\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 106, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/106\hfil Pyramidal central configurations]
{Pyramidal central configurations and perverse solutions}

\author[T. Ouyang, Z. Xie, S. Zhang\hfil EJDE-2004/106\hfilneg]
{Tiancheng Ouyang, Zhifu Xie, Shiqing Zhang} % in alphabetical order

\address{Tiancheng Ouyang \hfill\break
Department of Mathematics, Brigham Young University\\
Provo, Utah 84604, USA} 
\email{ouyang@math.byu.edu}

\address{Zhifu Xie\hfill\break
Department of Mathematics, Brigham Young University\\
Provo, Utah 84604, USA} 
\email{zhifu@math.byu.edu}

\address{Shiqing Zhang  \hfill\break
Department of Mathematics, Yangzhou University, Yangzhou, China}
\email{shiqing2001@163.net}

\date{}
\thanks{Submitted December 6, 2003. Published September 10, 2004.}
\subjclass[2000]{37N05, 70F10, 70F15}
\keywords{$n$-body problems; pyramidal central configuration; \hfill\break\indent
 regular polygonal base; perverse solutions}

\begin{abstract}
  For $n$-body problems, a central configuration (CC) plays an
  important role. In this paper, we establish the relation between
  the spatial pyramidal central configuration (PCC) and the
  planar central configuration. We prove that the base of PCC is
  also a CC and we also prove that for some given conditions a
  planar CC can be extended to a PCC. In particular, if the
  pyramidal central configuration has a regular polygon base, then
  the masses of base are equal and the distance between the top
  vertex and the base is fixed  and the mass of the top vertex is
  selective. Furthermore, the pyramidal central configuration gives
  rise to an example of a perverse solution in $\mathbb{R}^3$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}

\section{ Introduction and Main Results}

In this paper, we investigate  the quantitative relationship between
the spatial pyramidal  central configuration and its base.
We also investigate perverse solution in  $\mathbb{R}^3$.
The Newtonian $n$-body problem concerns the motion of $n$ point particles
with masses  $m_j \in \mathbb{R}^+$ and positions $\bar{q}_j \in \mathbb{R}^3$
$(j=1,\dots,n)$. This motion is governed by the Newton's law
\begin{equation}
m_j\ddot{\bar{q_j}}=\frac{\partial U(\bar{q})}{\partial
\bar{q}_j}, \label{eq1.1}
\end{equation}
where $\bar{q}=(\bar{q}_1,\dots,\bar{q}_n)$ and
the Newtonian potential is
\begin{equation}
U(\bar{q})=\sum_{1\leq k<j\leq n}\frac{m_k
m_j}{|\bar{q}_k-\bar{q}_j|}. \label{eq1.2}
\end{equation}
Consider the space
$$
X=\big\{ \bar{q}=(\bar{q}_1,\dots,\bar{q}_n )\in \mathbb{R}^{3n} :
\sum_{k=1}^{n}m_{k}\bar{q}_{k}=0 \big\}
$$
i.e. suppose that the center of mass is fixed at the origin of the
space. Because the potential is singular when two particles have
the same position, it is natural to assume that the configuration
avoids the set $\triangle =\{ \bar{q}: \bar{q}_k =\bar{q}_j $ for
some $ k\neq  j \}$. The set
$X\backslash \triangle$ is called the configuration space.

\noindent{\bf Definition }%  1.1
 A configuration $\bar{q}=(\bar{q}_1,\dots,\bar{q}_n) \in X\backslash \triangle$
is called a central configuration (CC) if there exists a constant
$\lambda$ such that
\begin{equation}
\sum_{j=1,j\neq  k}^{n} \frac{m_j
m_k}{|\bar{q}_j-\bar{q}_k|^3}(\bar{q}_j-\bar{q}_k)=-\lambda
m_k\bar{q}_k, \quad 1\leq k\leq n. \label{eq1.3}
\end{equation}
The value of the constant $\lambda$ in \eqref{eq1.3} is uniquely determined by
\begin{equation}
\lambda = \frac{U}{I}, \label{eq1.4}
\end{equation}
where $I=\sum_{k=1}^{n}m_k |\bar{q}_k|^2$.

