# Classical $n$-body system in volume variables. II. Four-body case

A. M. Escobar-Ruiz<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

###### Abstract

It is evident that the positions of 4 bodies in $d>2$ dimensional space can be
identified with vertices of a tetrahedron. Square of volume of the
tetrahedron, weighted sum of squared areas of four facets and weighted sum of
squared edges are called the volume variables. A family of translation-
invariant potentials which depend on volume variables alone is considered as
well as solutions of the Newton equations which solely depend on volume
variables. For the case of zero angular momentum $L=0$ the corresponding
Hamiltonian, which describes these solutions, is derived. Three examples are
studied in detail: (I) the (super)integrable 4-body closed chain of harmonic
oscillators for $d>2$ (the harmonic molecule), (II) a generic, two volume
variable dependent potential whose trajectories possess a constant moment of
inertia ($d>1$), and (III) the 4-body anharmonic oscillator for $d\geq 1$.
This work is the second of the sequel: the first one [IJMPA 36, No. 18 (2021)]
was dedicated to study the 3-body classical problem in volume variables.

Classical mechanics, 4-body system,symmetric reduction, harmonic pairwise
interaction

##  I Introduction

For two interacting point-like particles of masses $m_{1}$ and $m_{2}$ in
$\mathbb{R}^{d}$ ($d>1$), with scalar potential $V=V(r_{12})$ that depends on
the relative distance $r_{12}=\mid{\bf r}_{1}-{\bf r}_{2}\mid$, the Lagrangian
is given by

${\cal L}_{2}\ =\ \frac{1}{2}m_{1}\,{\dot{{\bf r}}_{1}}^{2}\ +\
\frac{1}{2}m_{2}\,{\dot{{\bf r}}_{2}}^{2}\ -\ V(r_{12})\ ,$ (1)

here ${\bf r}_{i}\in\mathbb{R}^{d}$ is the position vector of the $i$th
particle and $\dot{{\bf r}}_{i}\equiv\frac{d}{dt}{\bf r}_{i}$ is the velocity,
$i=1,2$. The $d$-dimensional center-of-mass (cms) motion can be separated out,
and we arrive at the $d$-dimensional problem in the _space of relative
motion_. In the case of _zero total angular momentum_ the $d$-dimensional
2-body dynamics, which the Lagrangian (1) governs, is reduced to one-
dimensional dynamics in variable $r_{12}$, governed by the Lagrangian

$\tilde{\cal L}_{2}\ =\ \frac{1}{2}\mu\,{\dot{r}}_{12}^{2}\ -\ V(r_{12})\ ,$
(2)

where $\mu=\frac{m_{1}\,m_{2}}{m_{1}+m_{2}}$ stands for the reduced mass. The
symmetry reduction ${\cal L}_{2}\rightarrow\tilde{\cal L}_{2}$ from
$(2d)$-dimensional coordinate space to one-dimensional space of relative
radial motion is well described in textbooks on Classical Mechanics, see e.g.
[1]. We should only emphasize that the _generalized coordinate_ $r_{12}$ in
(2) plays a fundamental role in this symmetry reduction, it possesses a clear
geometrical meaning: it corresponds to the length of the interval which
connects two point masses. This variable can be naturally called the _interval
(length) of interaction_. Formal $Z_{2}$ invariance $r_{12}\rightarrow-r_{12}$
of the kinetic energy suggests to introduce the volume variable
$\rho_{12}=r_{12}^{2}$.

For the three-body case in $\mathbb{R}^{d}$ ($d>1$) with three arbitrary
masses, the emergent geometrical object is the triangle of interaction formed
by the body positions as vertices. The lengths of its edges are the 3
relatives distances $r_{ij}=\mid{\bf r}_{i}-{\bf r}_{j}\mid$ between
particles. Following the proposal by J.L. Lagrange [2], in our previous paper
[3] it was shown that the classical dynamics of the system at zero total
angular momentum is reduced to the dynamics of the triangle with fixed
baricenter: it is described by a reduced Hamiltonian ${\cal H}_{L=0}$ with
three degrees of freedom $r_{12},r_{13},r_{23}$, see also [4, 5]. It was
called the $r-$representation of 3-body at zero angular momentum, where the
variables $r_{ij}$ (complemented by their corresponding canonical momenta) are
used. As a realization of the formal $Z_{2}$ invariance of the Hamiltonian
${\cal H}_{0}$: $r_{ij}\rightarrow-r_{ij}$ in [3] it was proposed to introduce
another set of generalized coordinates $\rho_{ij}=r_{ij}^{2}$. It was called
the $\rho-$representation of ${\cal H}_{L=0}$. As a next step, for the
particular case of three equal masses $m_{1}=m_{2}=m_{3}\equiv m$, the lowest
order symmetric polynomial invariants

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

were proposed to use as new generalized coordinates, they were called the
geometrical variables. These coordinates encode the entire discrete symmetry
$\mathbb{Z}_{2}^{\otimes 3}\oplus{S}_{3}\oplus{S}_{3}$ of the kinetic energy
in ${\cal H}_{0}$, they are simultaneously invariant under the reflections
$r_{ij}\rightarrow-r_{ij}$, the permutations of the bodies and the permutation
of the edges of the triangle of interaction. Moreover, for potentials that
depends on first two variables only in (3), $V=V(z_{1},z_{2})$, which are the
volume variables for 3-body problem, a further reduction to two degrees of
freedom was made and the effective (reduced) Hamiltonian was constructed as
well. Note that the volume variable ${\cal S}_{2}=(4z_{2}-z_{1}^{2})/16$ is
the square of the area of the triangle of interaction whereas the volume
variable ${\cal P}_{2}=z_{1}$ is sum of its three edges squared. The main
properties of the aforementioned representations were illustrated in [3] by
several examples explicitly such as the three-body chain of harmonic
oscillators ($d>2$), the celebrated three-body Newton problem in Celestial
Mechanics and the three-body choreography in $d=2$ on the algebraic
lemniscate.

In the case of four bodies with $d$-degrees of freedom ($d>2$), the six
relative distances $r_{ij}$ are nothing but the edges of the tetrahedron of
interaction which is formed by taking the bodies positions as vertices. For
arbitrary potentials $V=V(r_{ij})$, if assuming zero total angular momentum, a
reduced Hamiltonian ${\cal H}_{\rm red}\equiv{\cal H}_{L=0}$ in the space of
relative motion has six degrees of freedom, it will be constructed in both
$r-$ and $\rho-$representations. Interestingly, it describes a six-dimensional
classical particle moving in a curved space, (see also [6] for the
corresponding quantum problem). It is worth mentioning that as early as in
1900 O. Dziobek [7] already considered the central configurations in $n$-body
Newtonian gravity in the $r-$representation. As explained before, this
representation employs elements of dimension one (edges of the polytope) as
generalized coordinates: it continues to be used, attracting the attention in
contemporary Celestial Mechanics (see Refs.[8, 9, 10, 11] and references
therein). Here, we explore alternative generalized variables of different
dimensionalities.

The goal of the present study is to construct the six (geometrical) variables
that encode the discrete symmetries of the tetrahedron of interaction, then
build three volume variables out of those and for which the kinetic energy of
the effective (reduced) free Hamiltonian is a polynomial function in the phase
space. We will consider a (sub)-family of potentials which depend on the
volume variables alone, which includes many physically important particular
cases, three of those will be studied explicitly. The novelty of the present
approach is the explicit introduction of elements of different dimensionality
of the tetrahedron of interaction as generalized coordinates. By doing so, an
interesting link between the dynamics of the system and the theory of
polytopes occurs. It realizes a symmetry reduction from $(4d)$-dimensional
coordinate space to six-dimensional space.

##  II Four-body system: case $d>2$

We consider for $d>2$ a classical system of four interacting point-like
particles with masses $m_{1},m_{2},m_{3}$ and $m_{4}$, respectively. The non-
relativistic Lagrangian is of the form,

${\cal L}\ =\ {\cal T}\ -\
V(r_{12},\,r_{13},\,r_{14},\,r_{23},\,r_{24},\,r_{34})\ ,$ (4)

where the kinetic energy is given by

${\cal T}\ =\ \frac{1}{2}m_{1}\,{\dot{{\bf r}}_{1}}^{2}\ +\
\frac{1}{2}m_{2}\,{\dot{{\bf r}}_{2}}^{2}\ +\ \frac{1}{2}m_{3}\,{\dot{{\bf
r}}_{3}}^{2}\ +\ \frac{1}{2}m_{4}\,{\dot{{\bf r}}_{4}}^{2}\ ,$ (5)

here ${\bf r}_{i}\in\mathbb{R}^{d}$ is the vector position of the $i$th body,
$\dot{{\bf r}_{i}}\equiv\frac{d}{dt}{\bf r}_{i}$ and the scalar 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
and translationally invariant. The configuration space is $4d$-dimensional
$\mathbb{R}^{d}({\bf r}_{1})\times\mathbb{R}^{d}({\bf
r}_{2})\times\mathbb{R}^{d}({\bf r}_{3})\times\mathbb{R}^{d}({\bf r}_{4})$.

Equivalently, the Hamiltonian of the system reads

${\cal H}\ =\ \frac{{{\bf p}}_{1}^{2}}{2\,m_{1}}\ +\ \frac{{{\bf
p}}_{2}^{2}}{2\,m_{2}}\ +\ \frac{{{\bf p}}_{3}^{2}}{2\,m_{3}}\ +\ \frac{{{\bf
p}}_{4}^{2}}{2\,m_{4}}\ +\
V(r_{12},\,r_{13},\,r_{14},\,r_{23},\,r_{24},\,r_{34})\ ,$ (6)

with ${{\bf p}}_{j}=\frac{\partial\,{\cal L}}{\partial\dot{{\bf r}}_{j}}$,
$j=1,2,3,4$, being the canonical momentum vectorial variables, respectively.
Hence, the phase space is $8d$-dimensional. Since the center-of-mass motion
can be separated completely, the internal relative motion of the system is
characterized by $3d$ degrees of freedom. In this space of relative motion,
let be

$\\{q_{1},\,q_{2},\,q_{3},\ldots,\,q_{3d}\\}\ ,$ (7)

any set of generalized coordinates. For instance, one possibility is to take
the three $d$-dimensional vectorial Jacobi coordinates defined by the linear
relations

$\displaystyle{\bf r}^{(J)}_{1}\ =\
\sqrt{\frac{m_{1}\,m_{2}}{m_{1}+m_{2}}}\left({\bf r}_{2}\ -\ {\bf
r}_{1}\right)$ (8) $\displaystyle{\bf r}^{(J)}_{2}\ =\
\sqrt{\frac{(m_{1}+m_{2})\,m_{3}}{m_{1}+m_{2}+m_{3}}}\left({\bf r}_{3}\ -\
\frac{m_{1}\,{\bf r}_{1}+m_{2}\,{\bf r}_{2}}{m_{1}+m_{2}}\right)$
$\displaystyle{\bf r}^{(J)}_{3}\ =\
\sqrt{\frac{(m_{1}+m_{2}+m_{3})\,m_{4}}{m_{1}+m_{2}+m_{3}+m_{4}}}\left({\bf
r}_{4}\ -\ \frac{m_{1}\,{\bf r}_{1}+m_{2}\,{\bf r}_{2}+m_{3}\,{\bf
r}_{3}}{m_{1}+m_{2}+m_{3}}\right)\ .$

In these coordinates the $3d$-dimensional kinetic energy term ${\cal T}_{\rm
rel}^{(3d)}$ of the relative motion becomes diagonal. Explicitly,

$2\,{\cal T}\ =\ {\cal T}_{\rm center-of-mass}^{(d)}\ +\ {\cal T}_{\rm
rel}^{(3d)}\ =\ {({{\bf P}}_{{}_{\bf Y}})}^{2}\ +\ {({\bf p}^{(J)}_{1})}^{2}\
+\ {({\bf p}^{(J)}_{2})}^{2}\ +\ {({\bf p}^{(J)}_{3})}^{2}\ .$ (9)

The kinetic energy of relative motion is the sum of kinetic energies in the
Jacobi coordinate directions. This property will be exploited in the study of
the 4-body harmonic chain.

For future convenience, we select the center-of-mass system as the inertial
frame, namely, ${\bf Y}=0,\,{{{\bf P}}_{{}_{\rm Y}}=M\,{\dot{\bf Y}}=0}$. Here

${\bf Y}\ \equiv\ \frac{m_{1}\,{\bf r}_{1}+m_{2}\,{\bf r}_{2}+m_{3}\,{\bf
r}_{3}+m_{4}\,{\bf r}_{4}}{M}\quad,\qquad M=m_{1}+m_{2}+m_{3}+m_{4}\ .$

Accordingly, from the original $4d-$dimensional problem we obtain a reduced
Hamiltonian, ${\cal H}\rightarrow{\cal H}_{\rm rel}$, in the space of relative
motion

${\cal H}_{\rm rel}\ =\ {({\bf p}^{(J)}_{1})}^{2}\ +\ {({\bf
p}^{(J)}_{2})}^{2}\ +\ {({\bf p}^{(J)}_{3})}^{2}\ +\ V({\bf
r}^{(J)}_{1},\,{\bf r}^{(J)}_{2},\,{\bf r}^{(J)}_{3})\ .$ (10)

##  III Reduced Hamiltonian at zero angular momentum

###  III.1 $r$-representation

Let us focus on the description of the trajectories that solely depend on the
relative distances $r_{ij}=\mid{\bf r}_{i}-{\bf r}_{j}\mid$ and possess zero
total angular momentum. The phase space associated to the relative motion is
$6d-$dimensional. Let us denote it by $\Gamma_{\rm rel}$. Now, in the space of
relative motion parametrized by Jacobi-vectorial variables we introduce a
further change of coordinates

$\\{{\bf r}^{(J)}_{1},\,{\bf r}^{(J)}_{2},\,{\bf r}^{(J)}_{3}\\}\ \Rightarrow\
\\{r_{12},\,r_{13},\,r_{14},\,r_{23},\,r_{24},\,r_{34},\,\phi_{1},\,\phi_{2},\,\ldots,\,\phi_{(3d-6)}\\}\
,$ (11)

where the collection {$\phi_{i}$}, $i=1,2,3,\ldots,(3d-6)$, stands for any set
of variables such that the transformation (11) is canonical. For the potential
$V$ does not involve the $\phi-$variables, it follows that the hypersurface

$\Gamma_{r}\ \equiv\ \Gamma_{\rm rel}|_{\
p_{{}_{\phi_{1}}}=\,p_{{}_{\phi_{2}}}=\,p_{{}_{\phi_{3}}}=\,\ldots\,=\,p_{{}_{\phi_{(3d-6)}}}=0}\quad;\,\qquad\Gamma_{r}\
\subset\ \Gamma_{\rm rel}$ (12)

is an invariant manifold[12]. In other words, any trajectory of ${\cal H}_{\rm
rel}$ (10) with initial conditions $\
p_{{}_{\phi_{1}}}=\,p_{{}_{\phi_{2}}}=\,p_{{}_{\phi_{3}}}=\,\ldots\,=\,p_{{}_{\phi_{(3d-6)}}}=0$
will remain in $\Gamma_{r}$ under time evolution. It should be noted that the
$\phi-$variables are not cyclic coordinates. In general, they will appear in
the kinetic energy term ${\cal T}_{\rm rel}^{(3d)}$ explicitly. Hence, the
associated momenta $p_{\phi}$ are not first Liouville integrals but particular
integrals of motion[13]. They are conserved quantities for specific initial
conditions only.

