs, $\pm i$,

$\displaystyle V^{(1)}_{\gamma}\ =\
\gamma\,\bigg{(}\,-\frac{1}{\sqrt{\rho_{12}}}\ +\ \frac{1}{\sqrt{\rho_{23}}}\
+\ \frac{1}{\sqrt{\rho_{31}}}\,\bigg{)}\ ,$ (42) $\displaystyle
V^{(2)}_{\gamma}\ =\ \gamma\,\bigg{(}\,\frac{1}{\sqrt{\rho_{12}}}\ -\
\frac{1}{\sqrt{\rho_{23}}}\ +\ \frac{1}{\sqrt{\rho_{31}}}\,\bigg{)}\ ,$
$\displaystyle V^{(3)}_{\gamma}\ =\
\gamma\,\bigg{(}\,\frac{1}{\sqrt{\rho_{12}}}\ +\ \frac{1}{\sqrt{\rho_{23}}}\
-\ \frac{1}{\sqrt{\rho_{31}}}\,\bigg{)}\ .$

Hence, the above equation (41) describes all four possible 3-body
Newton/Coulomb potentials with imaginary unit charges with constant of
interaction $\gamma>0$.

The discriminant of the equation (41) admits factorization

$D_{\gamma}\ =\ 4096\,\gamma^{12}\,{T}^{8}\,\left(4\,P\,{T}\,\left(36\,S\ +\
P^{2}\right)\ -\ 16\,S\,\left(4\,S\ +\ P^{2}\right)^{2}\ -\
27\,{T}^{2}\right)\ ,$ (43)

and its last factor coincides with the last factor in the factorized
expression for determinant $D_{\rm geo}$ of the metric (30). For any isosceles
triangle both determinant and discriminant vanish, $D_{\gamma}=D_{\rm geo}=0$.
Furthermore, in the case of an equilateral triangle of interaction the
potential (40) simplifies,

$V_{\gamma}\ =\ -\frac{3^{\frac{3}{2}}\gamma}{\sqrt{2P}}\ .$

It corresponds to the solution found long ago by Lagrange Lagrange , it
describes to the so called central configuration, see e.g. Refs.Moeckel ;
Saari ; Albouy  and reference therein. Also, the remarkable periodic Figure
Eight solution of the three-body problem found by Moore Moore  and confirmed
by Chenciner and Montgomery Chenciner-Montgomery  \- the so-called 3-body
choreography - gets a natural presentation in the geometrical variables. On
this trajectory, the motion of the bodies alternates between six collinear
configurations and six isosceles-triangular ones.

####  IV.2.2 Choreographic motion

As for the choreographic trajectories in $\mathbb{R}^{2}$ ($d=2$), when three
bodies follow one to each other being on the same curve, Fujiwara et al.
Fujiwara  solved the inverse problem of choreographic motion on a Figure-8
given by the algebraic lemniscate of Jacob Bernoulli (1694) on $(x,y)$-plane

$(x^{2}\,+\,y^{2})^{2}\ =\ c^{2}\,(x^{2}-y^{2})\ ,$ (44)

where without loss of generality one can put $c=1$, and found 3-body pairwise
potential. It was shown that such a figure eight is the choreographic
trajectory for three unit mass, point-like particles with zero angular
momentum with two-parametric potential

$V_{\infty}\ =\ \frac{1}{4}\,\ln{T}-\frac{\sqrt{3}}{12}\,P\ ,$ (45)

with the volume variables $P,T$ given by (28), ATJC . The first attractive
terms in (45) represents a 3-body $\mathbb{R}^{2}$ Newtonian potential for
gravitational constant $\gamma=1/2$, while the second repulsive term is the
square of the hyperradius in the space of relative motion. It is worth
mentioning that the Figure-8 algebraic lemniscate with potential (45) is
_close_ to the transcendental Figure-8 trajectory found by C MooreMoore , see
also Ref.Chenciner-Montgomery , for the $\mathbb{R}^{3}$ Newtonian gravity
potential.