\noindent{\bf Definition }% 1.2}
A central configuration of $N+1$ bodies, $N$ of which are
coplanar, the $(N+1)th$ being off the plane, is called a pyramidal
central configuration (PCC). Equivalently, we will say that the CC
has the shape of a pyramid, where the $N$ bodies or the $N$
positions are called the base of the corresponding pyramidal
central configuration.

 Central configurations give rise to
 simple, explicit solutions of the N-body problem \cite{m3}.
 If the bodies are placed in a central configuration and released
 with zero initial velocity, they will collapse homothetically to
 a collision at center of mass. If the central configuration is
 planar, one can also choose initial velocities which lead to a
 periodic solution for which the configuration rigidly rotates
 around center of mass with angular velocity $\sqrt{\lambda}$.

A complete understanding of the nature of the central configurations
is of fundamental importance to the $n$-body
problem of celestial mechanics as these configurations play an
essential role in the global structures of the solutions of the
$n$-body problem.

Although three centuries have passed
 since Euler, Lagrange, etc. studied these problems,
the classification of the central configuration  is
 still unknown even for 4 bodies. It continues to attract much attention and some
 marvellous results have been obtained \cite{f1,l1}.  In the
 celebrated work \cite{a1} of 1996, Albouy was able to establish a
 symmetry and prove that there are exactly three central
 configurations for the planar 4-body problem with equal masses.
 In 2002, Yiming Long and Sanzhong Sun studied the central configuration
 for the 4-body problem under the weaker condition that only the opposite
 masses are equal.

In 1996, Nelly Faycal established a classification of all PCC
  of the five-body problem with its base admitting a plane of
  reflexive symmetry. She studied the four cases which corresponds to
  the base of the pyramid of five bodies that admits one axis of
  symmetry, two axes of symmetry, or more axes of symmetry. The
  four cases are: pyramid with a square base, pyramid with a
  rectangular base, pyramid with a kite-shaped base and pyramid
  with a trapezoid base. She also generalized some of the results
  in the case of five masses to N+1 masses. She proved that the
  coplanar masses are concyclic (i.e. all lie on the same circle),
and that the mass off the plane
  is equidistant from them \cite[Theorem 6.1.1]{f2}. She also proved
  that in a pyramidal central configuration the mass off the plane
  is arbitrary \cite[Theorem 6.2.2]{f2}. She also investigated the
  relation between the pyramidal central configuration and its
  base \cite[Corollary 6.2.1]{f2}.

 This paper is distributed as follows. In
    section 2, we collect some basic properties of PCC that will
    be useful in the proof of the main theorem in section 3 and
    section 4. In section 3 we show the relation between spatial
    pyramidal central configuration and its base and also find the
    quantitative formulas of masses and distance for a PCC with regular polygon
    base. In section 4 we construct an example which gives rise to
    a perverse solution in $\mathbb{R}^3$. Although some results in
    section 2 and 3 follow straight ahead from the main theorems
    of Nelly Faycal \cite{f2}, we have decided to include
    our proofs here so that our paper will be
    completely self-contained.

\section{Some General Lemmas}

The proof to the following Lemmas can be
found in Nelly Faycal \cite{f1,f2}.

\begin{lemma}[{\cite[Theorem 6.1.1]{f2}}] \label{lem2.1}
If $\bar{q}=(\bar{q}_{1},\dots, \bar{q}_{N+1})$ is a PCC such
that $\bar{q}_{N+1}$ is at the top vertex which is off the plane
containing
 $m_{1},\dots,m_{N}$, then $m_{N+1}$ is equidistant from $m_{1},\dots,m_{N}$.
\end{lemma}