The trajectories possessing zero total angular momentum necessarily lie in
$\Gamma_{r}$. Therefore, by taking the restriction of ${\cal H}_{\rm rel}$ on
$\Gamma_{r}$, we form an effective (reduced) Hamiltonian (${\cal H}_{\rm
rel}\rightarrow H_{r}$)

$\displaystyle{H}_{r}$ $\displaystyle=\
\frac{1}{2}\bigg{[}\,\frac{p_{12}^{2}}{m_{12}}\ +\ \frac{p_{13}^{2}}{m_{13}}\
+\ \frac{p_{14}^{2}}{m_{14}}\ +\ \frac{p_{23}^{2}}{m_{23}}\ +\
\frac{p_{24}^{2}}{m_{24}}\ +\ \frac{p_{34}^{2}}{m_{34}}\ +\
\frac{r_{12}^{2}+r_{13}^{2}-r_{23}^{2}}{m_{1}\,r_{12}\,r_{13}}\,p_{12}\,p_{13}$
(13) $\displaystyle\qquad\ +\
\frac{r_{12}^{2}+r_{23}^{2}-r_{13}^{2}}{m_{2}\,r_{12}\,r_{23}}\,p_{12}\,p_{23}\
+\
\frac{r_{23}^{2}+r_{13}^{2}-r_{12}^{2}}{m_{3}\,r_{23}\,r_{13}}\,p_{23}\,p_{13}\
+\
\frac{r_{24}^{2}+r_{14}^{2}-r_{12}^{2}}{m_{4}\,r_{24}\,r_{14}}\,p_{24}\,p_{14}$
$\displaystyle\qquad\ +\
\frac{r_{12}^{2}+r_{14}^{2}-r_{24}^{2}}{m_{1}\,r_{12}\,r_{14}}\,p_{12}\,p_{14}\
+\
\frac{r_{12}^{2}+r_{24}^{2}-r_{14}^{2}}{m_{2}\,r_{12}\,r_{24}}\,p_{12}\,p_{24}\
+\
\frac{r_{13}^{2}+r_{34}^{2}-r_{14}^{2}}{m_{3}\,r_{13}\,r_{34}}\,p_{13}\,p_{34}$
$\displaystyle\qquad\ +\
\frac{r_{14}^{2}+r_{34}^{2}-r_{13}^{2}}{m_{4}\,r_{14}\,r_{34}}\,p_{14}\,p_{34}\
+\
\frac{r_{13}^{2}+r_{14}^{2}-r_{34}^{2}}{m_{1}\,r_{13}\,r_{14}}\,p_{13}\,p_{14}\
+\
\frac{r_{23}^{2}+r_{24}^{2}-r_{34}^{2}}{m_{2}\,r_{23}\,r_{24}}\,p_{23}\,p_{24}$
$\displaystyle\qquad\ +\
\frac{r_{23}^{2}+r_{34}^{2}-r_{24}^{2}}{m_{3}\,r_{23}\,r_{34}}\,p_{23}\,p_{34}\
+\
\frac{r_{24}^{2}+r_{34}^{2}-r_{23}^{2}}{m_{4}\,r_{24}\,r_{34}}\,p_{24}\,p_{34}\,\bigg{]}\
+\ V\ ,$

here $p_{ij}$ is the canonical momentum associated with $r_{ij}$. The
expression (13) we call the $r-$representation of the reduced Hamiltonian. The
kinetic energy term (equivalently, the free Hamiltonian) is a polynomial
function of the momentum variables with rational $r-$dependent coefficients.

Figure 1: Tetrahedron of interaction in $d=3$: the individual coordinate
vectors ${\bf r}_{i}$ mark positions of vertices of the tetrahedron with edges
$r_{ij}$. Accordingly, its four faces are triangles. The center-of-mass is
marked by a (blue) bubble.

####  III.1.1 The metric $g^{\mu\nu}(r)$

In (13), we identify the components of the associated cometric $g^{\mu\nu}(r)$
with the coefficients in front of the quadratic terms in momentum variables
$(p_{12},\,p_{31},\,p_{23})$. Explicitly,

$g^{\mu\nu}(r)=\left(\begin{array}[]{cccccc}\frac{m_{1}+m_{2}}{2m_{1}m_{2}}&\frac{r_{12}^{2}+r_{13}^{2}-r_{23}^{2}}{4m_{1}r_{12}r_{13}}&\frac{r_{12}^{2}+r_{14}^{2}-r_{24}^{2}}{4m_{1}r_{12}r_{14}}&\frac{r_{12}^{2}-r_{13}^{2}+r_{23}^{2}}{4m_{2}r_{12}r_{23}}&\frac{r_{12}^{2}-r_{14}^{2}+r_{24}^{2}}{4m_{2}r_{12}r_{24}}&0\\\
\frac{r_{12}^{2}+r_{13}^{2}-r_{23}^{2}}{4m_{1}r_{12}r_{13}}&\frac{m_{1}+m_{3}}{2m_{1}m_{3}}&\frac{r_{13}^{2}+r_{14}^{2}-r_{34}^{2}}{4m_{1}r_{13}r_{14}}&\frac{-r_{12}^{2}+r_{13}^{2}+r_{23}^{2}}{4m_{3}r_{13}r_{23}}&0&\frac{r_{13}^{2}-r_{14}^{2}+r_{34}^{2}}{4m_{3}r_{13}r_{34}}\\\
\frac{r_{12}^{2}+r_{14}^{2}-r_{24}^{2}}{4m_{1}r_{12}r_{14}}&\frac{r_{13}^{2}+r_{14}^{2}-r_{34}^{2}}{4m_{1}r_{13}r_{14}}&\frac{m_{1}+m_{4}}{2m_{1}m_{4}}&0&\frac{-r_{12}^{2}+r_{14}^{2}+r_{24}^{2}}{4m_{4}r_{14}r_{24}}&\frac{-r_{13}^{2}+r_{14}^{2}+r_{34}^{2}}{4m_{4}r_{14}r_{34}}\\\
\frac{r_{12}^{2}-r_{13}^{2}+r_{23}^{2}}{4m_{2}r_{12}r_{23}}&\frac{-r_{12}^{2}+r_{13}^{2}+r_{23}^{2}}{4m_{3}r_{13}r_{23}}&0&\frac{m_{2}+m_{3}}{2m_{2}m_{3}}&\frac{r_{23}^{2}+r_{24}^{2}-r_{34}^{2}}{4m_{2}r_{23}r_{24}}&\frac{r_{23}^{2}-r_{24}^{2}+r_{34}^{2}}{4m_{3}r_{23}r_{34}}\\\
\frac{r_{12}^{2}-r_{14}^{2}+r_{24}^{2}}{4m_{2}r_{12}r_{24}}&0&\frac{-r_{12}^{2}+r_{14}^{2}+r_{24}^{2}}{4m_{4}r_{14}r_{24}}&\frac{r_{23}^{2}+r_{24}^{2}-r_{34}^{2}}{4m_{2}r_{23}r_{24}}&\frac{m_{2}+m_{4}}{2m_{2}m_{4}}&\frac{-r_{23}^{2}+r_{24}^{2}+r_{34}^{2}}{4m_{4}r_{24}r_{34}}\\\
0&\frac{r_{13}^{2}-r_{14}^{2}+r_{34}^{2}}{4m_{3}r_{13}r_{34}}&\frac{-r_{13}^{2}+r_{14}^{2}+r_{34}^{2}}{4m_{4}r_{14}r_{34}}&\frac{r_{23}^{2}-r_{24}^{2}+r_{34}^{2}}{4m_{3}r_{23}r_{34}}&\frac{-r_{23}^{2}+r_{24}^{2}+r_{34}^{2}}{4m_{4}r_{24}r_{34}}&\frac{m_{3}+m_{4}}{2m_{3}m_{4}}\\\
\end{array}\right)\ .$ (14)

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

$\mathcal{D}_{m}\ =\ \frac{9\,{(m_{1}+m_{2}+m_{3}+m_{4})}^{2}}{64\
m_{1}^{3}\,m_{2}^{3}\,m_{3}^{3}\,{m_{4}^{3}}\
r_{12}^{2}\,r_{13}^{2}\,r_{14}^{2}\,r_{23}^{2}\,r_{24}^{2}\,r_{34}^{2}}\,{\cal
V}\,(\,{\cal S}\,{\cal P}\ -\ 144\,m_{1}\,m_{2}\,m_{3}\,m_{4}\,{\cal V}\,)\ ,$
(15)

where

$\displaystyle{\cal V}$ $\displaystyle=\
\frac{1}{288}\bigg{[}\,-2r_{34}^{2}r_{12}^{4}-2r_{34}^{4}r_{12}^{2}-2r_{13}^{2}r_{23}^{2}r_{12}^{2}+2r_{14}^{2}r_{23}^{2}r_{12}^{2}+2r_{13}^{2}r_{24}^{2}r_{12}^{2}-2r_{14}^{2}r_{24}^{2}r_{12}^{2}\
+\ $ (16) $\displaystyle
2r_{13}^{2}r_{34}^{2}r_{12}^{2}+2r_{14}^{2}r_{34}^{2}r_{12}^{2}+2r_{23}^{2}r_{34}^{2}r_{12}^{2}+2r_{24}^{2}r_{34}^{2}r_{12}^{2}-2r_{14}^{2}r_{23}^{4}-2r_{13}^{2}r_{24}^{4}-2r_{14}^{4}r_{23}^{2}\
+\ $ $\displaystyle
2r_{13}^{2}r_{14}^{2}r_{23}^{2}-2r_{13}^{4}r_{24}^{2}+2r_{13}^{2}r_{14}^{2}r_{24}^{2}+2r_{13}^{2}r_{23}^{2}r_{24}^{2}+2r_{14}^{2}r_{23}^{2}r_{24}^{2}-2r_{13}^{2}r_{14}^{2}r_{34}^{2}+2r_{14}^{2}r_{23}^{2}r_{34}^{2}$
$\displaystyle\ +\ 2r_{13}^{2}r_{24}^{2}r_{34}^{2}\ -\
2r_{23}^{2}r_{24}^{2}r_{34}^{2}\,\bigg{]}\ ,$

$\displaystyle{\cal S}\ =$ $\displaystyle
m_{1}\,m_{2}\,m_{3}\,\big{(}2r_{13}^{2}r_{12}^{2}+2r_{23}^{2}r_{12}^{2}+2r_{13}^{2}r_{23}^{2}-r_{13}^{4}-r_{23}^{4}-r_{12}^{4}\big{)}\
+\ $ (17) $\displaystyle
m_{1}\,m_{2}\,m_{4}\,\big{(}2r_{14}^{2}r_{12}^{2}+2r_{24}^{2}r_{12}^{2}+2r_{14}^{2}r_{24}^{2}-r_{12}^{4}-r_{14}^{4}-r_{24}^{4}\big{)}\
+\ $ $\displaystyle
m_{1}\,m_{3}\,m_{4}\,\big{(}2r_{14}^{2}r_{13}^{2}+2r_{34}^{2}r_{13}^{2}+2r_{14}^{2}r_{34}^{2}-r_{14}^{4}-r_{34}^{4}-r_{13}^{4}\big{)}\
+\ $ $\displaystyle
m_{2}\,m_{3}\,m_{4}\,\big{(}2r_{24}^{2}r_{23}^{2}+2r_{34}^{2}r_{23}^{2}+2r_{24}^{2}r_{34}^{2}-r_{24}^{4}-r_{34}^{4}-r_{23}^{4}\big{)}\
,$
${\cal P}\ =\
\frac{m_{1}m_{2}r_{12}^{2}+m_{1}m_{3}r_{13}^{2}+m_{1}m_{4}r_{14}^{2}+m_{2}m_{3}r_{23}^{2}+m_{2}m_{4}r_{24}^{2}+m_{3}m_{4}r_{34}^{2}}{m_{1}+m_{2}+m_{3}+m_{4}}\
,$ (18)

are called volume variables. The variable $\cal V$ is the square of the volume
of the tetrahedron of interaction, $\cal S$ is the weighted sum of the 4 areas
(squared) of the faces (triangles) of the tetrahedron whilst $\cal P$ is the
weighted sum of the 6 edges (squared) of the tetrahedron. In fact, $\cal P$
coincides with the moment of inertia of the system.

Remark. The above determinant $\mathcal{D}_{m}$ (15) is a rational function in
$r$-variables. It effectively depends on the four coordinates $\cal V$, $\cal
S$, $\cal P$ and $T\equiv
r_{12}^{2}\,r_{13}^{2}\,r_{14}^{2}\,r_{23}^{2}\,r_{24}^{2}\,r_{35}^{2}$. Note
that it vanishes, $\mathcal{D}_{m}=0$, when i) the volume of the tetrahedron
of interaction is equal to zero, ii) the tetrahedron is regular, or iii) a
triple-body collision occurs. It is singular at the two-body collision point.