Recently, in Ref.ATJC  it was found that the 3-body choreographic motion on
the algebraic lemniscate is maximally (particularly) superintegrable.
Moreover, the two variables $P$ and $T$ become particular integrals (see also
Ref.Vieyra ). Along this Figure-8 trajectory, these variables take the
constant value

$P\ =\ {T}\ =\ \frac{3\,\sqrt{3}}{2}\ .$ (46)

Therefore, it is natural to work with the Hamiltonian $(\ref{Hredgeo})$. It
can be shown that three equations of motion (34)-(36) constrained by the
conditions (46) lead to non-linear ODE

$S\,\left(\,256\,S^{2}\,+\,864\,S\,-\,243\,\right)\ +\
48\,\sqrt{3}\,\dot{S}^{2}\ =\ 0\ ,$ (47)

which defines the elliptic curve, see Fig.2, with variables $P,T$ playing role
of elliptic invariants ATJC . The solution of (47) is given by the Weierstrass
function $\wp(t;P,T)$. Hence, in the configuration space parametrized by
geometrical variables the 3-body choreography on algebraic lemniscate by Jacob
Bernoulli is given by a (planar) elliptic curve!

Figure 2: Time evolution of the volume variable $S$, area (squared) of the
triangle of interaction, for the 3-body choreographic Figure-8 on the
algebraic lemniscate by Bernoulli. The points $S=0$ correspond to the so-
called Euler line for which all three bodies are on the line.

##  V Volume variables representation

###  V.1 Case of equal masses

Now we consider the case of three unit masses $m_{1}=m_{2}=m_{3}=1$ and focus
on potentials that depend solely on the volume variables $P$ and $S$, see
(28),

$V\ =\ V(P,\,S)\ .$ (48)

In this case the equations of motion (37) that emerge from the Hamiltonian
${\cal H}_{\rm geo}$ (29) admit a number of solutions with $P_{T}=0$. This
fact is not trivial since the momentum $P_{T}$ is not a Liouville integral of
motion, $\\{P_{T},\,{\cal H}_{\rm geo}\\}_{P.B.}\neq 0$.

Equivalently, in the six-dimensional phase space $(P,S,T,P_{P},P_{S},P_{T})$
the hypersurface $P_{T}=c$ becomes an invariant manifold Wiggins  at $c=0$:
$\\{P_{T},\,{\cal H}_{\rm geo}\\}_{P.B.}|_{{}_{P_{T}=0}}\ =\ 0$. This implies
that any trajectory of ${\cal H}_{\rm geo}$ for which the initial condition
$P_{T}=0$ is imposed will remain on the hypersurface ${\cal H}_{\rm
geo}(P,S,T,P_{P},P_{S},P_{T}=0)=E$ during the evolution. Inserting $P_{T}=0$
directly into the Hamiltonian (29) we arrive at

${\cal H}_{\rm vol}\ \equiv\ {\cal H}_{\rm geo}\mid_{{}_{\small P_{T}=0}}\ =\
3\,P\,P_{P}^{2}\ +\ P\,S\,P_{S}^{2}\ +\ 12\,S\,P_{P}\,P_{S}\ +\ V(P,S)\ ,$
(49)

which describes a two-dimensional particle in a curve space, see below. Notice
that $T-$dependence in (49) is absent as well.

From ${\cal H}_{\rm vol}$ we obtain the Newton equations

$\displaystyle\ddot{P}\ =\ \frac{1}{2\,D_{\rm
vol}}\bigg{[}\,3\,P\,S\,\dot{P}^{2}\ -\ 36\,S\,\dot{P}\,\dot{S}\ +\
9\,P\,\dot{S}^{2}\,\bigg{]}\ -\ 6\left(\,2\,S\,\partial_{S}\,V\ +\
P\,\partial_{P}\,V\,\right)$ (50) $\displaystyle\ddot{S}\ =\
\frac{1}{2\,D_{\rm vol}}\bigg{[}\,3\,\dot{S}^{2}\,\left(P^{2}-18S\right)\ -\
6\,S^{2}\,\dot{P}^{2}\ +\ 6\,P\,S\,\dot{P}\,\dot{S}\,\bigg{]}\ -\
2\,S\,\left(\,P\,\partial_{S}\,V\ +\ 6\,\partial_{P}\,V\,\right)\ ,$

cf.(34),(35).