\begin{proof}
Since $\bar{q}=(\bar{q}_{1},\dots, \bar{q}_{N+1})$ forms a CC,
then there exists a scalar $\lambda$ such that
\begin{equation}
\sum_{j=1,j\neq i}^{N+1}\frac{m_{j}m_{i}}{|\bar{q}_{j}-\bar{q}_{i}|^3}
(\bar{q}_{j}-\bar{q}_{i})=-\lambda m_{i}\bar{q}_{i}  ,1\leq i\leq N+1.\label{eq2.1}
\end{equation}
Writing $\bar{q}_{i}=(\bar{x}_{i},\bar{y}_{i},\bar{z}_{i})\in \mathbb{R}^{3}$
in terms of its coordinate $G\bar{x}\bar{y}\bar{z}$, and
$D_{j,i}=|\bar{q}_{j}-\bar{q}_{i}|$ for $1\leq i,j\leq N+1$. Since
 the masses $m_{1},\dots,m_{N}$ lie on a common plane, we may assume then, without
 loss of generality, that this plane is parallel to $G\bar{x} \bar{z}$.
Hence $\bar{y}_{1}=\bar{y}_{2}=\dots=\bar{y}_{N}$. Multiplying
\eqref{eq2.1} by $\bar{y}$ which is the unit vector  of
$\bar{y}-$direction. We obtain
\begin{equation}
\sum_{j=1,j\neq  i}^{N+1}\frac{m_{j}m_{i}}{D_{j,i}^3}(\bar{q}_{j}-\bar{q}_{i})
\bar{y}=-\lambda m_{i}\bar{q}_{i}\bar{y},1\leq i\leq N+1. \label{eq2.2}
\end{equation}
 From \eqref{eq2.2}, for $i=1,2$, we obtain
\begin{gather}
\frac{m_{N+1}m_{1}}{D_{N+1,1}^{3}}(\bar{y}_{N+1}-\bar{y}_{1})
=-\lambda m_{1}\bar{y}_{1}. \label{eq2.3}\\
\frac{m_{N+1}m_{2}}{D_{N+1,2}^{3}}(\bar{y}_{N+1}-\bar{y}_{2})
=-\lambda m_{2}\bar{y}_{2}. \label{eq2.4}
\end{gather}
Hence \eqref{eq2.3}, \eqref{eq2.4} give
\begin{equation}
m_{N+1}\big(\frac{1}{D_{N+1,1}^{3}}-\frac{1}{D_{N+1,2}^{3}}\big)
(\bar{y}_{N+1}-\bar{y}_{1})=0. \label{eq2.5}
\end{equation}
Since $\bar{y}_{N+1}-\bar{y}_{1}\neq  0$ otherwise $m_{1},\dots,m_{N+1}$
are coplanar
which contradicts to the definition of yramidal central configuration, then
$$ D_{N+1,1}=D_{N+1,2}.$$
Similarly, we readily obtain
$$ D_{N+1,i}=D_{N+1,j} 1\leq i,j\leq N.$$
So $m_{N+1}$ is equidistant from $m_{1},\dots,m_{N}$.
\end{proof}

\begin{remark} \label{remk1} \rm
The position $\bar{q}_{1},\dots,\bar{q}_{N}$ are concyclic.
In fact, they lie on the intersection of a plane with a sphere, since they are
 coplanar by assumption and they belong to a sphere centered at $m_{N+1}$
by Lemma \ref{lem2.1}.
\end{remark}

\begin{remark} \label{rmk2} \rm
For $N=3, \bar{q}_1,\dots,\bar{q}_4$ form a PCC in addition to the symmetry
of the positions then $\bar{q}_1,\dots,\bar{q}_4$
are at the vertices of regular tetrahedron.
\end{remark}

\begin{lemma}[{\cite[Theorem 6.2.1]{f2}}] \label{lem2.2}
If $\bar{q}=(\bar{q}_{1},\dots, \bar{q}_{N+1})$ is a PCC then
$\lambda =M_{N+1}g$, where $M_{N+1}=m_{1}+\dots+m_{N+1}$ is the
total masses and $g=\frac{1}{D_{N+1,i}^{3}} 1\leq i\leq N$.
\end{lemma}