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

The cometric (14) is invariant formally under reflections
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$,

$r_{ij}\,\rightarrow\,-r_{ij}\qquad\ i,j=1,2,3,4\quad(i\neq j)\ .$

If we introduce new variables,

$\rho_{ij}\ =\ r_{ij}^{2}\ ,$ (19)

with the corresponding canonical momenta

$P_{ij}\,=\,\frac{1}{2\,r_{ij}}\,p_{ij}\ ,$ (20)

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

$\displaystyle{H}_{\rho}$ $\displaystyle=$ (21) $\displaystyle\
2\,\bigg{[}\,\frac{\rho_{12}\,P_{12}^{2}}{m_{12}}\ +\
\frac{\rho_{13}\,P_{13}^{2}}{m_{13}}\ +\ \frac{\rho_{14}\,P_{14}^{2}}{m_{14}}\
+\ \frac{\rho_{23}\,P_{23}^{2}}{m_{23}}\ +\
\frac{\rho_{24}\,P_{24}^{2}}{m_{24}}\ +\ \frac{\rho_{34}\,P_{34}^{2}}{m_{34}}\
+\ \frac{\rho_{12}+\rho_{13}-\rho_{23}}{m_{1}}\,P_{12}\,P_{13}$
$\displaystyle\qquad\ +\
\frac{\rho_{12}+\rho_{23}-\rho_{13}}{m_{2}}\,P_{12}\,P_{23}\ +\
\frac{\rho_{23}+\rho_{13}-\rho_{12}}{m_{3}}\,P_{23}\,P_{13}\ +\
\frac{\rho_{24}+\rho_{14}-\rho_{12}}{m_{4}}\,P_{24}\,P_{14}$
$\displaystyle\qquad\ +\
\frac{\rho_{12}+\rho_{14}-\rho_{24}}{m_{1}}\,P_{12}\,P_{14}\ +\
\frac{\rho_{12}+\rho_{24}-\rho_{14}}{m_{2}}\,P_{12}\,P_{24}\ +\
\frac{\rho_{13}+\rho_{34}-\rho_{14}}{m_{3}}\,P_{13}\,P_{34}$
$\displaystyle\qquad\ +\
\frac{\rho_{14}+\rho_{34}-\rho_{13}}{m_{4}}\,P_{14}\,P_{34}\ +\
\frac{\rho_{13}+\rho_{14}-\rho_{34}}{m_{1}}\,P_{13}\,P_{14}\ +\
\frac{\rho_{23}+\rho_{24}-\rho_{34}}{m_{2}}\,P_{23}\,P_{24}$
$\displaystyle\qquad\ +\
\frac{\rho_{23}+\rho_{34}-\rho_{24}}{m_{3}}\,P_{23}\,P_{34}\ +\
\frac{\rho_{24}+\rho_{34}-\rho_{23}}{m_{4}}\,P_{24}\,P_{34}\,\bigg{]}\ +\ V\
.$

The Hamiltonian (21) is written in what we call the $\rho-$representation (cf.
(13)). Its kinetic energy is a linear polynomial function of the
$\rho-$coordinates. Also, the cross terms in momentum variables associated
with non-adjacent edges of the polytope of interaction are absent. In the free
system ($V=0$), the Hamilton’s equations for the momentum variables $P_{ij}$
do not depend on the $\rho-$variables at all.

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

From (21) we obtain the associated cometric

$g^{\mu\nu}(\rho)\
=\left(\begin{array}[]{cccccc}\frac{2\left(m_{1}+m_{2}\right)\rho_{12}}{m_{1}m_{2}}&\frac{\rho_{12}+\rho_{13}-\rho_{23}}{m_{1}}&\frac{\rho_{12}+\rho_{14}-\rho_{24}}{m_{1}}&\frac{\rho_{12}-\rho_{13}+\rho_{23}}{m_{2}}&\frac{\rho_{12}-\rho_{14}+\rho_{24}}{m_{2}}&0\\\
\frac{\rho_{12}+\rho_{13}-\rho_{23}}{m_{1}}&\frac{2\left(m_{1}+m_{3}\right)\rho_{13}}{m_{1}m_{3}}&\frac{\rho_{13}+\rho_{14}-\rho_{34}}{m_{1}}&\frac{-\rho_{12}+\rho_{13}+\rho_{23}}{m_{3}}&0&\frac{\rho_{13}-\rho_{14}+\rho_{34}}{m_{3}}\\\
\frac{\rho_{12}+\rho_{14}-\rho_{24}}{m_{1}}&\frac{\rho_{13}+\rho_{14}-\rho_{34}}{m_{1}}&\frac{2\left(m_{1}+m_{4}\right)\rho_{14}}{m_{1}m_{4}}&0&\frac{-\rho_{12}+\rho_{14}+\rho_{24}}{m_{4}}&\frac{-\rho_{13}+\rho_{14}+\rho_{34}}{m_{4}}\\\
\frac{\rho_{12}-\rho_{13}+\rho_{23}}{m_{2}}&\frac{-\rho_{12}+\rho_{13}+\rho_{23}}{m_{3}}&0&\frac{2\left(m_{2}+m_{3}\right)\rho_{23}}{m_{2}m_{3}}&\frac{\rho_{23}+\rho_{24}-\rho_{34}}{m_{2}}&\frac{\rho_{23}-\rho_{24}+\rho_{34}}{m_{3}}\\\
\frac{\rho_{12}-\rho_{14}+\rho_{24}}{m_{2}}&0&\frac{-\rho_{12}+\rho_{14}+\rho_{24}}{m_{4}}&\frac{\rho_{23}+\rho_{24}-\rho_{34}}{m_{2}}&\frac{2\left(m_{2}+m_{4}\right)\rho_{24}}{m_{2}m_{4}}&\frac{-\rho_{23}+\rho_{24}+\rho_{34}}{m_{4}}\\\
0&\frac{\rho_{13}-\rho_{14}+\rho_{34}}{m_{3}}&\frac{-\rho_{13}+\rho_{14}+\rho_{34}}{m_{4}}&\frac{\rho_{23}-\rho_{24}+\rho_{34}}{m_{3}}&\frac{-\rho_{23}+\rho_{24}+\rho_{34}}{m_{4}}&\frac{2\left(m_{3}+m_{4}\right)\rho_{34}}{m_{3}m_{4}}\\\
\end{array}\right)\ ,$ (22)

which is linear in $\rho$-coordinates, cf.(14). Its determinant reads

$D_{m}\ =\
\frac{576\,{(m_{1}+m_{2}+m_{3}+m_{4})}^{2}}{m_{1}^{3}\,m_{2}^{3}\,m_{3}^{3}\,{m_{4}^{3}}}\,{\cal
V}\,(\,{\cal S}\,{\cal P}\ -\ 144\,m_{1}\,m_{2}\,m_{3}\,m_{4}\,{\cal V}\,)\ .$

Hence, $D_{m}$ is a polynomial function of the $\rho$-variables and it depends
on three coordinates alone, namely the volume variables $\cal V$, $\cal S$ and
$\cal P$. Explicitly, the volume variables $\cal S$ and $\cal P$ in the
$\rho-$representation read as follows:

$\displaystyle{\cal S}\ =$ $\displaystyle
m_{1}\,m_{2}\,m_{3}\,\big{(}2\rho_{13}\,\rho_{12}+2\rho_{23}\,\rho_{12}+2\rho_{13}\,\rho_{23}-\rho_{13}^{2}-\rho_{23}^{2}-\rho_{12}^{2}\big{)}\
+\ $ (23) $\displaystyle
m_{1}\,m_{2}\,m_{4}\,\big{(}2\rho_{14}\,\rho_{12}+2\rho_{24}\,\rho_{12}+2\rho_{14}\,\rho_{24}-\rho_{12}^{2}-\rho_{14}^{2}-\rho_{24}^{2}\big{)}\
+\ $ $\displaystyle
m_{1}\,m_{3}\,m_{4}\,\big{(}2\rho_{14}\,\rho_{13}+2\rho_{34}\,\rho_{13}+2\rho_{14}\,\rho_{34}-\rho_{14}^{2}-\rho_{34}^{2}-\rho_{13}^{2}\big{)}\
+\ $ $\displaystyle
m_{2}\,m_{3}\,m_{4}\,\big{(}2\rho_{24}\,\rho_{23}+2\rho_{34}\,\rho_{23}+2\rho_{24}\,\rho_{34}-\rho_{24}^{2}-\rho_{34}^{2}-\rho_{23}^{2}\big{)}\
,$ ${\cal P}\ =\
\frac{m_{1}m_{2}\,\rho_{12}+m_{1}m_{3}\,\rho_{13}+m_{1}m_{4}\,\rho_{14}+m_{2}m_{3}\,\rho_{23}+m_{2}m_{4}\,\rho_{24}+m_{3}m_{4}\,\rho_{34}}{m_{1}+m_{2}+m_{3}+m_{4}}\
,$ (24)

see (17)-(18), and similarly for $\cal V$.

The six dimensional $\rho$-space with cometric (22) is not flat. The classical
Hamiltonian ${H}_{\rho}$ (21) is in complete agreement with the result
obtained through the procedure of _dequantization_ from the corresponding
quantum Hamiltonian operator.

###  III.3 Four-body closed chain of interactive harmonic oscillators

As an application of the $\rho-$representation, we consider the case of a
4-body oscillator system. The defining potential $V^{(\text{har})}$ is given
by

$V^{(\text{har})}\ =\ 2\,\omega^{2}\bigg{[}\nu_{12}\,\rho_{12}\ +\
\nu_{13}\,\rho_{13}\ +\ \nu_{14}\,\rho_{14}\ +\ \nu_{23}\,\rho_{23}\ +\
\nu_{24}\,\rho_{24}\ +\ \nu_{34}\,\rho_{34}\bigg{]}\ ,$ (25)

being $\omega>0$ the frequency and
$\nu_{12},\nu_{13},\nu_{14},\nu_{23},\nu_{24},\nu_{34}\geq 0$ determine the
spring constants. In general, the Hamiltonian (21) with $V^{(\text{har})}$
does not admit separation of variables in $\rho-$coordinates. However, unlike
the $r$-representation, the potential is linear in $\rho-$variables. Upon a
gauge rotation of the Hamiltonian, the corresponding $S-$states eigenfunctions
are simply polynomial functions in $\rho-$variables. Also, for the simplest
case of equal masses and equal spring constants the above potential
$V^{(\text{har})}$ becomes a function of the volume variable $\cal P$ alone.

Since the kinetic energy in the original Hamiltonian (6) decouples in Jacobi
vectorial coordinates (8), it is natural to look for separation of variables
in such coordinates. A direct computation shows that, in general, the
potential (25) is a non-diagonal quadratic expression in the Jacobi vectorial
coordinates (8). Nevertheless, the Principal Axes Theorem guarantees that it
is always diagonalizable in certain variables. These variables are linear
combinations of (8) and the coefficients can be described by a rotation matrix
(Euler angles) that preserves the kinetic energy term in diagonal form. The
corresponding expressions are not particularly illuminating (rather
cumbersome) and will not be displayed.

Remark. For arbitrary masses and frequencies, the 4-body oscillator system in
$\mathbb{R}^{d}$ ($d>2$) can be transformed to a system of three
$d-$dimensional decoupled isotropic harmonic oscillators where the
frequencies, in general, are not commensurable. It implies that the original
Hamiltonian (6) with potential $V^{(\text{har})}$ is an integrable system in
full generality (even for non-zero values of angular momentum). In the case of
3-bodies with pairwise harmonic interactions at $d=3$, the integrability was
demonstrated in Ref.[14] explicitly.

It can be shown that in the special case of the three constraints imposed

$m_{1}\,\nu_{34}\ =\ m_{3}\,\nu_{14}\quad,\qquad m_{2}\,\nu_{34}\ =\
m_{3}\,\nu_{24}\quad,\qquad m_{2}\,\nu_{13}\ =\ m_{1}\,\nu_{23}\ ,$ (26)

the original Hamiltonian (6) with potential $V^{(\text{har})}$ reduces, in the
center-of-mass reference frame, to the 3-body Jacobi oscillator system

${\cal H}_{J}\ \equiv\frac{1}{2}\bigg{[}{({\bf p}^{(J)}_{1})}^{2}\ +\ {({\bf
p}^{(J)}_{2})}^{2}\ +\ {({\bf p}^{(J)}_{3})}^{2}\,\bigg{]}\ +\ V^{(J)}\ ,$
(27) $V^{(J)}\ =\ A_{1}\,{({r}^{(J)}_{1})}^{2}\ +\
A_{2}\,{({r}^{(J)}_{2})}^{2}\ +\ A_{3}\,{({r}^{(J)}_{3})}^{2}\ ,$

${r}^{(J)}_{i}=\mid{\bf r}^{(J)}_{i}\mid$, $i=1,2,3$, see (8); here we make
the identifications

$\displaystyle A_{1}\ =\
\frac{2\,\omega^{2}\,\left(m_{2}\,\nu_{12}+m_{1}\,\left(\nu_{12}+\nu_{23}+\nu_{24}\right)\
\right)}{m_{1}\,m_{2}}\ ,\quad A_{2}\ =\
\frac{2\,\omega^{2}\,\left(m_{1}\,\nu_{23}+m_{2}\,\nu_{23}+m_{3}\,\left(\nu_{23}+\nu_{24}\right)\
\right)}{m_{2}\,m_{3}}$ (28) $\displaystyle\qquad A_{3}\ =\
\frac{2\,\nu_{24}\,\omega^{2}\,\left(m_{1}+m_{2}+m_{3}+m_{4}\right)}{m_{2}\,m_{4}}\
.$

The symmetries of ${\cal H}_{J}$, which describes formally three
$d-$dimensional isotropic oscillators (equivalently, an integrable anisotropic
harmonic oscillator in the $3d-$dimensional space), were analyzed in [15, 16]
(and references therein). Now, if the additional constraint

$m_{4}\,\nu_{23}\ =\ m_{3}\,\nu_{24}\ ,$

is imposed in (27), then two frequencies coincide $A_{2}=A_{3}$. In this case,
the 4-body Jacobi oscillator system as well as $V^{(\text{har})}$ becomes
minimally superintegrable. Finally, the extra constraint

$m_{3}\,\nu_{12}\ =\ m_{1}\,\nu_{23}\ ,$

renders the system maximally superintegrable, all three angular frequencies
coincide $A_{1}=A_{2}=A_{3}$. It may be mentioned that the quantum counterpart
of the model was studied in detail in [17]. Here, the relation of
$V^{(\text{har})}$ to the 4-body Jacobi oscillator system clarifies and shows
(in a simpler manner) the (super)integrability properties of the system. Such
an identification was not realized in Ref.[17] being unknown in that time.