Remark. It is worth clarifying the connection between the trajectories of
${\cal H}_{\rm vol}$ (49) and those of ${\cal H}_{\rm geo}$ (29). The
evolution $P=P(t)$ and $S=S(t)$ in (50) also satisfies the equations of motion
(34)-(36), where $T=T(t)$ is fully determined by $P(t)$ and $S(t)$ if the
condition $P_{T}=0$ is imposed, see (39). In this way we reduced effectively
the dimensionality of the system (29) from six to four.

The Hamiltonian (49) is written in what we call the _volume variables
representation_.

####  V.1.1 The potential $V=\frac{{F[P/\sqrt{S}]}}{\sqrt{S}}$

Let us consider an arbitrary potential $V=V(P,S)$ on the submanifold of phase
space for which ${\cal H}_{\rm vol}=0$ (zero energy level) and $\dot{P}=0$ (a
constant moment of inertia). In this case, from (49) and (50) it follows that
the potential must obey the equation

$2\,S\,\partial_{S}\,V\ +\ P\,\partial_{P}\,V\ =\ -V\ .$ (51)

The solution of (51) is given by

$V\ =\ \frac{{U[P/\sqrt{S}]}}{\sqrt{S}}\ ,$

where $U[z]$ is an arbitrary function of the argument $z$. The particular case
$U_{0}=-\gamma\frac{P}{\sqrt{S}}$, thus $V=-\gamma\frac{P}{S}$ has been fully
analyzed in Ref.Montgomery  where the author constructs the hyperbolic plane
with its geodesic flow as the scale plus symmetry reduction of a three-body
problem in the Euclidean plane. In fact, in Ref.Montgomery  the generalization
of $V=-\gamma\frac{P}{S}$ to the case of arbitrary masses was considered as
well. Later on, we will show that the mass-dependent case can be easily
obtained from the present equal-mass case.

For an arbitrary function $U[P/\sqrt{S}]$ we obtain the first-order nonlinear
ODE for the volume variable $S$

$P\,{\dot{S}}^{2}\ +\ 4\,U\,(P^{2}\ -\ 12\,S)\,\sqrt{S}\ =\ 0\ ,$ (52)

where $P>0$ plays the role of an external parameter. In the case of the
potential $V=-\gamma\frac{P}{S}$ we arrive to the solution

$S(t)\ =\ \frac{P^{2}}{12}\ -\ 3\,\gamma\,{(c_{1}\ +\ 2\,t)}^{2}>0\ ,$ (53)

here $c_{1}$ is a constant of integration. Notice that the potential
$V\propto\frac{1}{P}$ leads to the solutions $S(t)=12\,P^{2}$ (an equilateral
triangle) and $S(t)=0$ (three particles on a line).

####  V.1.2 The metric $g_{\rm vol}^{\mu\nu}$

Now, the metric (or, equivalently, the tensor of inertia) for (49) is of the
form

$g_{\rm vol}^{\mu\nu}\ =\left|\begin{array}[]{cc}3\,{P}&\ 6\,S\\\
6\,S&P\,S\end{array}\right|\ ,$ (54)

with determinant

$D_{\rm vol}\ =\ 3\,S\,(\,{P}^{2}\ -\ 12\,S\,)\ .$

In the case of an _equilateral triangle of interaction_ the relation ${P}^{2}\
=\ 12\,S$ holds and this determinant vanishes, ${D}_{\rm vol}=0$. It also
follows from (54) that the corresponding Ricci scalar $\rm{Rs}$ is given by

${\rm Rs}_{\rm vol}\ =\ \frac{3\,S\,P\,(\,S\ -\ 3\,)}{{D_{\rm vol}}^{2}}\ ,$
(55)

see below, (68). ${\rm Rs}_{\rm vol}$ is singular for equilateral triangle of
interaction.