\begin{proof} Denote by Oxyz, the coordinate system obtained from
G$\bar{x}\bar{ y} \bar{z}$ by
 parallel translation to a new origin $O\in P$, where $O$ belongs to the plane $P$ containing $m_{1},\dots,m_{N}.$ Let $q_{1},\dots,q_{N+1}$ be
the position vectors of $m_{1},\dots,m_{N+1}$ in Oxyz. Obviously
\begin{equation}
\overline{OG}=\frac{1}{m}\sum_{j=1}^{N+1}m_{j}q_{j}.\label{eq2.6}
\end{equation}
Since $\bar{q}=(\bar{q}_{1},\dots, \bar{q}_{N+1})$ is a CC,
there exists a $\lambda$ such that
\begin{equation}
\sum_{j=1,j\neq  i}^{N+1}\frac{m_{j}m_{i}}{|\bar{q}_{j}-\bar{q}_{i}|^3}
(\bar{q}_{j}-\bar{q}_{i})=-\lambda m_{i}\bar{q}_{i}  ,1\leq i\leq N+1.\label{eq2.7}
\end{equation}
Take for the scalar multiple of equation \eqref{eq2.7} with $\bar{y}$ a
unit vector in $\bar{y}$-direction. For $i=1,\dots,N+1$, we use
$\bar{q}_{i}=q_{i}-\overline{OG}$ to
 get
\begin{equation}
 \sum_{j=1 j\neq  i}^{N+1}\frac{m_{j}m_{i}}{|q_{j}-q_{i}|^3}(q_{j}-q_{i})
=-\lambda m_{i}(q_{i}-\overline{OG}),\label{eq2.8}
\end{equation}
that is
\[
 \sum_{j=1,j\neq  i}^{N+1}\frac{m_{j}m_{i}}{|q_{j}-q_{i}|^3}(q_{j}-q_{i})=
-\lambda m_{i}\Big(\frac{1}{M_{N+1}}\sum_{j=1}^{N+1}m_{j}q_{i}-
\frac{1}{M_{N+1}}\sum_{j=1}^{N+1}m_{j}q_{j}\Big), %\label{eq2.9}
\]
or
\begin{equation}
 \sum_{j=1,j\neq  i}^{N+1}m_{j}m_{i}\Big(\frac{1}{D_{j,i}^{3}}
 -\frac{\lambda}{M_{N+1}}\Big)(q_{j}-q_{i})=0, \label{eq2.10}
 \end{equation}
then
\begin{equation}
 \sum_{j=1,j\neq  i}^{N+1}m_{j}m_{i}\Big(\frac{1}{D_{j,i}^{3}}
 -\frac{\lambda}{M_{N+1}}\Big)(q_{j}-q_{i})\bar{y}=0. \label{eq2.11}
 \end{equation}
But $\bar{y}$ is perpendicular to the plane P containing the vectors
$q_{1},\dots,q_{N}$ then
\[
m_{N+1}m_{i}\Big(\frac{1}{D_{N+1,i}^{3}}-\frac{\lambda}{M_{N+1}}\Big)(q_{N+1}-q_i)
\bar{y}=0 %\label{eq2.12}
\]
Hence
\[
\lambda =\frac{M_{N+1}}{D_{N+1,i}^{3}} \quad 1\leq i\leq N. %\label{eq2.13}
\]
Note \eqref{eq2.7} holds if and only if \eqref{eq2.10} holds.
\end{proof}

\section{Relation Between Pyramidal Central Configuration and Its Base}

 The following theorem is an extension to
arbitrary masses of the Faycal five-body results
\cite[Corollary 2.3.1 and Theorem 2.3.1]{f2}.

\begin{theorem} \label{thm3.1}
If $\bar{q}=(\bar{q}_1,\dots,\bar{q}_{N+1})(N\geq 3)$ is a
PCC, such that $\bar{q}_{N+1}$  is at the top vertex which is
off the plane containing $m_1, \dots, m_{N}$, then the particles of the base
$m_1, \dots, m_{N}$ also form a CC.