###  III.4 Generalized chain of interactive 4-body harmonic oscillators

It is worth mentioning a generalization of the previous 4-body oscillator
system (25). The corresponding potential $V^{(\text{har})}$, in the
$\rho-$representation, reads

$\begin{split}V_{R}^{(\text{har})}\ =\
&2\,\omega^{2}\bigg{[}\nu_{12}\,{(\sqrt{\rho_{12}}-R_{12})}^{2}\ +\
\nu_{13}\,{(\sqrt{\rho_{13}}-R_{13})}^{2}\ +\
\nu_{14}\,{(\sqrt{\rho_{14}}-R_{14})}^{2}\\\ &+\
\nu_{23}\,{(\sqrt{\rho_{23}}-R_{23})}^{2}\ +\
\nu_{24}\,{(\sqrt{\rho_{24}}-R_{24})}^{2}\ +\
\nu_{34}\,{(\sqrt{\rho_{34}}-R_{34})}^{2}\bigg{]}\ ,\end{split}$ (29)

where $R_{ij}>0$ play the role of rest lengths. We call it generalized
harmonic oscillator system or 4-body harmonic molecule. Especially, the
tetrahedron of interaction with sides $R_{ij}$ defines equilibrium
configuration. In this case, the Hamiltonian (21) with potential
$V_{R}^{(\text{har})}$ does not admit a separation of variables neither in
$\rho-$coordinates nor in other coordinate system. In the simpler 3-body case
on the plane $d=2$, identical particles and equal spring and rest lengths
constants, the corresponding generalized harmonic system exhibits a rich
dynamics as a function of the values of parameters and the energy. For
instance, a power-law statistics, that fits the Levi-walk model [18], develops
[19, 20]. Recently, an experimental realization of this 3-body system was
assembled using analog electrical elements [21].

By substituting $\rho_{12}={(R+A_{12}\,e^{i\,\Omega\,t})}^{2}$,
$\rho_{13}={(R+A_{13}\,e^{i\,\Omega\,t})}^{2}$,
$\rho_{23}={(R+A_{23}\,e^{i\,\Omega\,t})}^{2}$ into the Newton’s equations of
motion for the Hamiltonian (21) with potential $V_{R}^{(\text{har})}$ (29) and
requiring all the linear terms in the constants $A_{ij}$ to vanish, it can be
shown that in the regime of small oscillations ($A_{ij}\ll 1$), the symmetric
4-body harmonic oscillator system with equal masses $m_{i}=1$, equal spring
constants $\nu_{ij}=1$ and equal rest lengths $R_{ij}=R>0$ ($i\neq j=1,2,3,4$)
possesses 6 normal modes with frequencies $\Omega=\sqrt{8}\,\omega$ (symmetric
stretch), $\Omega=\sqrt{\frac{8}{5}}\,\omega$ (triple degenerate) and
$\Omega=\sqrt{\frac{8}{7}}\,\omega$ (doubly degenerate), respectively. The
symmetric stretch represents an exact periodic trajectory of the system beyond
small oscillations.

##  IV Volume variables representation ($d>2$)

In this Section we consider the special family of potentials $V$ that can be
written in terms of the volume variables ${\cal V}$, $\cal S$ and $\cal P$
defined in (16), (17), (18), respectively. Namely,

$V\ =\ V({\cal V},\,{\cal S},\,{\cal P})\ .$

Let us recall that variable $\cal V$ is the square of the volume of the
tetrahedron of interaction, $\cal S$ is the weighted sum of the 4 areas
(squared) of the faces (triangles) of the tetrahedron whilst $\cal P$ is the
weighted sum of the 6 edges (squared) of the tetrahedron. Clearly, $\cal P$ is
an homogeneous polynomial of degree 1 in $\rho$-variables whereas $\cal S$ and
$\cal V$ are polynomials of degree 2 and 3, respectively.

Remark. Each of these three volume variables is $S_{4}$ invariant under the
permutation of any pair of the four particles, volume variables remain
unchanged under the permutation $({\bf r}_{i},\,m_{i})\Leftrightarrow({\bf
r}_{j},\,m_{j})$. However, even in the simplest case of equal masses
$m_{1}=m_{2}=m_{3}=m_{4}$, the variables $\cal V$ and $\cal S$ are not $S_{6}$
invariant under the permutation of any pair of the six $\rho-$coordinates.

Explicitly, in the reduced 12-dimensional phase space $\Gamma_{r}$ of relative
motion, we perform the following change of variables

$\\{r_{12},\,r_{13},\,r_{14},\,r_{23},\,r_{24},\,r_{34}\\}\ \Rightarrow\
\\{{\cal V},\,{\cal S},\,{\cal
P},\,\varphi_{1},\,\varphi_{2},\,\varphi_{3}\\}\ ,$ (30)

here the generalized coordinates $\varphi_{i}$ ($i=1,2,3$) denote any set of
variables such that the transformation (30) is canonical. Again, if the
potential $V$ does not depend on the $\varphi-$variables, it follows that the
hypersurface

$\Gamma_{\rm vol}\ \equiv\ \Gamma_{r}|_{\
p_{{}_{\varphi_{1}}}=\,p_{{}_{\varphi_{2}}}=\,p_{{}_{\varphi_{3}}}=0}\quad;\,\qquad\Gamma_{\rm
vol}\ \subset\ \Gamma_{r}\ \subset\ \Gamma_{\rm rel}$ (31)

becomes an invariant manifold[12].

Thus, by taking the restriction of ${H}_{r}$ on $\Gamma_{\rm vol}$ we obtain a
further effective Hamiltonian (${H}_{r}\rightarrow{\cal H}_{\rm vol}$).

As a result of standard computation the Hamiltonian that governs the
trajectories that solely depend on the three volume coordinates is of the form

$\displaystyle{\cal H}_{\rm vol}\ =$ $\displaystyle\frac{1}{72\,m}{\cal
V}\,{\cal S}\,P^{2}_{{\cal V}}\ +\ \big{(}3456\,m\,M\,{\cal V}\,+\,8\,M\,{\cal
S}\,{\cal P}\big{)}\,P^{2}_{{\cal S}}\ +\ 2\,{\cal P}\,P^{2}_{{\cal P}}$ (32)
$\displaystyle+\ 32\,M\,{\cal V}\,{\cal P}\,P_{\cal V}\,\,P_{\cal S}\ +\
12\,{\cal V}\,P_{\cal V}\,\,P_{\cal P}\ +\ 8\,{\cal S}\,P_{\cal S}\,P_{\cal
P}\ +\ V({\cal V},\,{\cal S},\,{\cal P})\ ,$

here $m\equiv m_{1}\,m_{2}\,m_{3}\,m_{4}$, $M\equiv m_{1}+m_{2}+m_{3}+m_{4}$
and $P_{\cal V}$, $P_{\cal S}$, $P_{\cal P}$ are the canonical momentum
variables conjugate to coordinates ${\cal V,S,P}$, respectively. The
Hamiltonian (32) describes a three-dimensional particle moving in a curved
space. Tensor of inertia can be identified with cometric in this case as well.
The associated cometric for (32) is given by

$g^{\mu\nu}({\cal V},\,{\cal S},\,{\cal P})\
=\left|\begin{array}[]{ccc}\frac{1}{72\,m}{\cal V}\,{\cal S}&\ 16\,M\,{\cal
V}\,{\cal P}&\ 6\,{\cal V}\\\ &&\\\ 16\,M\,{\cal V}\,{\cal P}&\
3456\,m\,M\,{\cal V}\,+\,8\,M\,{\cal S}\,{\cal P}&\ 4\,{\cal S}\\\ &&\\\
6\,{\cal V}&\ 4\,{\cal S}&\ 2\,{\cal P}\end{array}\right|\ .$ (33)

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

$\text{Det}\big{[}\,g^{\mu\nu}({\cal V},\,{\cal S},\,{\cal P})\,\big{]}\ =\
-\frac{2{\cal V}\left(559872m^{2}M{\cal V}^{2}+2304mM^{2}{\cal P}^{3}{\cal
V}-2592mM{\cal P}{\cal S}{\cal V}-M{\cal P}^{2}{\cal S}^{2}+{\cal
S}^{3}\right)}{9\,m}\ .$ (34)

This determinant vanishes at

${\cal V}\ =\ 0\ ,$

i.e. when the four particles move on a plane or are collinear. Moreover, the
second factor in (34) also vanishes for certain configurations with non-zero
$\cal V$. In particular, if all four masses are equal then $\cal V$ vanishes
for the configuration formed by an equilateral triangle base and three equal
isosceles triangle sides as well as for a regular tetrahedron. The space with
cometric (33) is not flat.

###  IV.1 Equations of motion

From the Hamiltonian (32), we obtain the following Hamilton’s equations of
motion:

$\displaystyle\dot{P}_{\cal P}\ =\ -2\,(\,P_{\cal P}^{2}\ +\ 4\,M\,P_{\cal
S}(\,P_{\cal S}\,S\ +\ 4\,P_{\cal V}\,{\cal V})\,)\ -\ \partial_{\cal P}V$
(35) $\displaystyle\dot{P}_{\cal S}\ =\ -8\,P_{\cal P}\,P_{\cal S}\ -\
8\,M\,{\cal P}\,P_{\cal S}^{2}\ -\ \frac{{\cal V}\,P_{\cal V}^{2}}{72\,m}\ -\
\partial_{\cal S}V$ $\displaystyle\dot{P}_{\cal V}\ =\ -3456\,m\,M\,P_{\cal
S}^{2}\ -\ 4\,(\,3\,P_{\cal P}\ +\ 8\,M\,{\cal P}\,P_{\cal S}\,)\,P_{\cal V}\
-\ \frac{{\cal S}\,P_{\cal V}^{2}}{72\,m}\ -\ \partial_{\cal V}V$
$\displaystyle\dot{\cal P}\ =\ 4\,(\,{\cal P}\,P_{\cal P}\ +\ 2\,{\cal
S}\,P_{\cal S}\ +\ 3\,{\cal V}\,P_{\cal V}\,)$ $\displaystyle\dot{\cal S}\ =\
8\,(\,{\cal S}\,P_{\cal P}\ +\ 2\,M\,({\cal P}\,{\cal S}\,P_{\cal S}\ +\
432\,m\,{\cal V}\,P_{\cal S})\ +\ 2\,{\cal P}\,{\cal V}\,P_{\cal V}\,)$
$\displaystyle\dot{\cal V}\ =\ 12\,{\cal V}\,P_{\cal P}\ +\ 32\,M\,{\cal
P}\,{\cal V}\,P_{\cal S}\ +\ \frac{{\cal S}\,{\cal V}\,P_{\cal V}}{36\,m}$

###  IV.2 Two-variable potentials $V=V({\cal S},\,{\cal P})$: planar case
$d=2$

For the 2-variable potentials of the form

$V\ =\ V({\cal S},\,{\cal P})\ ,$ (36)

the above equations of motion (35) admit particular special solutions with
${\cal V}=0$ where the problem becomes effectively two-dimensional (2 degrees
of freedom). The corresponding reduced Hamiltonian is given by

$\displaystyle{h}_{\rm vol}\ =$ $\displaystyle 8\,M\,{\cal S}\,{\cal
P}\,P^{2}_{{\cal S}}\ +\ 2\,{\cal P}\,P^{2}_{{\cal P}}\ +\ 8\,{\cal
S}\,P_{\cal S}\,P_{\cal P}\ +\ V({\cal S},\,{\cal P})\ .$ (37)

The condition ${\cal V}=0$ implies that the volume of the tetrahedron of
interaction vanishes. On the plane $d=2$, this is the case necessarily. We
call (37) the (${\cal P},\,{\cal S}$)-representation.

####  IV.2.1 The potential $V=\frac{{U[P/\sqrt{S}]}}{\sqrt{S}}$

Now, within the (${\cal P},\,{\cal S}$)-representation we consider the
physically relevant case of

$\dot{\cal P}\ =\ 0\ ,$

namely, a system with constant moment of inertia.

Specifically, we consider an arbitrary potential $V=V({\cal P},{\cal S})$ on
the submanifold of phase space for which ${h}_{\rm vol}=0$ (zero energy level)
and $\dot{{\cal P}}=0$ (a constant moment of inertia). In this case, from (35)
it follows that the potential must obey the following linear partial
differential equation

$2\,{\cal S}\,\partial_{\cal S}\,V\ +\ {\cal P}\,\partial_{\cal P}\,V\ =\ -V\
.$ (38)

The solution of (38) is given by

$V\ =\ \frac{{U[{\cal P}/\sqrt{{\cal S}}]}}{\sqrt{{\cal S}}}\ ,$ (39)

where $U[z]$ is an arbitrary function of the argument $z$.

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

${\cal P}\,{\dot{{\cal S}}}^{2}\ -\ 32\,U\,(M\,{\cal P}^{2}\ -\ {\cal
S})\,\sqrt{{\cal S}}\ =\ 0\ ,$ (40)

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

${\cal S}(t)\ =\ M\,{\cal P}^{2}\ +\
\frac{1}{4}\,\gamma\,\left(\,32\,t^{2}\,-\,8\,\sqrt{2}\,c_{1}\,t\,+\,c_{1}^{2}\right)\
>\ 0\ ,$ (41)

$c_{1}$ is a constant of integration. Notice that the potential
$V\propto\frac{1}{{\cal P}}$ leads to the solutions ${\cal S}(t)=M\,{\cal
P}^{2}$ (for four equal unit masses $M=4$, hence, a regular tetrahedron) and
${\cal S}(t)=0$ (four particles on a line).