An interesting particular case occurs when the potential (48) does not depend
on $S$ and is a function of the volume variable $P$ alone

$V\ =\ V(P)\ .$

It follows from (49) that there exist trajectories for the original
Hamiltonian (29) which depend on $P$ only. Such trajectories lie on the
intersection between the hypersurfaces $P_{T}=0$ and $P_{S}=0$, they are
described by the two-dimensional Hamiltonian

${\cal H}_{\rm vol}\mid_{{}_{\small P_{S}=0}}\ \equiv\ {\cal H}_{P}\ =\
3\,{P}\,P_{P}^{2}\ +\ V(P)\ ,$ (56)

cf.(49). The connection between the trajectories of ${\cal H}_{P}$ (56) with
those of ${\cal H}_{\rm vol}$ (49) and ${\cal H}_{\rm geo}$ (29) is the
following. Taking a non-trivial solution $P=P(t)\neq 0$ of (56) one can define
the function $S=S(t)$ for which the canonical momentum $P_{S}$ vanishes in
${\cal H}_{\rm vol}$. Explicitly, the condition can be obtained from (49),

$P_{S}\ =\ \frac{2\,{\dot{P}}\,S\,-\,P\,{\dot{S}}}{2\,S\,(12\,S\,-\,P^{2})}\
=\ 0\ .$ (57)

Such $P=P(t)$ and $S=S(t)=\lambda\,P^{2}(t)$, with $\lambda\neq 12$ a non-
negative parameter, satisfy the equations (50) for ${\cal H}_{\rm vol}$.
Moreover, they also obey the equations (34)-(36) for ${\cal H}_{\rm geo}$
where at $\lambda\neq 0$ the variable $T$ turns out to be identically zero due
to the condition $P_{T}=0$, hence, the 3body problem degenerates into a 2body
system. At $\lambda=0$, thus $S=0$, in general $T\neq 0$ and the domain of the
problem degenerates to the line.

####  V.1.3 Anharmonic Oscillator potential

As a concrete example, let us consider the following potential

$V^{(AO)}\ =\ A\,P\ +\ B\,P^{2}\ =\
\frac{A}{2}(\,\rho_{12}\,+\,\rho_{23}\,+\,\rho_{31}\,)\ +\
\frac{B}{4}{(\,\rho_{12}\,+\,\rho_{23}\,+\,\rho_{31}\,)}^{2}\ ,$ (58)

where $B\geq 0$ and $A>0$ if $B=0$. In terms of variable $P$ the potential
(58) corresponds to a shifted one-dimensional harmonic oscillator on the half
line $\mathbb{R}_{+}$ with non-standard kinetic energy. In the
$\rho-$representation it corresponds to a certain three-dimensional quadratic
potential on the first octant $\mathbb{R}^{3}_{+}$ whilst in the
$r-$representation it describes a three-dimensional isotropic harmonic
oscillator with quartic anharmonicity 555Note that in present formalism, a
more general anharmonic potential $V(P,S)\ =\ A\,P+B\,P^{2}+C\,S$ (the general
$S_{3}$-symmetric three-variable quartic potential in $r$-variables) can be
studied in straightforward manner..

It is worth mentioning that even symmetric harmonic 3body planar systems, with
equal spring rest lengths $L$, equal spring constants $k$ and equal masses $m$
despite its apparent simplicity, display a wide array of interesting dynamics
for different energy values, (see Ref.Katz  and references therein). In
Ref.Katz  the authors demonstrate that for such systems the orientation of the
triangle is a nontrivial, history-dependent variable of the system, and serves
as a sensitive measurable for the type of dynamics the system follows.

For the potential (58), the Hamiltonian (56) becomes

$\displaystyle{\cal H}^{(AO)}_{P}$ $\displaystyle=\ 3\,{P}\,P_{P}^{2}\ +\
A\,P\ +\ B\,P^{2}\ .$ (59)

The time-evolution of the system is given by

$P(t)\ =\ x^{2}\ =\
\frac{A\,k^{2}}{B\,(1-k^{2})}\,\text{sn}^{2}(y,ik)\quad,\qquad
y\,=\,\pm\sqrt{\frac{3\,A}{1-k^{2}}}\ t\ ,$ (60)

($B\neq 0$, $k\neq\pm 1$) where sn$(y,ik)$ is the Jacobi elliptic function
with imaginary elliptic modulus $(ik)$. In the case of physical systems the
function $P(t)$ should be positive and the parameter $k\neq\pm 1$ is real.