 Conversely, if $m_1, m_2, \dots, m_N$ with
position $q_1,q_2,\dots,q_N,$ are coplanar and form a CC with
multiplier $\lambda$ and if there exists a position $q_{N+1}$ such
that
$$ \frac{1}{|q_{N+1}-q_i|^{3}}=\frac{\lambda}{M_N}, \quad 1\leq i\leq N,
$$
where $M_N=\sum_{i=1}^{N}m_i$, then for any mass $m_{N+1}$ with position $q_{N+1}$,
$m_1,m_2$, \dots, $m_{N+1}$ form a PCC
\end{theorem}

\begin{proof} If $\bar{q}=(\bar{q}_1,\dots,\bar{q}_{N+1})$ is a PCC, similar
to the proof of Lemma \ref{lem2.2} and according to the results \eqref{eq2.10} of
Lemma \ref{lem2.2}, we have
\begin{equation}
\sum_{j=1,j\neq
i}^{N}m_j(\frac{1}{D_{j,i}^3}-\frac{1}{D_{N+1,1}^3})(q_j-q_i)=0\,. \label{eq3.1}
\end{equation}
Furthermore, we choose the new origin $O$ in Lemma \ref{lem2.2} as the
center of masses $m_1,\dots,m_N,$ (i.e.
$\sum_{j=1}^{N}m_jq_j=0$). Then we have
\begin{align*}
 \sum_{j=1,j\neq  i}^{N}\frac{m_j}{D_{j,i}^3}(q_j-q_i)
&=\sum_{j=1,j\neq  i}^{N}\frac{m_j}{D_{N+1,1}^3}(q_j-q_i)\\
&=\frac{1}{D_{N+1,1}^3}\sum_{j=1,j\neq  i}^{N}m_j(q_j-q_i)\\
&=\frac{1}{D_{N+1,1}^3}\sum_{j=1}^{N}m_j(q_j-q_i)\\
&=\frac{1}{D_{N+1,1}^3}\sum_{j=1}^{N}m_jq_j-\frac{1}{D_{N+1,1}^3}
\sum_{j=1}^{N}m_jq_i\\
&=-\frac{\sum_{j=1}^{N}m_j}{D_{N+1,1}^3}q_i=-\frac{M_N}{D_{N+1,1}^3}q_i.
\end{align*}
Let $\lambda =(\sum_{j=1}^{N}m_j)/D_{N+1,1}^3$. Then
$q_1,\dots,q_N$ form a central configuration.
Conversely because $m_1,m_2,\dots,m_N$ with positions
$q_1,q_2,\dots,q_N$, form a CC then
\begin{equation}
\sum_{j=1,j\neq  i}^{N} \frac{m_j
m_i}{|q_j-q_i|^3}(q_j-q_i)=-\lambda m_iq_i, \quad 1\leq
i\leq N. \label{eq3.2}
\end{equation}
Let
\begin{equation}
z_0=\frac{1}{M_{N+1}}\sum_{j=1}^{N+1}m_jq_j, \quad %\label{eq3.3}
\bar{q}_j=q_j-z_0.
\end{equation}
Then $\sum_{j=1}^{N}m_j\bar{q}_j=-m_{N+1}\bar{q}_{N+1}$. For
$i\neq  N+1$, we obtain
\begin{align*}
& \sum_{j=1,j\neq  i}^{N+1}\frac{m_{j}m_{i}}{|\bar{q}_{j}-\bar{q}_{i}|^3}
(\bar{q}_{j}-\bar{q}_{i})+\frac{\lambda M_{N+1}}{M_N} m_{i}\bar{q}_{i}\\
&=\sum_{j=1,j\neq  i}^{N+1}\frac{m_{j}m_{i}}{|\bar{q}_{j}-\bar{q}_{i}|^3}
(\bar{q}_{j}-\bar{q}_{i})+\frac{\lambda}{M_N} m_{i}
\sum_{j=1,j\neq  i}^{N+1}m_j(\bar{q}_{i}-\bar{q}_j)\\
&=\sum_{j=1,j\neq  i}^{N+1}m_{j}m_{i}
\Big(\frac{1}{|\bar{q}_{j}-\bar{q}_{i}|^3}-\frac{\lambda}{M_N}\Big)
(\bar{q}_{j}-\bar{q}_{i})\\
&=\sum_{j=1,j\neq  i}^{N}m_{j}m_{i}\Big(\frac{1}{D_{j,i}^3}-\frac{\lambda}{M_N}\Big)
(\bar{q}_{j}-\bar{q}_{i})\\
&=\sum_{j=1,j\neq i}^{N}m_{j}m_{i}\Big(\frac{1}{D_{j,i}^3}-\frac{\lambda}{M_N}\Big)
(q_{j}-q_{i})=0.
\end{align*}
That is
$$
\sum_{j=1,j\neq  i}^{N+1}\frac{m_{j}m_{i}}{|\bar{q}_{j}-\bar{q}_{i}|^3}
(\bar{q}_{j}-\bar{q}_{i})=-\frac{\lambda M_{N+1}}{M_N} m_{i}\bar{q}_{i}
=-\lambda' m_{i}\bar{q}_{i},
$$
where
$$ \lambda'=\frac{M_{N+1}}{|q_j-q_{N+1}|^3}.
$$
And
for $i=N+1$,
\begin{align*}
\sum_{j=1}^{N}\frac{m_{j}m_{N+1}}{|\bar{q}_{j}-\bar{q}_{N+1}|^3}
(\bar{q}_{j}-\bar{q}_{N+1})
&= \frac{m_{N+1}}{D_{N+1,j}^3}\sum_{j=1}^{N} m_j(\bar{q}_{j}-\bar{q}_{N+1})\\
&=\frac{m_{N+1}}{D_{N+1,j}^3}(-m_{N+1}\bar{q}_{N+1}-M_{N}\bar{q}_{N+1})\\
&=-\frac{M_{N+1}}{D_{N+1,j}^3}m_{N+1}\bar{q}_{N+1}\\
&=-\lambda' m_{N+1}\bar{q}_{N+1}.
\end{align*}
The proof is complete. \end{proof}