As a function of the two lowest order volume variables, those that are linear
and quadratic expressions in $\rho-$variables, respectively, the same
potential $V=-\gamma\frac{{\cal P}}{{\cal S}}$ occurs in the three-body case.
It was originally studied in Ref.[22] where the author builds the hyperbolic
plane with its geodesic flow as the scale plus symmetry reduction of a 3-body
problem in the Euclidean plane.

####  IV.2.2 The metric $g_{\rm vol}^{\mu\nu}$

Now, the metric (or, equivalently, the tensor of inertia) for (37) is of the
form

$g_{\rm vol}^{\mu\nu}\ =\left|\begin{array}[]{cc}2\,{{\cal P}}&\ 4\,{\cal
S}\\\ 4\,{\cal S}&8\,M\,{\cal P}\,{\cal S}\end{array}\right|\ ,$ (42)

with determinant

$D_{\rm vol}\ =\ 16\,{\cal S}\,(\,M\,{{\cal P}}^{2}\ -\ {\cal S}\,)\ .$

In the case of a _regular tetrahedron_ with four equal unit masses the
relation $4\,{{\cal P}}^{2}\ =\ {\cal S}$ holds and this determinant vanishes,
${D}_{\rm vol}=0$. It also follows from (42) that the corresponding Ricci
scalar $\rm{Rs}$ is given by

${\rm Rs}_{\rm vol}\ =\ \frac{32\,M\,{\cal P}\,{\cal S}\,(\,4\,M\,{\cal S}\ -\
1\,)}{{D_{\rm vol}}^{2}}\ .$ (43)

###  IV.3 One-variable potentials: $V=V({\cal P})$

An interesting particular case occurs when the potential (36) does not depend
on $\cal S$ and is a function of the volume variable $\cal P$ alone

$V\ =\ V({\cal P})\ .$

It follows from (37) that there exist trajectories which depend on $\cal P$
only. Such trajectories lie on the intersection between the hypersurfaces
${\cal V}=0$ and $P_{\cal S}=0$, they are described by the Hamiltonian

${h}_{\rm vol}\mid_{{}_{\small P_{\cal S}=0}}\ \equiv\ {\cal H}_{\cal P}\ =\
2\,{\cal P}\,P_{\cal P}^{2}\ +\ V({\cal P})\ ,$ (44)

The connection between the trajectories of ${\cal H}_{\cal P}$ (44) with those
of ${h}_{\rm vol}$ (37) is the following. Taking a non-trivial solution ${\cal
P}={\cal P}(t)\neq 0$ of (44) one can define the function ${\cal S}={\cal
S}(t)$ for which the canonical momentum $P_{\cal S}$ vanishes in ${h}_{\rm
vol}$. Explicitly, the condition can be obtained from (37),

$P_{\cal S}\ =\ \frac{2\,{\dot{P}}\,{\cal S}\,-\,P\,{\dot{\cal S}}}{16\,{\cal
S}\,({\cal S}\,-\,M\,{\cal P}^{2})}\ =\ 0\ .$ (45)

Such ${\cal P}={\cal P}(t)$ and ${\cal S}={\cal S}(t)=\lambda\,{\cal
P}^{2}(t)$, with $\lambda\neq M$ a non-negative parameter, satisfy the
equations of motion for ${h}_{\rm vol}$. Moreover, they also obey the
equations (35) for ${\cal H}_{\rm vol}$ with ${\cal V}=0$, hence, the system
degenerates into a planar system. At $\lambda=0$, thus ${\cal S}=0$, the
problem further degenerates to the line.

####  IV.3.1 Anharmonic Oscillator potential

As a concrete example, let us consider the following one-variable potential

$\displaystyle V^{(AO)}$ $\displaystyle\ =\ A\,{\cal P}\ +\ B\,{\cal P}^{2}\
=$ (46) $\displaystyle
A\,\frac{m_{1}m_{2}\,\rho_{12}+m_{1}m_{3}\,\rho_{13}+m_{1}m_{4}\,\rho_{14}+m_{2}m_{3}\,\rho_{23}+m_{2}m_{4}\,\rho_{24}+m_{3}m_{4}\,\rho_{34}}{M}$
$\displaystyle\
+B\,{\bigg{[}\frac{m_{1}m_{2}\,\rho_{12}+m_{1}m_{3}\,\rho_{13}+m_{1}m_{4}\,\rho_{14}+m_{2}m_{3}\,\rho_{23}+m_{2}m_{4}\,\rho_{24}+m_{3}m_{4}\,\rho_{34}}{M}\bigg{]}}^{2}\
,$

where (i) $B\geq 0$ and (ii) if $B=0$ the parameter $A>0$. In terms of
variable $P$ the potential (46) corresponds to a shifted one-dimensional
harmonic oscillator on the half line $\mathbb{R}_{+}$ with non-standard
kinetic energy $2\,{{\cal P}}\,P_{{\cal P}}^{2}$, see below. In the
$\rho-$representation it corresponds to a certain six-dimensional quadratic
potential, defined on a certain subspace of $\mathbb{R}^{6}_{+}$, whilst in
the $r-$representation it describes a six-dimensional anisotropic harmonic
oscillator with the quartic anharmonicity.

For the potential (46), the Hamiltonian (44) becomes

$\displaystyle{\cal H}^{(AO)}_{\cal P}$ $\displaystyle=\ 2\,{{\cal
P}}\,P_{{\cal P}}^{2}\ +\ A\,{\cal P}\ +\ B\,{\cal P}^{2}\ .$ (47)

The time-evolution of the system is given by

${\cal P}(t)\ =\ x^{2}\ =\
\frac{A\,k^{2}}{B\,(1-k^{2})}\,\text{sn}^{2}(y,ik)\quad,\qquad
y\,=\,\pm\sqrt{\frac{2\,A}{1-k^{2}}}\ t\ ,$ (48)

($B\neq 0$, $k\neq\pm 1$) 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 (48) 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 (47) we obtain the harmonic oscillator potential
$V=A\,{\cal P}$, which is quadratic polynomial in the $r$-representation. In
this case the trajectories are trigonometric vibrations, ${\cal
P}(t)=c_{1}\,\cos^{2}(\sqrt{9\,A/2}\,\,t+c_{2})$ with energy ${\cal
H}^{(AO)}_{\cal P}\,=\,\ A\,c_{1}$ where $c_{1}>0,c_{2}$ are real constants of
integration. In general, due to the energy conservation ${\cal H}^{(AO)}_{\cal
P}=E$ the trajectories in the phase space are always cubic curves,

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

Note that at fixed values of the energy $E$ and $A$ the presence of
anharmonicity ($B>0$) in (47) tends to decrease the amplitude of the harmonic
motion $(B=0)$. Also, the harmonic potential $V=A\,{\cal P}$ corresponds to
the aforementioned 4-body chain of harmonic oscillators. From the definition
of $\cal P$, it follows that it can be written as a 4-body maximally
superintegrable Jacobi oscillator as well.

##  V u-variables representation ($d>1$)

In this Section we consider 4 identical particles,
$m_{1}=m_{2}=m_{3}=m_{4}=1$, and focus on the three-variable family of
potentials $V$ of the form,

$V\ =\ V(u_{1},\,u_{2},\,u_{3})\ ,$

where

$u_{1}\ =\ \rho_{12}\ +\ \rho_{34}\quad,\qquad u_{2}\ =\ \rho_{13}\ +\
\rho_{24}\quad,\qquad u_{3}\ =\ \rho_{14}\ +\ \rho_{23}\ ,$ (49)

are linear functions of the $\rho-$variables and each of them corresponds to
the sum of two disconnected edges squared of the tetrahedron of interaction.
Clearly, they also are $S_{4}-$permutationally invariant under the
permutations of any pair of bodies. The sum of all three $u-$variables is
given by the volume variable ${\cal P}=u_{1}+u_{2}+u_{3}$.

The Hamiltonian that governs the trajectories that solely depend on the three
$u-$coordinates reads

$\displaystyle{\cal H}_{\rm u}\ =$ $\displaystyle 4\,(\,u_{1}\,P_{u_{1}}\ +\
u_{2}\,P_{u_{2}}\ +\ u_{3}\,P_{u_{3}}\,)\ +\
4\,(u_{1}\,+\,u_{2}\,-\,u_{3})\,P_{u_{1}}\,P_{u_{2}}$ (50) $\displaystyle\ +\
4\,(u_{1}\,+\,u_{3}\,-\,u_{2})\,P_{u_{1}}\,P_{u_{3}}\ +\
4\,(u_{2}\,+\,u_{3}\,-\,u_{1})\,P_{u_{2}}\,P_{u_{3}}\ +\
V(u_{1},\,u_{2},\,u_{2})\ ,$

here $P_{u_{2}}$, $P_{u_{2}}$, and $P_{u_{3}}$ are the corresponding canonical
momentum variables. The Hamiltonian (50) describes a three-dimensional
particle moving in a curved space. Tensor of inertia can be identified with
cometric in this case as well. Unlike the volume variables, for the planar
system $d=2$ all three $u-$variables are, in general, always different from
zero.

##  VI Conclusions

In the present study the four-body classical system of four arbitrary masses
in Euclidean space $\mathbb{R}^{4d}$ as the coordinate space is considered. We
assume the generic potentials $V=V(r_{ij})$, not necessarily pairwise
interactions, depending on the 6 relative mutual distances $r_{ij}$ between
bodies are considered.

For trajectories of the total angular momentum equal to zero the original
$4d-$dimensional problem is reduced to a six-dimensional problem defined in
the space of relative (internal) radial motion. It realizes a symmetry
reduction from $(4d)$-dimensional coordinate space to six-dimensional space of
relative distances. The corresponding reduced Hamiltonian is constructed
explicitly using two representations: (I) the $r-$representation where the
generalized coordinates are the six relative distances $r_{ij}$ and where the
kinetic energy term (free Hamiltonian) is a second order polynomial in the
associated canonical momenta $p_{ij}$ with rational coefficients in the
variables $r_{ij}$, and (II) the $\rho-$representation where the free
Hamiltonian is a polynomial function of both the $\rho-$variables,
$\rho_{ij}=r_{ij}^{2}$, and their canonical momenta $P_{ij}$. The cometric
appearing in the corresponding Hamiltonians $H_{r}$ (13) and $H_{\rho}$ (21),
respectively, describes a six-dimensional particle moving in a curved space.

As a particular example, the 4-body harmonic oscillator system was studied
within the $\rho-$representation in detail. Moreover, in Jacobi vectorial
variables due to the separation of variables, the original Hamiltonian can be
reduced to a 3-body Jacobi oscillator integrable system, where the potential
is the sum of squares of Jacobi vectorial coordinates. We emphasize that this
property is exclusive of the Jacobi variables and is valid even for a non-zero
value of the angular momentum. Furthermore, for specific relations between the
masses and the frequencies occurring in the harmonic oscillator potential, the
system becomes superintegrable. A non-integrable generalization (4-body
harmonic molecule) of this model was briefly analyzed in the regime of small
oscillations.

At $d>2$ the positions of the 4 bodies form a Platonic solid, the tetrahedron.
This geometric object naturally defines the volume variables ${\cal V}$ (16),
$\cal S$ (17) and $\cal P$ (18). In fact, the variable $\cal P$ coincides with
the moment of inertia of the system. Moreover, in the case of four equal
masses $m_{1}=m_{2}=m_{3}=m_{4}$ each of the volume variables is
$\mathcal{S}_{4}$-invariant under the interchange of any pair of the particles
(vertices) in the tetrahedron.

Assuming $d>2$ and the potential depends on three volume variables $V=V({\cal
V},\,{\cal S},\,{\cal P})$, the class of trajectories that depend only on the
volume variables is governed by a six-dimensional Hamiltonian ${\cal H}_{\rm
vol}$ (32) in the phase space. We call it _volume-representation of the
Hamiltonian._ It describes a three-dimensional particle moving in a space
which is non-flat. For $d=2$, where ${\cal V}=0$, we indicate a generic two-
variable potential $V=V({\cal S},\,{\cal P})$, for which the moment of inertia
${\cal P}$ is an integral of motion. Also, a further reduction can be
performed in the case $V=V({\cal P})$. Here, the system admits a family of
special ${\cal P}-$dependable trajectories, it described by a certain two-
dimensional reduced Hamiltonian in the phase space which is derived.

For the case of identical particles with equal spring constants, the special
$u-$representation was developed, if the potential depends only on the
$u-$variables (each of them being the sum of two disconnected edges squared of
the tetrahedron of interaction). Each of these three coordinates is
$\mathcal{S}_{4}$-invariant under the permutation of any pair of the particles
(vertices) in the tetrahedron. They are evidently related with the volume
variable ${\cal P}=u_{1}+u_{2}+u_{3}$.

The presented 4-body studies at $n=4$ together with previous 3-body ones at
$n=3$, reported in [3], lead us to a certain conjectures for the general
$n$-body case in $\mathbb{R}^{d}$ at $d>n-2$:

  * •

The $n-$body harmonic oscillator potential

$V^{(\rm har)}_{n}\ =\ \omega^{2}\,\sum_{i>j}^{n}\nu_{ij}\,r_{ij}^{2}\ =\
\omega^{2}\,\sum_{i>j}^{n}\nu_{ij}\,\rho_{ij}\quad,\qquad\nu_{ij}>0\ ,$

is integrable, it admits a complete separation of variables when they are
taken as linear combinations of the Jacobi vectorial coordinates. The original
Hamiltonian, in the center-of-mass reference frame, corresponds to
$(n-1)-$variables Jacobi oscillator system. By imposing the conditions on the
masses and string constants $\frac{m_{i}}{m_{j}}=\frac{\nu_{ik}}{\nu_{jk}}$
($i\neq j\neq k$), converts the system into maximally superintegrable. In this
case, the harmonic potential has the form $V^{(\rm har)}_{n}\propto{\cal
P}_{n}$, hence, it is proportional to the moment of inertia of the system

${\cal P}_{n}\ =\ \frac{\sum_{i>j}m_{i}\,m_{j}\,r_{ij}^{2}}{\sum_{i}m_{i}}\ .$

  * •

At fixed $n$, the polytope of interaction (formed by taking bodies positions
as vertices) defines $(n-1)$ volume variables. There exist trajectories, that
depend solely on these volume variables, the corresponding (reduced)
Hamiltonian is a polynomial function.