For the solution (60) the Hamiltonian has a meaning of energy and it takes the
value

${\cal H}^{(AO)}_{P}\ =\ \frac{A^{2}\,k^{2}}{B\left(1-k^{2}\right)^{2}}\ .$

Putting the $B=0$ in (59) we obtain the harmonic oscillator potential
$V=A\,P$, which is quadratic polynomial in the $r$-representation. In this
case the trajectories are trigonometric vibrations,
$P(t)=c_{1}\,\cos^{2}(\sqrt{3\,A}\,t+c_{2})$ with energy ${\cal
H}^{(AO)}_{P}\,=\,\ c_{1}\,A$ where $c_{1}>0,c_{2}$ are real constants of
integration. In general, due to the energy conservation ${\cal
H}^{(AO)}_{P}=E$ the trajectories in the phase space are always cubic curves,

$3\,{P}\,P_{P}^{2}\ +\ A\,P\ +\ B\,P^{2}\ =\ E\ .$

Note that at fixed values of the energy $E$ and $A$ the presence of
anharmonicity ($B>0$) in (59) tends to decrease the amplitude of the harmonic
motion $(B=0)$.

###  V.2 Case of unequal masses: modified volume variables

In this Section the case of three bodies with arbitrary masses
$(m_{1},m_{2},m_{3})$ is considered. From the general Hamiltonian ${\cal
H}_{0}$ at zero angular momentum, see (21), it will be constructed a four-
dimensional _reduced_ Hamiltonian, which depends on the modified _volume
variables_ $P_{m}$ and $S_{m}$ alone and their respective canonical momenta,
for two-variable potentials $V(P_{m},\,S_{m})$.

As a first step, let us make the change of variables in the Hamiltonian ${\cal
H}_{0}$ (21)

$(\,\rho_{12},\,\rho_{31},\,\rho_{23}\,)\ \rightarrow\
(\,P_{m},\,S_{m},\,Q\,)\ ,$ (61)

where the volume variable

$S_{m}\ =\ \frac{3\,m_{1}\,m_{2}\,m_{3}}{m_{1}+m_{2}+m_{3}}\,S\ ,$ (62)

is proportional to the square of the area of the triangle of interaction $S$
(28), and

$P_{m}\ =\ \frac{1}{2}\bigg{[}\frac{1}{m_{3}}\,\rho_{12}\ +\
\frac{1}{m_{1}}\,\rho_{23}\ +\ \frac{1}{m_{2}}\,\rho_{31}\,\bigg{]}\ ,$ (63)

is the _weighted sum_ of the _edges (squared)_ of the triangle of interaction.
At equal masses $m_{1}=m_{2}=m_{3}=1$ the modified volume variables coincide
to the original ones $P$ and $S$: $P_{m}\rightarrow P$, $S_{m}\rightarrow S$.
The third variable $Q=Q(\rho_{12},\rho_{31},\rho_{23})$ can be any function of
$\rho$’s with condition that the Jacobian of the transformation (61) is
invertible (nonsingular) in the domain $\mathbb{R}^{3}_{+}$, where the problem
is defined. In these variables, the Hamiltonian (21) takes the form

$\displaystyle{\cal H}_{Q}$ $\displaystyle\ =\
\frac{m_{1}+m_{2}+m_{3}}{m_{1}\,m_{2}\,m_{3}}\,{P_{m}}\,P_{{\small
P_{m}}}^{2}\ +\ P_{Q}\,(F_{1}\,P_{{\small P_{m}}}\ +\ F_{2}\,P_{S_{m}}\ +\
F_{3}\,P_{Q})$ (64) $\displaystyle\ +\
\frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}{P_{m}}\,S_{m}\,P_{S_{m}}^{2}\
+\ 4\,\frac{m_{1}+m_{2}+m_{3}}{m_{1}\,m_{2}\,m_{3}}\,S_{m}\,P_{{\small
P_{m}}}\,P_{S_{m}}\ +\ V\ ,$

where, in general, the coefficients $F_{1}$, $F_{2}$ and $F_{3}$ are functions
of $(P_{m},S_{m},Q)$. Let us consider that the family of two-variable
potentials in (64) depends on the volume variables