The following theorem is an extension to the case
of arbitrary masses of the Faycal five-body result
\cite[Theorem 3.1.1]{f2}.

\begin{theorem} \label{thm3.2}
For $N\geq 3$ the $N+1$ body problem with masses $m_1,m_2,\dots,m_{N+1}$
in $\mathbb{R}^{+}$, and
positions $\bar{q}_1,\dots,\bar{q}_{N+1} \in \mathbb{R}^3$,
assume $\bar{q}_1,\dots,\bar{q}_N$
 are coplanar and lie at the vertices of a regular polygon inscribed on
a unit circle, and the $(N+1)th$
is off the plane. Then the $N+1$ bodies form a PCC if and only
if the distance between top vertex and the vertices of the base
satisfies
\begin{equation}
\frac{1}{D_{N+1,k}^3}=\frac{1}{4N}\sum_{j=1}^{N-1}\csc
\big(\frac{\pi j}{N}\big)<1,\quad 1\leq k\leq N, \label{eq3.5}
\end{equation}
where $D_{k,j}=|\bar{q}_k-\bar{q}_j|,$  and the masses in the base
are equal $m_1=m_2=\dots=m_{N}$, the masse $m_{N+1}$ in the top
vertex is arbitrary.
\end{theorem}

We remark that there is no loss of generality in assuming
that the regular polygon is inscribed on the unit circle since the
CC \eqref{eq1.3} is invariant under the transformation
$\bar{q}_k\to \bar{q}_k/a, \lambda \to a^2\lambda$.
In addition, the distance between top vertex and the
vertices of the base doesn't depend on the masses and is
completely determined by the base. The central configurations of N
bodies cannot be extended to any pyramidal central configuration
for $N\geq 473$ because for $N\geq 473$,
$\frac{1}{4N}\sum_{j=1}^{N-1}\csc (\frac{\pi j}{N})>1$. So a
planar central configuration can not always be extended to a
pyramidal central configuration.You can find more comments in
Moeckel \cite{m3}.

