#  New Periodic Solutions for Some Planar
$N+3$-Body Problems with Newtonian Potentials ††thanks: Supported partially by
NSF of China.

Pengfei Yuan and Shiqing Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Mathematics, Sichuan University, Chengdu 610064, China

###### Abstract

For some planar Newtonian $N+3$-body problems, we use variational minimization
methods to prove the existence of new periodic solutions satisfying that $N$
bodies chase each other on a curve, and the other $3$ bodies chase each other
on another curve. From the definition of the group action in equations
$(3.1)-(3.3)$, we can find that they are new solutions which are also
different from all the examples of Ferrario and Terracini (2004)$[22]$.
Key Words: $N+3$-body problems, periodic solutions, winding numbers,
variational minimizers.
2000 Mathematicals Subject Classification: 34C15, 34C25, 58F

##  1 Introduction and Main Results

In recent years, many authors used methods of minimizing the Lagrangian action
on a symmetric space to study the periodic solutions for Newtonian $N$-body
problem $([2],[4]-[6],[8]-[29],[31]-[40])$. Especially,
A.Chenciner-R.Montgomery $[16]$ proved the existence of the remarkable figure
eight type periodic solution for Newtonian three-body problem with equal
masses, C.Simó $[32]$ discovered many new periodic solutions for Newtonian
$N$-body problem using numerical methods. C.Machal $[27]$ studied the fixed-
ends (Bolza) problem for Newtonian $N$-body problem and proved that the
minimizer for the Lagrangian action has no interior collision; A.Chenciner
$[12]$, D.Ferario and S.Terracini $[22]$ simplified and developed C.Marchal’s
important works; S.Q.Zhang $[36]$, S.Q.Zhang, Q.Zhou $([37]-[40])$ decomposed
the Lagrangian action for $N$-body problem into some sum for two-body problem
and compared the lower bound for the lagrangian action on test orbits with the
upper bound on collision set to avoid collisions under some cases. Motivated
by the works of A.Chenciner and R.Montgomery, C.Simó, C.Marchal, S.Q.Zhang and
Q.Zhou, K.C. Chen $([8]-[11])$ studied some planar $N$-body problems and got
some new planar non-collision periodic and quasi-periodic solutions.

The equations for the motion of the Newtonian $N$-body problem are:

$\displaystyle m_{i}\ddot{q}_{i}=\frac{\partial U(q)}{\partial q_{i}},\quad
i=1,\ldots,N,$ (1.1)

where $q_{i}\in\mathbb{R}^{k}$ denotes the position of $m_{i}$, and the
potential function is :

$U=\sum_{1\leq i<j\leq N}\dfrac{m_{i}m_{j}}{|q_{i}-q_{j}|}.$

It is well known that critical points of the action functional $f$:

$f(q)=\int_{0}^{T}(\frac{1}{2}\sum_{i=1}^{N}m_{i}|\dot{q}_{i}|^{2}+U(q))dt,\quad
q\in E,$ $None$

are $T$ periodic solutions of the $N$-body problem $(1.1)$,

where

$E=\\{q=(q_{1},q_{2},\ldots,q_{N})\,|\,q_{i}(t)\in
W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{k}),\,\sum_{i=1}^{N}m_{i}q_{i}(t)=0,\,q_{i}(t)\neq
q_{j}(t),\forall i\neq j,\forall t\in\mathbb{R}\\},$ $None$

$W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{k})=\\{x(t)\,|\,x(t)\in
L^{2}(\mathbb{R},\mathbb{R}^{k}),\dot{x}(t)\in
L^{2}(\mathbb{R},\mathbb{R}^{k}),x(t+T)=x(t)\\}.$ $None$

Definition 1.1 Let $\Gamma:x(t),\,t\in[a,\,b]$ be a given oriented continuous
closed curve, and $p$ a point of the plane, not on the curve. Then the mapping
$\varphi:\Gamma\rightarrow S^{1}$ given by

$\varphi(x(t))=\dfrac{x(t)-p}{|x(t)-p|},\quad t\in[a,b],$ $None$

is defined to be the position mapping of the curve $\Gamma$ relative to $p$.
When the point on $\Gamma$ goes around the given oriented curve once, its
image point $\varphi(x)$ will go around $S^{1}$ in the same direction with
$\Gamma$ a number of times. When moving counter-clockwise or clockwise, we set
the sign $+$ or $-$, and we denote it by $deg(\Gamma,\,\,p)$. If $p$ is the
origin, we denote it by $deg(\Gamma)$.