$V\ =\ \frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}\,V_{m}(P_{m},\,S_{m})\
,$ (65)

alone. In this case it is easy to see that the Hamiltonian (64) admits
trajectories with $P_{Q}=0$ in the six-dimensional phase space
$(P_{m},S_{m},Q,P_{P_{m}},P_{S_{m}},P_{Q})$: $\\{P_{Q},\,{\cal H}_{\rm
Q}\\}_{P.B.}\mid_{{}_{P_{Q}=0}}\ =\ 0$. Such trajectories are described by the
Hamiltonian

$\displaystyle{\cal H}_{m}$ $\displaystyle\ \equiv\ {\cal
H}_{0}\mid_{P_{Q}=0}$ (66) $\displaystyle\ =\
\frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}\bigg{[}\,3\,{P_{m}}\,P_{{\small
P_{m}}}^{2}\ +\ {P_{m}}\,S_{m}\,P_{S_{m}}^{2}\ +\ 12\,{S_{m}}\,P_{{\small
P_{m}}}\,P_{S_{m}}\ +\ V_{m}({P_{m}},\,{S_{m}})\,\bigg{]}\ ,$

which generalizes ${\cal H}_{\rm vol}$ (49) to the non-equal mass case.
Multiplying ${\cal H}_{m}$ (66) by
$\frac{3\,m_{1}\,m_{2}\,m_{3}}{m_{1}+m_{2}+m_{3}}$ we obtain the Hamiltonian
(49). Hence, we arrive to the interesting result

> _The zero total angular momentum trajectories of a 3-body system with equal
> masses in a two-variable potential $V(P,S)$ and those of 3 bodies with
> arbitrary masses and potential
> $\frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}V(P_{m},S_{m})$ do
> coincide!_
>

The Hamiltonian (66) describes a two-dimensional particle moving in curved
space or, equivalently, two-dimensional solid body in external potential $V$.
The trajectories of ${\cal H}_{m}$ are solutions to the equations of motion
for ${\cal H}_{Q}$ (64) as well.

The corresponding metric for (66) takes the form

$g_{m}^{\mu\nu}\
=\left|\begin{array}[]{cc}\frac{m_{1}+m_{2}+m_{3}}{m_{1}\,m_{2}\,m_{3}}\,{P_{m}}&\
2\,\frac{m_{1}+m_{2}+m_{3}}{m_{1}\,m_{2}\,m_{3}}\,S_{m}\\\
2\,\frac{m_{1}+m_{2}+m_{3}}{m_{1}\,m_{2}\,m_{3}}\,S_{m}&\frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}\,{P_{m}}\,S_{m}\end{array}\right|\
,$ (67)

with determinant

${\tilde{D}}_{m}\ =\
\frac{{(m_{1}\,+\,m_{2}\,+\,m_{3})}^{2}}{{3\,(m_{1}\,m_{2}\,m_{3})}^{2}}\,S_{m}\,[\,{P_{m}}^{2}\,-\,12\,S_{m}\,]\
,$

cf.(54). It is worth emphasizing that the metrics (67) and (54) coincide up to
a multiplicative factor. From (67) it follows that the corresponding Ricci
scalar $\rm{Rs}$ is given by

${\rm\tilde{R}s}\ =\
\frac{m_{1}\,m_{2}\,m_{3}\,{P_{m}}\,(\,S_{m}\,-\,3)}{(m_{1}\,+\,m_{2}\,+\,m_{3})\,S_{m}\,{[\,{P_{m}}^{2}\,-\,12\,S_{m}\,]}^{2}}\
=\
\frac{{(m_{1}\,+\,m_{2}\,+\,m_{3})}^{3}\,S_{m}\,{P_{m}}\,(\,S_{m}\,-\,3)}{9\,{(m_{1}\,m_{2}\,m_{3})}^{3}\,{\tilde{D}}_{m}^{2}}\
,$ (68)

cf.(55).