  * •

For $d=2$, the potential of the form

$V({\cal P}_{n},\,{\cal S}_{n})\ =\ \frac{U[{\cal P}_{n}/{\cal
S}_{n}]}{\sqrt{{\cal S}_{n}}}\ ,$

where ${\cal S}_{n}$ is the weighted sum of the squares of the area of the
faces (triangles) of the degenerate polytope of interaction and ${\cal P}_{n}$
is the sum squared of lengths of its edges, and $U[z]$ is an arbitrary
function of $z$, possesses an infinite family of planar trajectories for which
the variable ${\cal P}_{n}$ is an integral of motion.

  * •

At any dimension $d$ the potential $V=A\,{\cal P}_{n}\,+\,B\,{\cal P}_{n}^{2}$
(a type of the quartic anharmonic potential in $r-$variables), admits
particular solutions where

${\cal P}_{n}(t)\ \propto\
\frac{A\,k^{2}}{B\,(1-k^{2})}\,\text{sn}^{2}(y,ik)\quad,\qquad
y\,=\,\pm\sqrt{\frac{A}{1-k^{2}}}\ t\ ,$ (51)

($A,B\neq 0$, $k\neq\pm 1$) is given by a Jacobi elliptic function with
imaginary elliptic modulus $(ik)$.

At $d\leq n-2$, as for example 3 bodies on a line or 4 bodies on the plane,
the polytope of interaction is degenerate. Thus, not all the
$\frac{n(n-1)}{2}$ $r$-variables are independent and the polynomiality of the
$L=0$ reduced Hamiltonian in $\rho-$representation is absent. In this case,
the number of non-vanishing volume-variables is equal to $d$.

##  VII Author’s contributions

All authors contributed equally to this work.

##  VIII Acknowledgments

We thank Willard Miller for useful discussions in the early stages of this
study. This work was supported partially by CONACyT grant A1-S-17364 and DGAPA
grant IN113022 (Mexico). A.V.T. thanks PASPA-UNAM for a support during his
sabbatical stay at University of Miami, where this work was completed.

##  IX DATA AVAILABILITY

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

## References

  * [1] Landau L D, Lifshitz, 1976 Mechanics Vol. 1 $\S$41 Pergamon Press
  * [2] Lagrange J L, 1772 Essai sur le problème des trois corps, Oeuvres, Vol. 6.
  * [3] Escobar-Ruiz A. M., Linares R., Turbiner A. V. and Miller W. Jr., 2021 Classical $n$-body system in geometrical and volume variables: I. Three-body case
Int. J. Mod. Phys. A 36 2150140

  * [4] Murnaghan F D, 1936 A Symmetric Reduction of the Planar Three-Body Problem
Am. J. Math. 58 829

  * [5] Van Kampen E R and Wintner Aurel, 1937 On a Symmetrical Canonical Reduction of the Problem of Three Bodies
Am. J. Math. 59 153

  * [6] Escobar-Ruiz M. A., Miller W. Jr., and Turbiner A. V., 2019 Four-body problem in $d$-dimensional space: Ground state, (quasi)-exact-solvability. IV
J. Math. Phys. 60 062101

  * [7] Dziobek O., 1900 Uber einen merkwürdigen fall des vielkörperproblems
Astron. Nachr. 152 33–46

  * [8] Santoprete M, 2018 Four-body central configurations with one pair of opposite sides parallel
J. Math. Anal. Appl. 464 421–434

  * [9] Cors J. M. and Roberts G. E., 2012 Four-body co-circular central configurations
Nonlinearity 25 343

  * [10] Albouy A and Chenciner A, 1998 Le problème des n corps et les distances mutuelles
Inv. Math. 131 151-184

  * [11] Saari D G, 1980 On the role and the properties of central configurations in the $n$-body problem
Celestial Mechanics 21 9-20

  * [12] Wiggins S, 1990 Invariant Manifolds: Linear and Nonlinear Systems $\S$1.1C in Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, pp. 14-25
  * [13] Turbiner A V, 2013 Particular Integrability and (Quasi)-exact-solvability
J. Phys. A: Math. Theor. 46 025203

  * [14] Soares de Castro A. and Sugaya M., 1993 Exact solution for a three-dimensional three-body problem with harmonic interactions
European Journal of Physics 14 259-261

  * [15] Rodríguez M. A., Tempesta P. and Winternitz P., 2008 Reduction of superintegrable systems: The anisotropic harmonic oscillator
Phys. Rev. E 78 046608

  * [16] Jauch J. M. and Hill E. L., 1940 On the Problem of Degeneracy in Quantum Mechanics
Phys. Rev. 57 641

  * [17] Escobar-Ruiz A. M., Miller Willard Jr., and Turbiner A. V., 2021 Four-body (an)harmonic oscillator in $d-$dimensional space: S-states, (quasi)-exactsolvability, hidden algebra $sl(7)$
J. Math. Phys. 62 072101

  * [18] Zaburdaev, V. and Denisov, S. and Klafter, J., 2015 Lévy walks
Rev. Mod. Phys. 87 483

  * [19] Saporta Katz, Ori and Efrati, Efi, 2019 Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System
Phys. Rev. Lett. 122 024102

  * [20] Katz O. and Efrati E., 2020 Regular regimes of the harmonic three-mass system
Phys. Rev. E 101 032211

  * [21] Escobar-Ruiz, A. M. and Quiroz-Juarez, M. A. and Del Rio-Correa, J. L. and Aquino, N., 2022 Classical harmonic three-body system: An experimental electronic realization
Scientific Reports 12, 13346

  * [22] Montgomery R., 2017 Constructing the Hyperbolic Plane as the reduction of a three-body problem
Reg. and Chaotic Dynamical Systems 6, v. 22


# Propagation Measurements and Path Loss Models for sub-THz in Urban
Microcells

Yunchou Xing and Theodore S. Rappaport This research is supported by the NYU
WIRELESS Industrial Affiliates Program and National Science Foundation (NSF)
Research Grants: 1909206 and 2037845. NYU WIRELESS, NYU Tandon School of
Engineering, Brooklyn, NY, 11201, {ychou<EMAIL_ADDRESS>
###### Abstract

Terahertz frequency bands will likely be used for the next-generation wireless
communication systems to provide data rates of hundreds of Gbps or even Tbps
because of the wide swaths of unused and unexplored spectrum. This paper
presents two outdoor wideband measurement campaigns in downtown Brooklyn
(urban microcell environment) in the sub-THz band of 140 GHz with TX-RX
separation distance up to 117.4 m: i) terrestrial urban microcell measurement
campaign, and ii) rooftop surrogate satellite and backhaul measurement
campaign. Outdoor omnidirectional and directional path loss models for both
line-of-sight and non-line-of-sight scenarios, as well as foliage loss (signal
attenuation through foliage), are provided at 140 GHz for urban microcell
environments. These measurements and models provide an understanding of both
the outdoor terrestrial (e.g., 6G cellular and backhaul) and non-terrestrial
(e.g., satellite and unmanned aerial vehicle communications) wireless
channels, and prove the feasibility of using THz frequency bands for outdoor
fixed and mobile cellular communications. This paper can be used for future
outdoor wireless system design at frequencies above 100 GHz.

###### Index Terms:

THz; 6G; path loss; foliage loss; D band; air-to-ground; non-terrestrial
network; satellite; backhaul; UMi; channel model; sub-THz; 142 GHz; 73 GHz; 28
GHz.

##  I Introduction

One of the most prominent advancements in the fifth generation (5G) of mobile
communications, over prior generations (e.g., 4G LTE), is the use of much
wider bandwidth at millimeter wave (mmWave, 30-300 GHz) in comparison to the
limited spectrum available at sub-6 GHz frequencies [1, 2, 3]. The vast
bandwidth enables multi-Gbps data rate operations at mobile devices and
various new applications like wireless cognition and centimeter-level
positioning [1, 4, 5]. Both handset terminals and base stations will use
highly directional antenna arrays, resulting in huge differences in antenna
beamforming and adaptation to wireless channels at mmWave compared to sub-6
GHz frequencies (e.g., narrower antenna beamwidth, higher penetration loss,
stronger reflections but much lossier diffractions) [1, 6, 3]. Extensive
research has been conducted at frequencies below 100 GHz and several channel
models have been developed by standards and different research organizations
such as 3GPP, 5GCM, NYUWIRELESS, METIS, and mmMAGIC, which has helped to
facilitate the deployment of 5G networks [7, 8, 9, 10, 11, 12].

Terahertz (THz) frequency bands (e.g., frequencies from 100 GHz to 3 THz) are
promising bands for the next generation of wireless communications (6G).
However, there are notable challenges seen for frequencies above 100 GHz
(e.g., high phase noise and Doppler shift, limited output power, and more
directional beams), which makes communications in THz bands more challenging
[1, 13]. Propagation measurements at THz frequency bands are imperative to
provide knowledge and understanding of the wireless channels above 100 GHz.

Most of the measurements at THz bands are focused on short-range indoor
scenarios [14, 15, 16, 17], which are limited by the dynamic range and the
cable connected between the transmitter (TX) and receiver (RX) of the vector
network analyzer (VNA) based system. Work in [18, 19] presented an indoor
wideband measurement campaign at 142 GHz looking at large indoor radio
propagation distances up to 40 m. There are very few outdoor measurement
campaigns at frequencies above 100 GHz [20, 21] and they primarily focus on
line-of-sight (LOS) propagation using either reflected materials [20] or a RF-
over-fiber extension [21] of a VNA based system.

This paper presents two outdoor measurement campaigns at 142 GHz in an urban
microcell (UMi) environment including both LOS and non-LOS (NLOS) scenarios in
downtown Brooklyn, NY, using a wideband sliding correlation-based channel
sounder system [22, 23]. The measurement system, procedures, and directional
and omnidirectional path losses and models of the terrestrial UMi measurements
are described in Section II. Section III presents the rooftop surrogate
satellite and backhaul measurements, as well as the foliage loss at 142 GHz,
and shows the isolation between surrogate passive satellite sensors and
terrestrial terminals, or between mobile links and terrestrial backhaul links
at frequencies above 100 GHz (little interference in the same or adjacent
bands). Section IV provides concluding remarks, showing that THz frequencies
can provide outdoor coverage up to 117.4 m in an UMi environment for both LOS
and NLOS scenarios with handset-type transmit power (1 mW and 39 dB processing
gain [22, 23]) for future mobile communications.

##  II Outdoor 142 GHz Terrestrial Urban Microcell Measurement Campaign

Propagation in the THz bands is difficult due to the severe path loss in the
first meter of propagation from the transmitting antenna and large penetration
losses caused by obstructions, which lead to fewer multipath components and
clusters in an indoor office compared to lower frequencies (e.g., 28 GHz) [13,
18, 19]. In Fall 2020, outdoor wideband measurements at 142 GHz were conducted
in New York University’s (NYU) downtown Brooklyn campus, which is a multipath-
rich urban environment [22]. Two measurement campaigns: i) terrestrial UMi
measurement campaign, and ii) rooftop surrogate satellite and backhaul
measurement campaign were conducted. These measurements can provide valuable
knowledge about the outdoor wireless channels in UMi scenarios and non-
terrestrial networks at frequencies above 100 GHz, which may be used for
future 6G and beyond communications [1, 24].

###  II-A Wideband 142 GHz Measurement system

Outdoor wireless channels were measured by transmitting and receiving an 11th
order pseudorandom m-sequence centered at 142 GHz with a broadband 1 GHz null-
to-null RF bandwidth. The baseband 500 MHz spread spectrum signal was first
mixed with an intermediate frequency (IF) of 7 GHz, and was then upconverted
with a driving local oscillator (LO) frequency of 135 GHz, resulting in the RF
center frequency of 142 GHz. Both the TX and RX used identical mechanically
steerable 27 dBi gain horn antennas with 8° half power beamwidths (HPBW) in
both the azimuth and elevation planes. The TX transmit power was 0 dBm applied
by the Virginia Diodes, Inc (VDI) upconverter (27 dBm effective isotropic
radiated power, EIRP), and the maximum measurable path loss range of the
channel sounder system was 152 dB, operating at a 5 dB SNR threshold [22, 25].
A detailed description of the 4th generation channel sounder system at 142 GHz
can be found in [22, 23].

###  II-B Terrestrial UMi Measurement Locations and Procedures

The terrestrial UMi measurement campaign was designed to study the wireless
channels and propagation characteristics in the sub-THz bands for 6G cellular
communications in urban microcell and small-cell scenarios. Fig. 1 shows a map
of six TX locations and 17 RX locations (with some RX locations reused for
more than one TX locations, such as RX1) around NYU’s downtown Brooklyn
campus. TX1 serves nine RX locations, where five of them are LOS (RX1, RX5, RX
23, RX 27, and RX31) and four of them are NLOS (RX9, RX14, RX16, and RX18).
TX2 serves four RX locations, where three of them are LOS (RX1, RX35, and
RX36) and one of them is NLOS (RX14). TX3 serves four RX locations, where
three of them are LOS (RX35, RX36, and RX37) and the other one is NLOS (RX1).
TX4 serves four RX locations, where two of them are LOS (RX3 and RX37) and the
other two are NLOS (RX1 and RX38). TX5 serves four RX locations, where two of
them are LOS (RX3 and RX35) and the other two are NLOS (RX1 and RX10). TX6
serves three RX locations, where one of them is LOS (RX1) and two of them are
NLOS (RX39 and RX40). In total, there are 16 LOS TX-RX location combinations
and 12 NLOS TX-RX location combinations with TX-RX separation distances up to
117.4 m. All the 16 LOS locations and 11 out of 12 NLOS locations could
successfully receive a signal and measure a power delay profile (PDP) through
the channel.

During the measurements, TXs were set at heights of 4 m above the ground
(similar height as lampposts) to emulate small-cell base stations (BS), and
the RXs were set at heights of 1.5 m above the ground to emulate mobile user
receivers. In the surrounding environment, there are metal lampposts, concrete
building walls, paved roads, trees, pedestrians, bare soil ground, concrete
pillars, glass windows, vehicles, and glass doors.