C.H.Deng and S.Q.Zhang $[20]$, X.Su and S.Q.Zhang $[33]$ studies periodic
solutions for a class of planar $N+2$-body problems, they defined the
following orbit spaces:

$\displaystyle\Lambda_{0}=\\{$ $\displaystyle q\in
E_{0}\,|\,q_{i}(t+\dfrac{T}{r})=O(\dfrac{2\pi}{r})q_{i}(t),\quad
i=1,\ldots,N+2;$ $\displaystyle
q_{i+1}(t)=q_{i}(t+\dfrac{T}{N}),\,\,i=1,\ldots,N-1,\,\,q_{1}(t)=q_{N}(t+\dfrac{T}{N});$
$\displaystyle q_{i}(t+\dfrac{T}{N})=q_{i}(t),\,\,i=N+1,\,N+2,\forall\,t>0\\}$
(1.6)

and

$\displaystyle\Lambda=\\{$ $\displaystyle q\in\Lambda_{0}\,|\,q_{i}(t)\neq
q_{j}(t),\,\forall i\neq j,\forall t\in\mathbb{R};$ $\displaystyle
deg(q_{i}(t)-q_{j}(t))=1,\,1\leq i\neq j\leq
N,deg(q_{N+1}(t)-q_{N+2}(t))=k_{1}\\},$ (1.7)

where

$E_{0}=\\{q=(q_{1},q_{2},\ldots,q_{N+2})\,|\,q_{i}(t)\in
W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{2}),\,\sum_{i=1}^{N+2}m_{i}q_{i}(t)=0\\},$
$None$ $O(\theta)=\left(\begin{array}[]{cc}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{array}\right).$

Motivated by their work, we consider $N+3$-body problems($N>3$, $N$ and $3$
are coprime), the equations of the motion are:

$m_{i}\ddot{q}_{i}(t)=\dfrac{\partial U(q)}{\partial
q_{i}},\,\,\,i=1,\,\ldots,\,N+3.$ $None$

We define the following orbit spaces :

$\displaystyle\Lambda_{1}=\\{q\in E_{1}\,|\,$ $\displaystyle
q_{i}(t+\frac{T}{r})=O(\frac{2\pi d}{r})q_{i}(t),\quad i=1,\,\ldots,\,N+3;$
$\displaystyle
q_{i+1}(t)=q_{i}(t+\frac{T}{N}),\,\,i=1,\,\ldots,\,N,\,\,q_{1}(t)=q_{N}(t+\frac{T}{N});$
$\displaystyle
q_{N+j}(t)=q_{N+j-1}(t+\frac{T}{3}),\,j=2,\,3,\,\,q_{N+1}(t)=q_{N+3}(t+\dfrac{T}{3});$
$\displaystyle q_{i}(t+\frac{T}{3})=q_{i}(t),\,i=1,\ldots,N;$ $\displaystyle
q_{j}(t+\frac{T}{N})=q_{j}(t),\,j=N+1,N+2,N+3\\},$ (1.10)

and

$\displaystyle\Lambda_{2}=\\{q\in\Lambda_{1}|$ $\displaystyle q_{i}(t)\neq
q_{j}(t),\forall i\neq j,\forall t\in R;$ $\displaystyle
deg(q_{i}(t)-q_{j}(t))=k_{1},1\leq i<j\leq N;$ $\displaystyle
deg(q_{i^{\prime}}(t)-q_{j^{\prime}}(t))=k_{2},N+1\leq
i^{\prime}<j^{\prime}\leq N+3\\},$

where

$E_{1}=\\{q=(q_{1},q_{2},\ldots,q_{N+3})|q_{i}(t)\in
W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{2}),\,\sum_{i=1}^{N+3}m_{i}q_{i}(t)=0\\}.$
$None$

Notice that $r,k_{1},k_{2},d$ satisfy the following compatible conditions:

$k_{1}=d(mod\,r)\,,k_{2}=d(mod\,r),k_{1}=3s_{1},k_{2}=Ns_{2},s_{1},s_{2}\in\mathbb{Z}.$
$None$

Since $N$ and $3$ are coprime, we have $(N,3)=1$. In this paper, we also
require $r$ and $3$ coprime, so $(r,3)=1$.

We get the following theorem:

Theorem 1.1 $(1)$ Consider the seven-body problems $(1.9)$ of equal masses,
for $r=7,\,k_{1}=3,\,k_{2}=-4,\,d=3$, then the global minimizer of $f$ on
$\bar{\Lambda}_{2}$ is a non-collision periodic solution of $(1.9)$.