In the particular case of a one-variable potential,

$V\ =\ \frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}\,V_{m}(P_{m})\ ,$

as it follows from (66), the Hamiltonian ${\cal H}_{Q}$ (64) admits
trajectories depending on $P_{m}$ only. They lie on the intersection between
the hypersurfaces $P_{Q}=0$ and $P_{S_{m}}=0$, they are governed by the two-
dimensional Hamiltonian

${\cal\tilde{H}}_{m}\ \equiv\ {\cal H}_{m}\mid_{P_{S_{m}}=0}\ =\
\frac{m_{1}+m_{2}+m_{3}}{3\,m_{1}\,m_{2}\,m_{3}}\,\bigg{[}\,3\,{P_{m}}\,P_{{\small
P_{m}}}^{2}\ +\ V({P_{m}})\,\bigg{]}\ ,$ (69)

cf.(56).

As concrete example one can consider the mass-dependent 3-body anharmonic
oscillator potential

$V_{m}^{(AHO)}\ =\ A\,P_{m}\ +\ B\,P_{m}^{2}\ ,$

cf.(58), where $A,B$ are parameters. As mentioned above, it is evident that
multiplying the Hamiltonian ${\cal\tilde{H}}_{m}$ (69) by
$\frac{3\,m_{1}\,m_{2}\,m_{3}}{m_{1}+m_{2}+m_{3}}$ we arrive at the same
Hamiltonian ${\cal H}_{P}$ (59). Hence, the trajectories of two 3-body
anharmonic oscillators defined by (59) and (69), respectively, coincide and,
in general, they are given by (60).

In the case of particular 3-body anisotropic harmonic oscillator with
different masses which occurs at $B=0$, see above,

$V_{m}^{(HO)}\ =\ A\,P_{m}\ =\ \frac{A}{2}\bigg{(}\,\frac{r_{12}^{2}}{m_{3}}\
+\ \frac{r_{31}^{2}}{m_{2}}\ +\ \frac{r_{23}^{2}}{m_{1}}\,\bigg{)}\ ,$

the spring constants in (25) take values
$\nu_{12}=\frac{1}{4\,\omega^{2}\,m_{3}}$,
$\nu_{13}=\frac{1}{4\,\omega^{2}\,m_{2}}$ and
$\nu_{23}=\frac{1}{4\,\omega^{2}\,m_{1}}$. It leads to the periodic
trigonometric solution

$P_{m}(t)\ =\ c_{1}\,\cos^{2}(\sqrt{3\,A}\,t+c_{2})\ ,$

where $c_{1}>0$ and $c_{2}$ are constants of integration.

##  VI Conclusions

In this paper, which is the first in a series, the 3-body classical system in
a $d$-dimensional coordinate space $\mathbb{R}^{d}$, $d\geq 2$, is considered.
The study is restricted to potentials $V=V(r_{12},r_{23},r_{13})$ that depend
solely on the relative distances between bodies. Assuming zero total angular
momentum in the center-of-mass frame, the original $3d-$dimensional problem in
configuration space $\mathbb{R}^{3d}$ is reduced to a $3-$dimensional space
(of relative motion) parametrized by mutual distances. The corresponding six-
dimensional Hamiltonian is constructed explicitly in the space of relative
motion. A new observation is that in the _$r-$ representation_ the Hamiltonian
$H_{r}$ (14) at zero potential (the free problem) possesses formally a
$\mathbb{Z}_{2}^{\otimes 3}$-symmetry $r_{ij}\rightarrow-r_{ij}$. This allows
us to code this symmetry by introducing the $\rho_{ij}=r_{ij}^{2}$ variables
and to proceed to the _$\rho-$ representation_ obtaining the Hamiltonian
${\cal H}_{0}$ (21). This Hamiltonian corresponds to the kinetic energy of
3-dimensional solid body. Making the identification of the coefficients of the
tensor of inertia in ${\cal H}_{0}$ as the entries of a contravariant metric
(cometric), the emerging Hamiltonian describes a 3-dimensional classical
particle moving in a curved space with cometric $g^{\mu\,\nu}(\rho)$ (19). Its
kinetic energy is a polynomial in coordinate and momentum variables and,
unlike the quantum case TME3-d , is not accompanied by effective potential.
The determinant of cometric Det$\big{[}g^{\mu\,\nu}(\rho)\big{]}$ is of
definite sign. It vanishes either at the triple-body collision point or on
collinear configurations only. This cometric does not become singular at the
two-body collision points unlike as it is in the $r-$representation, already
noticed in Ref.Murnaghan  and Ref.Kampen  for $d=2$ and $d=3$, respectively,
as un undesirable property. The free problem in both $r-$ and
$\rho-$representations is $\mathcal{S}_{3}$-permutationally invariant wrt the
interchange of body positions and their masses.