\begin{proof}[Proof of Theorem \ref{thm3.2}]
By lemma \ref{lem2.1}, for $1 \leq k,j \leq N$.
\begin{equation}
\frac{1}{D_{N+1,k}^3}=\frac{1}{D_{N+1,j}^3}\,. \label{eq4.1}
\end{equation}
By theorem \ref{thm3.1}, $\bar{q}_1, \bar{q}_2,\dots,\bar{q}_N$ form a
planar central configuration. Then these particles can rotate
about the center of masses by theorem (Perko-Walter \cite{p1} and
Xie-Zhang \cite{x1}).
\begin{equation}
\lambda=\frac{M_{N+1}\gamma}{N}=\frac{M_{N+1}}{D_{N+1,i}^3}, \label{eq4.2}
\end{equation}
where $\gamma=\frac{1}{4N}\sum_{j=1}^{N-1}\csc(\frac{\pi j}{N})$.
Then
\begin{equation}
\frac{1}{D_{N+1,i}^3}=\frac{1}{4N}\sum_{j=1}^{N-1}\csc\big(\frac{\pi
j}{N}\big).
\end{equation}
By Theorem \ref{thm3.1}, $\bar{q_1},\dots,\bar{q_N}$ form a planar central
configuration. As a result of \cite{p1,x1},  $m_1=m_2=\dots=m_N$.
Although the proof in \cite{x1} is not complete, the flaw pointed out
by Chenciner \cite{c1} does
not affect the conclusion,  $m_1=m_2=\dots=m_N$.
\end{proof}

Conversely, by Theorem \ref{thm3.1}, we know that we can put an arbitrary
mass body at the top vertex and the $N+1$ bodies form a pyramidal
central configuration.


\section{Perverse Solutions in $\mathbb{R}^3$}

Let $\bar{q}(t)=(\bar{q}_1(t),\bar{q}_2(t),\dots,\bar{q}_n(t))$ be a
solution of the $n$-body problem with Newtonian potential and masses
$m_1,m_2,\dots,m_n$. Chenciner \cite{c1} proposed the
following two questions:
\begin{enumerate}
\item  Does there exist
another system of masses, $(m_1', m_2',\dots,m_n')$, for which
$\bar{q}(t)$ is still a solution?

\item  The same as question 1 but
insisting that the sum $M=\sum_{i=1}^{n}m_i$ of the masses and the
center of mass
$C=\frac{1}{M}\sum_{i=1}^nm_i\bar{q}_i$ do not change.
\end{enumerate}

\noindent{\bf Definition.} If the answer to the
first (resp. second) question is yes, we shall say $\bar{q}(t)$ is
a perverse (resp. really perverse) solution and the allowed
systems of masses will
be called admissible.

Chenciner investigated the perverse
solutions in the planar case. He proved  for n=2 that no solution
is perverse, and for $n\geq 3$ that perverse solutions do exist by
constructing an example of a regular polygon rotating around the
body lying in the center of
the regular polygon.
Now, we  construct a perverse solution in $\mathbb{R}^3$. Let
$\bar{q}(t)=(\bar{q}_1(t),\bar{q}_2(t),\dots$,
$\bar{q}_N(t),\bar{q}_{N+1}(t),0)$ be a total collision solution
with $N+2$ masses $(m_1,m_2,\dots,m_N,m_{N+1},m_{N+2})$ and
satisfy the following initial conditions:
\begin{enumerate}
\item $(\bar{q}_1(0),\bar{q}_2(0),\dots,\bar{q}_N(0),\bar{q}_{N+1}(0))$
is a pyramidal central configuration such that $\bar{q}_{N+1}(0)$
is at the top vertex which is off the plane containing
$\bar{q}_1(0)$, $\bar{q}_2(0),\dots, \bar{q}_N(0)$

\item The center of mass is at the origin i.e. $m_1\bar{q}_1(0)+\dots
+m_{N+1}\bar{q}_{N+1}(0)+m_{N+2}\cdot 0=0$

\item $|\bar{q}_i|=|\bar{q}_j|, 1\leq i,j\leq N+1$

\item The initial velocity is zero i.e. $\bar{q}'(0)=0$.
\end{enumerate}

\begin{theorem} \label{thm4.1}
$\bar{q}(t)$ is a perverse solution with a one
parameter family of admissible sets of masses.
\end{theorem}