$(2)$ Consider the eight-body problems $(1.9)$ of equal masses, for
$r=8,\,k_{1}=3,\,k_{2}=-5,\,d=3$, then the global minimizer of $f$ on
$\bar{\Lambda}_{2}$ is a non-collision periodic solution of $(1.9)$.

$(3)$ Consider the ten-body problems $(1.9)$ of equal masses, for
$r=10,\,k_{1}=3,\,k_{2}=-7,\,d=3$, then the global minimizer of $f$ on
$\bar{\Lambda}_{2}$ is a non-collision periodic solution of $(1.9)$.

##  2  Some Lemmas

######  Lemma 2.1.

(Eberlein-Shmulyan$[7]$) A Banach space $X$ is reflexive if and only if any
bounded sequence in $X$ has a weakly convergent subsequence.

######  Lemma 2.2.

$([7])$ Let $X$ be a real reflexive Banach space, $M\subset X$ is a weakly
closed subset, $f:M\rightarrow R$ is weakly semi-continuous.If $f$ is
coercive, that is, $f(x)\rightarrow+\infty$ as $\parallel
x\parallel\rightarrow+\infty$, then $f(x)$ attains its infimum on $M$.

######  Lemma 2.3.

$([30])$ Let $G$ be a group acting orthogonally on a Hilbert space $H$. Define
the fixed point space $F_{G}=\\{x\in H|g\cdot x=x,\forall g\in G\\}$, if $f\in
C^{1}(H,R)$ and satisfies $f(g\cdot x)=f(x)$ for any $g\in G$ and $x\in H$,
then the critical point of $f$ restricted on $F_{G}$ is also a critical point
of $f$ on $H$.

######  Lemma 2.4.

$([41])$ Let $q\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{n})$ and
$\int_{0}^{T}q(t)\,\mathrm{d}t=0$, then we have
$(i)$. Poincare-Wirtinger’s inequality:

$\int_{0}^{T}|\dot{q}(t)|^{2}\,\mathrm{d}t\geq{\Big{(}\frac{2\pi}{T}\Big{)}}^{2}\int_{0}^{T}|{q}(t)|^{2}\,\mathrm{d}t.$
$None$

$(ii)$. Sobolev’s inequality:

$\underset{0\leq t\leq T}{\max}|q(t)|=\parallel
q\parallel_{\infty}\leq\sqrt{\frac{T}{12}}\big{(}\int_{0}^{T}|\dot{q}(t)|^{2}\mathrm{d}t\big{)}^{1/2}.$
$None$

######  Lemma 2.5.

(Gordon[24])(1) Let $x(t)\in W^{1,2}([t_{1},t_{2}],R^{k})$ and
$x(t_{1})=x(t_{2})=0$, Then for any $a>0$, we have

$\int_{t_{1}}^{t_{2}}(\frac{1}{2}|\dot{x}|^{2}+\frac{a}{|x|})dt\geq\frac{3}{2}(2\pi)^{\frac{2}{3}}a^{\frac{3}{2}}(t_{2}-t_{1})^{\frac{1}{3}}.$
$None$

(2)(Long and Zhang$[26]$) Let $x(t)\in W^{1,2}(R/TZ,R^{k})$,
$\int_{0}^{T}xdt=0$, then for any $a>0$, we have

$\int_{0}^{T}(\frac{1}{2}|\dot{x}(t)|^{2}+\frac{a}{|x|})dt\geq\frac{3}{2}(2\pi)^{\frac{2}{3}}a^{\frac{3}{2}}T^{\frac{1}{3}}.$
$None$

##  3 Proof of Theorem 1.1

we consider the system $(1.9)$ of equal masses. Without loss of generality, we
suppose that the masses $m_{1}=m_{2}=\cdots=m_{N+3}=1$, and the period $T=1$.