The positions of the 3 bodies form a triangle. In the case of three equal
masses $m_{1}=m_{2}=m_{3}=1$, the kinetic energy possesses an _accidental_
$\mathcal{S}_{3}$-symmetry wrt the interchange of the edges (intervals of
interaction) of this triangle. The full
$\mathbb{Z}_{2}^{3}\oplus\mathcal{S}_{3}^{2}$ symmetry of the free problem is
encoded in the set of generalized coordinates $P$, $S$ and $T$ obtaining the
Hamiltonian ${\cal H}_{\rm geo}$ (29) in the _geometrical-representation_. The
two coordinates $P$ and $S$ are called _volume variables_. The variable $P$
corresponds to the sum of squares of sides of the triangle whilst $S$ is the
area squared. In $(P,S,T)$-variables, the kinetic energy remains polynomial.
It also describes a three-dimensional particle moving in a curved space with a
$d$-independent metric $g^{\mu\,\nu}(P,S,T)$. The determinant
Det$\big{[}g^{\mu\,\nu}(P,S,T)\big{]}$ is again of definite sign. It vanishes
when the triangle of interaction is isosceles. The properties and simplicity
of geometrical-representation were illustrated in (I) the physically relevant
3-body Newtonian gravity potential ($d=3$) where (41) describes all four
possible 3-body Coulomb/Newton potentials in a unified manner, and (II) in the
planar 3-body choreographic motion on algebraic lemniscate ($d=2$) for which
the two geometrical variables $P$ and $T$ become _particular constants of
motion_ and the dynamics is parametrized by a planar elliptic curve
$S=\wp(t;P,T)$.

The volume variables admit a generalization, $S_{m}$ (62) and $P_{m}$ (63), to
systems with arbitrary masses. For potentials that solely depend on these
variables, $V=V(P_{m},S_{m})$, the trajectories in the _volume-representation_
are governed by a four-dimensional Hamiltonian ${\cal H}_{\rm vol}$ (49). It
describes a two-dimensional particle moving in a curved space. In the volume-
representation, the trajectories are mass-independent. In the particular case
of a one-variable potential, $V=V(P_{m})$, the system is described by a two-
dimensional Hamiltonian only. In this representation, a shifted harmonic
potential (equivalently, a 3-body mass-dependent anharmonic potential in
$r$-variables) was analyzed in detail. The volume-representation implies an
effective reduction of the problem beyond separation of variables.

Finally, in the general $n$-body case with $n\geq d-1$ the system can be
thought of as a type of ”breathing” _polytope of interaction_. The volume-
representation can be constructed immediately. At fixed $n$ there exist
($n-1$) volume-variables made out of elements with different dimensionality
(edges, faces, cells and so on) of the polytope of interaction. It reveals a
surprising link between the theory of regular polytopes and the dynamics of an
$n$-body system. This will be presented in a forthcoming paper.

##  VII Author’s contributions

All authors contributed equally to this work.

##  VIII Acknowledgments

A.M. thanks T. Fujiwara and E. Piña for important remarks and personal
discussions. R.L. is supported in part by CONACyT grant 237351 (Mexico).
A.V.T. thanks Don Saari for a clarification, he is supported in part by the
PAPIIT grant IN113819 (Mexico). W.M. is partially supported by a grant from
the Simons Foundation (# 412351 to Willard Miller, Jr.).

##  IX DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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