\begin{proof} $\bar{q}(t)$ is a solution of the Newton's equation
\begin{equation}
m_j\ddot{\bar{q}}_j=\sum_{k=1,k\neq  j}^{N+2} \frac{m_k
m_j}{|\bar{q}_k-\bar{q}_j|^3}(\bar{q}_k-\bar{q}_j), \quad
1\leq j\leq N+2.
\end{equation}
where $\bar{q}_{N+2}(t)=0$ for all t. Because $\bar{q}(t)$
satisfies the above initial conditions, $\bar{q}(t)$ will collapse
homothetically to a collision at the center of mass at zero while
keeping the shape in the whole motion. Therefore, $\bar{q}(t)$ is
a perverse solution and
$(m_1',m_2',\dots,m_N',m_{N+1}',m_{N+2}')$
is an admissible system of masses if and only if the accelerations
$\ddot{\bar{q}}_i(t)$ (for all $1\leq i\leq N+2$) do not change
with respect to the admissible masses. In fact, for $j\neq N+2$, we have
\begin{align*}
\ddot{\bar{q}}_j&=\sum_{k=1,k\neq  j}^{N+2} \frac{m_k
}{|\bar{q}_k-\bar{q}_j|^3}(\bar{q}_k-\bar{q}_j)\\
&=\sum_{k=1,k\neq  j}^{N+1} \frac{m_k
}{|\bar{q}_k-\bar{q}_j|^3}(\bar{q}_k-\bar{q}_j)+\frac{m_{N+2}
}{|\bar{q}_{N+2}-\bar{q}_j|^3}(\bar{q}_{N+2}-\bar{q}_j)\\
&=-\frac{M_{N+1}}{D_{N+1,j}^3}\bar{q}_j-\frac{m_{N+2}}{|\bar{q}_j|^3}\bar{q}_j
\quad\hbox{by lemma \ref{lem2.2}}\\
&=-\frac{\beta
M_{N+1}}{|\bar{q}_j|^3}\bar{q}_j-\frac{m_{N+2}}{|\bar{q}_j|^3}\bar{q}_j \\
&=-(\beta M_{N+1}+m_{N+2})\frac{\bar{q}_j}{|\bar{q}_j|^3}
\end{align*}
where $ \beta=|\bar{q}_j|^3/D_{N+1,j}^3$ is a constant for all
$1\leq j\leq N+1$ and for all t because $|\bar{q}_j|=|\bar{q}_k|$,
$D_{N+1,j}=D_{N+1,k}$ and the motion keeps the same shape. In
addition, for $j=N+2$, $\bar{q}_{N+2}$ is fixed at origin.
Therefore,
$(m_1',m_2',\dots,m_N',m_{N+1}',m_{N+2}')$
is an admissible masses if $\beta
M_{N+1}'+m_{N+2}'=\beta M_{N+1}+m_{N+2}$ and the
initial conditions are satisfied. For example, we can choose
$m_j'= \rho m_j$ for $1\leq j\leq N+1$ which leads the
initial conditions to be satisfied  and choose $m_{N+2}'
=\beta M_{N+1}+m_{N+2}-\beta \rho M_{N+1}$. It follows that $\rho$
may be chosen as a parameter of the set of admissible masses. In
particular, $\bar{q}(t)$ is perverse but not really perverse since
$\beta<1$.
\end{proof}

\begin{corollary} \label{coro1}
Under the same conditions as theorem \ref{thm4.1}, but inscribing
the base $\bar{q}_1(0),\dots,\bar{q}_N(0)$ on the vertex of a
unit regular polygon, the function $\bar{q}(t)$ is a perverse solution
for $N=3,4,5,6,7,8$.
\end{corollary}

\begin{proof}  We only need to check the conditions  (1) and
(2) are satisfied if we choose $m_1=\dots=m_N$ and the distance
$D_{N+1,k}$ between the $\bar{q}_{N+1}$ and $\bar{q}_k$ satisfying
\eqref{eq3.5}. For $N=3,4,5,6,7,8$, it could choose masses such
that (3) and (4) are satisfied. But for $N\geq 9$,
$D_{N+1,i}<1.394$ then it is impossible to make
$|\bar{q}_i|=|\bar{q}_j|$ for $1\leq i,j\leq N+1$.
\end{proof}

\subsection*{Acknowledgments}
 The authors wish to express their gratitude
to the anonymous referee for the advice to investigate perverse solutions of
N-body problems.


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\end{document}