At the beginning and the end of each measurement day, routine calibrations
[26] were conducted to make sure all the measurements were valid and accurate
[25, 23, 27]. For each TX-RX combination, two elevation angles were used at
the BS TX (the TX best pointing elevation angle that the maximum power is
received at the RX, and TX antenna downtilt 8° from the best TX pointing
elevation angle) and three elevation angles were used at each RX (the best RX
pointing elevation angle, and RX antenna uptilt and downtilt 8° from the best
RX pointing elevation angle). For each TX and RX elevation angle combination,
the antennas at both the TX and RX were exhaustively rotated by 8° HPBW in the
azimuth plane (e.g., 45 rotations to cover the 360° plane in azimuth) to
capture all the possible multipaths in any azimuth directions. A PDP was
recorded for each and every unique TX and RX pointing angle, and
omnidirectional PDPs were synthesized as introduced in [28]. Each measured PDP
consisted of an average of 20 consecutive instantaneous PDPs to reduce the
noise floor, and the averaging factor can be increased at the expense of
longer recording time [29].

Figure 1: Terrestrial urban microcell measurement campaign in NYU’s downtown
Brooklyn campus. Six TX locations are identified as stars with different
colors and the corresponding RX locations are identified as the same color
circles.

###  II-C 142 GHz Path Loss Models for Terrestrial UMi Measurements

The 1 m close-in (CI) free space reference distance path loss model (1) [30,
31, 25] is one of the most commonly used large-scale path loss models to
predict the signal strength over distance for various frequencies [30, 31, 7]:

$\small\begin{split}PL^{CI}(f_{c},d_{\text{3D}})\;\text{[dB]}&=\text{FSPL}(f_{c},1m)+10n\log_{10}\left(\dfrac{d_{3D}}{d_{0}}\right)+\chi_{\sigma},\\\
\text{FSPL}(f_{c},1m)&=32.4+20\log_{10}(\dfrac{f_{c}}{1\;\text{GHz}}),\end{split}$
(1)

where FSPL$(f_{c},1\;\text{m})$ is the large-scale free space path loss at
carrier frequency $f_{c}$ in GHz at 1 m, $n$ is the path loss exponent (PLE),
and $\chi_{\sigma}$ is the large-scale fading in dB (a zero mean Gaussian
random variable with a standard deviation $\sigma$ in dB) [25, 30, 31].

Figure 2: Urban UMi 142 GHz directional path loss scatter plot and outdoor
directional CI ($d_{0}$= 1 m) path loss model for both LOS and NLOS scenarios
using 27 dBi gain and 8° HPBW horn antennas at both the TX and RX (without
antenna gains included for path loss calculations). Each green circle
represents the LOS path loss at a LOS location, red crosses represent NLOS
path losses measured at arbitrary antenna pointing angles between the TX and
RX for NLOS scenarios, and each blue diamond represents the best antenna
pointing angles of both the TX and RX to receive the maximum power at the RX
for each NLOS TX-RX location combination.

Fig. 2 shows the directional path loss scatter plot and the best-fit
directional CI path loss model (1) at 142 GHz for both LOS and NLOS scenarios,
with antenna gains removed. Each data point in the plot represents one of the
directional PDPs for each TX-RX pointing combination. The best-fit minimum
mean square error (MMSE) LOS PLE is $n=$ 2.1 with a shadow fading standard
deviation of $\sigma=$ 2.8 dB at 142 GHz, showing that the LOS directional
channel is only very slightly lossier than the free space propagation ($n=$
2.0), this likely due to foliage attenuation [32] or antenna misalignment
between the TX and RX. The LOS directional PLE $n=$ 2.1 at 142 GHz is very
close to the LOS directional PLEs of $n=$ 1.9 and 2.0 with shadow fading
standard deviations of $\sigma=$ 1.1 dB and 1.9 dB in the identical urban
microcell environment at 28 and 73 GHz [33, 25, 34], respectively. The
slightly larger LOS PLEs at higher frequencies are likely due to the narrower
beam antennas (HPBW= 29°, 15°, and 8° antennas at 28, 73, and 142 GHz,
respectively), since the wider beam antenna will capture more multipath
components (e.g., reflections) within the LOS boresight beam and the narrower
beam antennas are harder to perfectly align.

The best-fit MMSE $NLOS_{\text{Best}}$ directional PLEs (when the TX and RX
antennas are pointing in the direction where the maximum power is received at
the RX) and $NLOS_{\text{Arbitrary}}$ directional PLEs are $n=$ 3.1 and 3.6,
with shadow fading standard deviations of $\sigma=$ 8.3 dB and 9.1 dB,
respectively. The smaller PLE of $NLOS_{\text{Best}}$ directional case
compared to the PLE of $NLOS_{\text{Arbitrary}}$ directional situations
indicates that there is usually a dominant path (usually a strong reflection)
that contains more power than the other multipaths in NLOS directional
scenarios. The $NLOS_{\text{Best}}$ and $NLOS_{\text{Arbitrary}}$ directional
PLEs $n$ are (3.5 and 4.1) at 28 GHz and (3.1 and 4.6) at 73 GHz [33, 25, 34].
The smaller $NLOS_{\text{Best}}$ PLE $n=$ 3.1 at both 73 and 142 GHz compared
to the $NLOS_{\text{Best}}$ PLE $n=$ 3.5 at 28 GHz indicates that reflected
paths are stronger at higher frequencies, which gives the surprising, non-
intuitive result that there is less path loss at higher frequencies when using
the best NLOS beam directions. This was also observed in an indoor office [13,
19].

Due to the limitation of the measurable dynamic range of the 142 GHz channel
sounder system (152 dB) [22], the signals with path loss exceeding 152 dB were
not detectable. Therefore, the 142 GHz PLE of NLOS arbitrary ($n=$ 3.60) may
not be accurate, as it is conditioned upon reception of a signal (see Fig. 2).

During the measurements, we observed that metal lampposts, concrete building
walls, and tinted glass were good reflectors at 142 GHz (providing only 2-8 dB
loss to the reflected multipath), which could provide first-order and second-
order reflections. However, the partition loss of the outdoor building
materials (e.g., penetration loss) at 142 GHz was very large such that
concrete walls and tinted windows usually provided 30-50 dB penetration loss.

Omnidirectional antenna pattern and omnidirectional received power were
synthesized by summing the received powers from every measured non-overlapping
directional HPBW antenna pointing angle combination, as described in [28].
Fig. 3 shows the best-fit omnidirectional CI path loss model and the scatter
plot of synthesized omnidirectional measured path loss (without antenna gains)
at 142 GHz at NYU courtyard in downtown Brooklyn, NY.

Figure 3: Urban UMi best-fit omnidirectional CI path loss model (without
antenna gains) at 142 GHz for both LOS and NLOS situations. The blue diamonds
represent the measured omnidirectional path loss at 142 GHz in the NLOS
environment and the green circles, conversely, represent the LOS situation.

The LOS omnidirectional PLE at 142 GHz is $n=$ 1.9 with a shadow fading
standard deviation $\sigma=$ 2.7 dB, which is slightly lower than the LOS
directional PLE of $n=$ 2.1, showing that the omnidirectional antennas would
capture slightly more power from all the directions than using boresight-
aligned directional antennas, and the boresight LOS multipath component is
clearly dominant compared to all other multipaths (e.g., reflected or
scattered) in the LOS scenarios. At lower frequencies, the LOS omnidirectional
PLEs are nearly identical with $n=$ 2.1 and 1.9 with shadow fading standard
deviations of $\sigma=$ 3.6 dB and 1.7 dB at 28 and 73 GHz [33, 25, 34],
respectively.

The 142 GHz NLOS omnidirectional PLE is $n=$ 2.9 with a shadow fading standard
deviation $\sigma=$ 8.2 dB which is smaller than but close to the
$\text{NLOS}_{\text{Best}}$ directional PLE of $n=$ 3.1 at 142 GHz, indicating
that the $NLOS_{Best}$ propagation path is generally the single dominant
propagation path among all the multipaths in NLOS scenarios. Thus, accurate
beamforming algorithms will be needed to find, capture, and combine the most
dominant multipath energy to maintain and extend the outdoor NLOS
communication range at frequencies above 100 GHz [34, 6]. The NLOS
omnidirectional PLEs are larger with more variability at lower frequencies,
such that $n=$ 3.4 and 2.8 with shadow fading standard deviations of $\sigma=$
9.7 dB and 8.7 dB at 28 and 73 GHz, respectively. Future data processing will
determine the relative strength of the strongest few multipath components in
THz NLOS channels. Comparing path loss models between different frequencies,
it is clear that reflections are stronger at 73 and 142 GHz compared to at 28
GHz, and THz channels of 142 GHz are very similar to 73 GHz except for the
path loss in the first meter of propagation when energy spreads into the far
field [19].

Figure 4: Omnidirectional PDP of TX1-RX23 which is in the LOS scenario with a
separation distance of 43.34 m. There are 10 multipath components in four time
clusters (TC), where the first multipath component is the LOS path, and the
others are reflected paths from the neighbor buildings and surrounding
lampposts as shown in Fig. 1.

A typical omnidirectional PDP measured between TX1 and RX23 in the LOS
scenario with a TX-RX separation distance of 43.34 m is shown in Fig.4
(omnidirectional PDPs in NLOS scenarios are being synthesized with the help of
the NYURay ray-tracer [5], and will be available in future work). Fig. 4 shows
four time clusters [35], where there are between 1 to 4 multipath components
in each time cluster and 10 multipath components in total. The first multipath
component is the boresight LOS path from TX1 to RX23 with a propagation time
delay of 144.47 ns. The other multipath components are either reflected from
the neighboring buildings, pillars, or surrounding lampposts, resulting in a
root mean square delay spread of 37.6 ns.

Conventional wisdom before this work is that LOS communication would be the
main use case (e.g., 6G cellular and backhaul) at 142 GHz in outdoor
environments. However, we observed that many outdoor construction materials
(e.g., metal lamppost, concrete building walls, and tinted glass) served as
excellent reflectors at 142 GHz which enabled good NLOS coverage up to 117.4
m, as shown in Figs. 2 and 3.

##  III Rooftop surrogate satellite and backhaul measurement campaign

The second measurement campaign used a rooftop surrogate satellite and
backhaul RX for non-terrestrial network scenarios (e.g., air-to-ground
communications and fixed point-to-point backhaul over the roof) [24]. This
measurement campaign attempted to emulate ground-to-satellite and ground-to-
unmanned aerial vehicle (UAV) communications, providing insights into
potential issues for ground-to-satellite interference and spectrum sharing and
coexistence techniques. This measurement campaign uses the same channel
sounder system and antennas as described in Section II-A but with ground
mounted TX locations using rotating antennas at 1.5 m height. Fig. 5 presents
the rooftop RX and ground TX locations in NYU’s downtown Brooklyn campus.

###  III-A Rooftop Surrogate Satellite and Backhaul Measurement Locations and
Procedures

For the rooftop surrogate satellite and backhaul measurement campaign, the RX
was placed on the rooftop corner of the 6 MetroTech building which is 38.2 m
above ground, emulating a surrogate satellite passive receiver. Horn antennas
with 8° half power beam width (HPBW) were used at both the rooftop RX and
ground-based TXs, where the rooftop RX antenna was mechanically steered and
extensively rotated to consider all possible pointing combinations in the
search of energy from the ground-based TXs that were systematically rotated
over all azimuth directions and several elevation directions described
subsequently [24]. In satellite communications, the received power level from
the ground to the satellite will be highly dependent upon the sidelobes of the
antennas and earth-based TX elevation angles, due to atmospheric absorption
and different slant path lengths [24]. Thus, eight ground-based TX (1.5 m
antenna height) locations (TX1-8) were chosen to have the LOS boresight
elevation angles to the roof-mounted RX in 10° decrements ranging from 80° to
10° to study the relationship between received power and elevation angles
[24]. Due to the space limitation of the campus measurement area, the farthest
TX provided a 15° elevation angle boresight to the RX instead of 10°. The
channel sounder requires a clear LOS link for calibration, but TX locations 1
through 8 were somewhat blocked by tree foliage. To overcome this issue, two
additional TX locations - TX 9 and 10 were chosen, which had the same link
lengths as TX 6 and 7, respectively, for free space calibration without any
link obstructions [24].

A multipath PDP for the (sometimes foliage-blocked) LOS boresight TX-RX
pointing combination was first measured at each TX location, and then the TX
antenna was rotated 360° in the azimuth plane by steps of 8°, and at elevation
angles of 0°, 8°, 16°, 24°, and 32°. For each TX pointing angle, the RX
searched every direction to capture any signals (e.g., direct path, reflected,
or scattered rays) [24].

Figure 5: Rooftop surrogate satellite and backhaul measurement campaign at NYU courtyard in downtown Brooklyn, NY. The surrogate satellite (and backhaul) receiver RX location is 38.2 m above the ground on the rooftop identified as a yellow star. Ten mobile TX locations on the ground are identified as purple circles. The LOS elevation pointing angles from TX1-8 to the RX location are 80° to 15°, respectively. TABLE I: Foliage-blocked LOS links (TX1-7 to RX) and clear LOS links (TX8-10 to RX) from the ground-mounted TXs (1.5 m ht) to the roof-mounted RX (38.2 m above the ground) at 142 GHz, with TX transmit power of -2 dBm and identical 27 dBi gain horn antennas at both of the TX and RX. The predicted received power (assuming free space propagation) $Pr_{FS}$ in dBm, measured received power through foliage-blocked links $Pr$ in dBm, and corresponding foliage loss $(Pr_{FS}-Pr)$ in dBm at different TX-RX separation distances and elevation angles are presented. The negligible difference of the predicted and measured received power of the clear LOS links at TX8, TX9, and TX10 validated the accuracy of the 142 GHz channel sounder system used in this paper. |  Foliage-blocked LOS Links | Clear LOS links
---|---|---
Ground-mounted TX Location | TX1 | TX2 | TX3 | TX4 | TX5 | TX6 | TX7  | TX8 | TX9 | TX10
TX-RX Separation Distance [m] | 38.5 | 40.4 | 43.9 | 49.3 | 58.6 | 70.6 | 102.7  | 178.9 | 70.6 | 102.7
LOS Boresight Elevation $\theta$ | 80° | 70° | 60° | 50° | 40° | 30° | 20°  | 15° | 30° | 20°
Predictied Free Space Received Power $Pr_{FS}$ [dBm] | -55.2 | -55.6 | -56.3 | -57.3 | -58.8 | -60.5 | -63.7  | -68.5 | -60.5 | -63.7
Measured Received Power $Pr$ [dBm] | -60.2 | -62.0 | -59.3 | -65.8 | -63.4 | -70.8 | -74.5  | -68.6 | -60.9 | -64.3
Foliage Loss $(Pr_{FS}-Pr)$ [dB] | 5.0 | 6.4 | 3.0 | 8.5 | 4.6 | 10.3 | 10.8  | 0.1 | 0.4 | 0.6
Average Foliage Loss and Variance | 6.9 dB, 3.0 dB  | Free Space Calibrations
Figure 6: The RX is at heights of 1.5 m above the roof corner, which prevents
the RX from being shadowed by the railing boundary, emulating a passive
receiver in a satellite. The TXs are at heights of 1.5 m above ground working
as mobile terminals.

###  III-B Rooftop Surrogate Satellite and Backhaul Measurement Results with
Foliage Loss Analysis

The received power at the roof-mounted RX (38.2 m above the ground) from the
ground-mounted TXs (1.5 m above the ground) at different distances and
elevation angles are shown in Fig. 7 and Table I (0° elevation angle signifies
the horizontal plane, and the positive values represent the TX elevation
boresight angles above the horizon). When the ground-based TX antenna is
pointing at a 0° elevation angle on the horizon (the blue curve in Fig.7),
there is virtually no power (other than sidelobe radiation) captured by the
rooftop RX even when the ground TX has antenna pattern energy leaking from its
main lobe antenna pattern while pointing nearly directly to the roof (e.g.,
with boresight elevation angle of 15°). The worst case of interference was
found when the TX is at Location 6 (70 m) and Fig.7 shows how raising the
elevation angle of the ground-based transmitter dramatically increases energy
detected by the roof-mounted RX, due to antenna pattern leakage and multipath
from surrounding buildings.

The heights of birch trees in the NYU courtyard, were 8-10 m as shown in Fig.
6. The foliage of the birch trees was between 5-10 m height above the ground
and began falling from the trees during the measurements in November 2020.
Slant link length (L) through foliage at different TX-RX separation distances
and elevation angles ($\theta$ ranges between 80°-20°) is approximately
$L=5~{}\text{m}/sin(\theta)=$ 5-15 m with an average slant link length of 7.9
m (3.4 m standard deviation) through the foliage.

The predicted received power (assuming free space propagation) $Pr_{FS}$ in
dBm, measured received power through foliage-blocked links $Pr$ in dBm (with
the TX transmit power of -2 dBm and identical 27 dBi gain horn antennas at
both of the TX and RX), and corresponding foliage loss $(Pr_{FS}-Pr)$ in dBm
at different TX-RX separation distances and elevation angles are presented in
Table I. The negligible difference of the predicted and measured received
power of the clear LOS links at TX8, TX9, and TX10 validated the accuracy of
the 142 GHz channel sounder system used in this paper. The foliage loss at 142
GHz ranges from 3-11 dB with a 6.9 dB average loss beyond free space and a
standard deviation of 3.0 dB, revealing an average foliage attenuation rate of
0.9 dB/m which is higher than the 0.4 dB/m foliage attenuation rate at 73 GHz
[32], indicating foliage loss (signal attenuation through foliage) increase
with carrier frequencies.

Figure 7: The rooftop base station (38.2 m above the ground) received power
vs. different distances and different elevation angles from ground users (1.5
m above the ground) at 142 GHz [24].

##  IV Conclusion

Two outdoor radio propagation measurement campaigns in urban microcells at 142
GHz are presented in this paper, using a wideband sliding correlation-based
channel sounder with identical narrow-beam 27 dBi gain horn antennas at both
the TX and RX. Omnidirectional and directional CI path loss models with a 1 m
reference distance are provided for both LOS and NLOS scenarios in the outdoor
terrestrial urban microcell environment at 142 GHz. The path loss results are
very encouraging and show that in NLOS scenarios there is usually one or a few
dominant paths (e.g., the best pointing beam $NLOS_{\text{Best}}$) compared to
the other multipath components (e.g., reflected or scattered), and accurate
beamforming algorithms will be needed to find, capture, and combine the most
dominant multipath energy to maintain and extend the outdoor NLOS
communication range at frequencies above 100 GHz. Suprisingly, metal lamppost,
concrete walls, and tinted glass perform as good reflectors at 142 GHz
(providing only 2-8 dB loss to the reflected multipath) which provide good
coverage using only moderate TX power up to 117.4 m in NLOS scenarios.
Elevation angles and antenna sidelobes play an important role in the received
power in air-to-ground channels. Reflected rays by neighboring buildings from
the ground to the air are about 20 dB lower than the direct LOS path, showing
that if the propagation of ground terminals is kept on the horizon (e.g.,
$\leq$ 15°), with reduced sidelobes there may not be any interference in the
same or adjacent bands between surrogate satellites and terrestrial terminals,
or between terrestrial backhaul links and mobile links at frequencies above
100 GHz. The average foliage loss in urban microcell environments is 6.9 dB (a
7-8 m slant link length through foliage) with the foliage attenuation rate of
0.9 dB/m at 142 GHz. Measurements and models presented here contribute to the
understanding that THz will be useful for urban wireless communication, even
in NLOS, as well as non-terrestrial (e.g., satellite and UAV communications)
wireless channels, and may help with spectrum sharing techniques between the
satellites and the ground terminals.

## References

  * [1] T. S. Rappaport _et al._ , “Wireless Communications and Applications Above 100 GHz: Opportunities and Challenges for 6G and Beyond (Invited),” _IEEE Access_ , vol. 7, pp. 78 729–78 757, Feb. 2019.
  * [2] H. Viswanathan and P. E. Mogensen, “Communications in the 6g era,” _IEEE Access_ , vol. 8, pp. 57 063–57 074, 2020.
  * [3] A. Ghosh, A. Maeder, M. Baker, and D. Chandramouli, “5g evolution: A view on 5g cellular technology beyond 3gpp release 15,” _IEEE Access_ , vol. 7, pp. 127 639–127 651, Sept. 2019.
  * [4] O. Kanhere and T. S. Rappaport, “Position Location for Futuristic Wireless Communications - 5G and Beyond,” _IEEE Communications Magazine_ , pp. 70–75, Feb. 2021.
  * [5] O. Kanhere and T. S. Rappaport, “Outdoor sub-thz position location and tracking using field measurements,” _IEEE 2021 International Conference on Communications_ , pp. 1–6, June 2021.
  * [6] S. Sun _et al._ , “MIMO for millimeter-wave wireless communications: beamforming, spatial multiplexing, or both?” _IEEE Communications Magazine_ , vol. 52, no. 12, pp. 110–121, Dec. 2014.
  * [7] 3GPP, “Technical specification group radio access network; study on channel model for frequencies from 0.5 to 100 GHz (Release 14),” 3rd Generation Partnership Project (3GPP), TR 38.901 V14.2.0, Sept. 2017. [Online]. Available: http://www.3gpp.org/DynaReport/38901.htm
  * [8] Aalto University, AT&T, BUPT, CMCC, Ericsson, Huawei, Intel, KT Corporation, Nokia, NTT DOCOMO, New York University, Qualcomm, Samsung, University of Bristol, and University of Southern California, “5G channel model for bands up to 100 GHz,” 2016, Oct. 21. [Online]. Available: http://www.5gworkshops.com/5GCM.html
  * [9] METIS, “METIS Channel Model,” METIS2020, Deliverable D1.4 v3, July 2015.
  * [10] mmMAGIC, “Measurement results and final mmmagic channel models,” H2020-ICT-671650-mmMAGIC/D2.2 v2.0, Dec. 2017. [Online]. Available: https://5g-ppp.eu/mmmagic/
  * [11] S. Sun _et al._ , “Propagation models and performance evaluation for 5G millimeter-wave bands,” _IEEE Transactions on Vehilcular Technology_ , July 2018.
  * [12] T. S. Rappaport _et al._ , “Overview of millimeter wave communications for fifth-generation (5G) wireless networks - with a focus on propagation models,” _IEEE Transactions on Antennas and Propagation_ , vol. 65, no. 12, pp. 6213–6230, Dec. 2017.
  * [13] Y. Xing and T. S. Rappaport, “Indoor Wireless Channel Properties at Millimeter Wave and Sub-Terahertz Frequencies,” in _IEEE 2019 Global Communications Conference_ , Dec. 2019, pp. 1–6.
  * [14] L. Pometcu and R. D. Errico, “Characterization of sub-thz and mmwave propagation channel for indoor scenarios,” in _12th European Conference on Antennas and Propagation (EuCAP 2018)_ , April 2018, pp. 1–4.
  * [15] M. Jacob and T. Kurner, “Radio channel characteristics for broadband WLAN/WPAN applications between 67 and 110 GHz,” in _2009 3rd European Conference on Antennas and Propagation_ , March 2009, pp. 2663–2667.
  * [16] K. Guan, B. Peng, D. He, J. M. Eckhardt, S. Rey, B. Ai, Z. Zhong, and T. Kurner, “Channel Characterization for Intra-Wagon Communication at 60 and 300 GHz Bands,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 6, pp. 5193–5207, June 2019.
  * [17] S. L. H. Nguyen _et al._ , “Comparing Radio Propagation Channels Between 28 and 140 GHz Bands in a Shopping Mall,” _European Conference on Antennas and Propagation_ , pp. 1–5, Apr. 2018.
  * [18] S. Ju, Y. Xing, O. Kanhere, and T. S. Rappaport, “Millimeter Wave and Sub-Terahertz Spatial Statistical Channel Model for an Indoor Office Building,” _IEEE Journal on Selected Areas in Communications, Special Issue on TeraHertz Communications and Networking_ , Feb. 2021.
  * [19] Y. Xing, T. S. Rappaport, and A. Ghosh, “Millimeter Wave and sub-THz Indoor Radio Propagation Channel Measurements, Models, and Comparisons in an Office Environment (Invited),” _in submission to IEEE Communications Letters_ , pp. 1–5, Feb. 2021.
  * [20] J. Ma, R. Shrestha, L. Moeller, and D. M. Mittleman, “Channel performance for indoor and outdoor terahertz wireless links,” _APL Photonics_ , vol. 3, no. 5, pp. 1–13, Feb. 2018.
  * [21] N. A. Abbasi, A. Hariharan, A. M. Nair, A. S. Almaiman, F. B. Rottenberg, A. E. Willner, and A. F. Molisch, “Double directional channel measurements for thz communications in an urban environment,” in _ICC 2020 - 2020 IEEE International Conference on Communications (ICC)_ , June 2020, pp. 1–6.
  * [22] Y. Xing and T. S. Rappaport, “Propagation Measurement System and Approach at 140 GHz–Moving to 6G and Above 100 GHz,” in _IEEE 2018 Global Communications Conference_ , Dec. 2018, pp. 1–6.
  * [23] G. R. MacCartney and T. S. Rappaport, “A flexible millimeter-wave channel sounder with absolute timing,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 6, pp. 1402–1418, June 2017.
  * [24] Y. Xing and T. S. Rappaport, “Spectrum Coexistence between Active and Passive Systems in Space and on the Ground at Frequencies above 100 GHz (Invited),” _in submission to IEEE Communications Letters_ , pp. 1–5, Feb. 2021.
  * [25] T. S. Rappaport _et al._ , “Wideband millimeter-wave propagation measurements and channel models for future wireless communication system design,” _IEEE Transactions on Communications_ , vol. 63, no. 9, pp. 3029–3056, Sept. 2015.
  * [26] Y. Xing _et al._ , “Verification and calibration of antenna cross-polarization discrimination and penetration loss for millimeter wave communications,” in _2018 IEEE 88th Vehicular Technology Conference (VTC-Fall)_ , Aug. 2018, pp. 1–6.
  * [27] T. S. Rappaport _et al._ , “Millimeter Wave Mobile Communications for 5G Cellular: It Will Work!” _IEEE Access_ , vol. 1, pp. 335–349, May 2013\.
  * [28] S. Sun _et al._ , “Synthesizing omnidirectional antenna patterns, received power and path loss from directional antennas for 5G millimeter-wave communications,” in _2015 IEEE Global Communications Conference (GLOBECOM)_ , Dec. 2015, pp. 3948–3953.
  * [29] T. S. Rappaport, _Wireless Communications: Principles and Practice_ , 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2002.
  * [30] G. R. MacCartney, Jr. _et al._ , “Indoor office wideband millimeter-wave propagation measurements and models at 28 GHz and 73 GHz for ultra-dense 5G wireless networks (Invited Paper),” _IEEE Access_ , vol. 3, pp. 2388–2424, Oct. 2015.
  * [31] S. Sun _et al._ , “Investigation of prediction accuracy, sensitivity, and parameter stability of large-scale propagation path loss models for 5G wireless communications (Invited Paper),” _IEEE Transactions on Vehicular Technology_ , vol. 65, no. 5, pp. 2843–2860, May 2016.
  * [32] T. S. Rappaport and S. Deng, “73 GHz wideband millimeter-wave foliage and ground reflection measurements and models,” in _IEEE International Conference on Communications Workshops (ICCW)_ , June 2015, pp. 1238–1243.
  * [33] G. R. MacCartney and T. S. Rappaport, “Millimeter-wave base station diversity for 5g coordinated multipoint (comp) applications,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 7, pp. 3395–3410, May 2019\.
  * [34] S. Sun _et al._ , “Millimeter wave multi-beam antenna combining for 5G cellular link improvement in New York City,” in _2014 IEEE International Conference on Communications (ICC)_ , June 2014, pp. 5468–5473.
  * [35] M. K. Samimi and T. S. Rappaport, “3-D millimeter-wave statistical channel model for 5G wireless system design,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 64, no. 7, pp. 2207–2225, July 2016.