Define $G=\mathbb{Z}_{r}\times\mathbb{Z}_{3}\times\mathbb{Z}_{N}$ and the
group action $g=\langle g_{1}\rangle\times\langle g_{2}\rangle\times\langle
g_{3}\rangle$ on the space $E_{1}$:

$\displaystyle g_{1}(q_{1}(t),\ldots,q_{N+3}(t))=(O(-\frac{2\pi
d}{r})q_{1}(t+\frac{1}{r}),\ldots,O(-\frac{2\pi d}{r})q_{N+3}(t+\frac{1}{r}))$
(3.1) $\displaystyle g_{2}(q_{1}(t),\ldots,q_{N+3}(t))$
$\displaystyle=(q_{1}(t+\frac{1}{3}),\ldots,q_{N}(t+\frac{1}{3}),q_{N+3}(t+\frac{1}{3}),q_{N+1}(t+\frac{1}{3}),q_{N+2}(t+\frac{1}{3}))$
(3.2) $\displaystyle g_{3}(q_{1}(t),\ldots,q_{N+3}(t))$
$\displaystyle=(q_{N}(t+\frac{1}{N}),q_{1}(t+\frac{1}{N})\ldots,q_{N-1}(t+\frac{1}{N}),q_{N+1}(t+\frac{1}{N}),q_{N+2}(t+\frac{1}{N}),q_{N+3}(t+\frac{1}{N}))$
(3.3)

This implies that $\Lambda_{1}$ is the fixed point space of $g$ on $E_{1}$.
Furthermore, for any $g_{i}$ and $q\in E_{1}$, we have $f(g_{i}\cdot q)=f(q)$
for $i=1,2,3$. Then the Palais symmetry principle implies that the critical
point of $f$ restricted on $\Lambda_{1}$ is also a critical point of $f$ on
$E_{1}$.

######  Lemma 3.1.

The critical point of minimizing the Lagrangian functional $f$ restricted on
$\Lambda_{2}$ (with winding number restriction) is also a critical point of
$f$ on $\Lambda_{1}$, then it is also the solution of $(1.9)$.

The proof is similar to that of Lemma $3.1$ in $[21]$, we omit it.

By $q_{i}(t)=O(-\dfrac{2\pi d}{r})q_{i}(t+\dfrac{1}{r})(i=1,\cdots,N+3)\,$, we
have

$\int_{0}^{1}q_{i}(t)dt=0.$

Then the Lemma $2.4$

$\int_{0}^{1}|\dot{q}_{i}(t)|^{2}dt\geq(2\pi)^{2}\int_{0}^{1}|q_{i}(t)|^{2}dt.$

Hence $f(q)$ is coercive on $\bar{\Lambda}_{2}$. It is easy to see that
$\bar{\Lambda}_{2}$ is a weakly closed subset.Fatou’s lemma implies that
$f(q)$ is a weakly lower semi-continuous. Then by Lemma $2.2$, $f(q)$ attains
$\inf{\\{f(q)|q\in\bar{\Lambda}_{2}\\}}$. Similar to Lemma $3.2$ in $[21]$, we
can obtain the following lemma.

######  Lemma 3.2.

The limit curve
$q(t)=(q_{1}(t),q_{2}(t),\ldots,q_{N+3}(t))\in\partial{\Lambda_{2}}$ of a
sequence
$q^{l}(t)=(q^{l}_{1}(t),q^{l}_{2}(t),\ldots,q^{l}_{N+3}(t))\in\Lambda_{2}$ may
either have collisions between some two point masses or has the same winding
number $(i.e.deg(q_{i}(t)-q_{j}(t))=k_{1},1\leq i\neq j\leq
N;deg(q_{i^{\prime}}(t)-q_{j^{\prime}}(t))=k_{2},N+1\leq i^{\prime}\neq
j^{\prime}\leq N+3).$

In the following, we prove that the minimizer of $f$ is a non-collision
solutions of the system $(1.9).$

Since $\sum_{i=1}^{N+3}q_{i}=0$, by the Lagrangian identity, we have

$f(q)=\frac{1}{N+3}\sum_{1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}-\dot{q}_{j}|^{2}+\frac{N+3}{|q_{i}-q_{j}|})dt$
$None$

Notice that each term on the right hand side of $(3.4)$ is a Lagrangian action
for a suitable two body problem, which is a key step for the lower bound
estimate on the collision set.

We estimate the infimum of the action functional on the collision set. Since
the symmetry for a two-body problem implies that the Lagrangian action on a
collision solution is greater than that on the non-collision solution, and the
more collisions there are, the greater the Lagrangian is. We only assume that
the two bodies collide at some moment $t_{0}$, without loss of generality, let
$t_{0}=0$, we will sufficiently use the symmetries of collision orbits.

since $q\in\bar{\Lambda}_{2}$, we have

$\displaystyle q_{i}(t+\dfrac{1}{r})=O(\dfrac{2\pi
d}{r})q_{i}(t),\,i=1,\ldots,N+3;$ (3.5) $\displaystyle
q_{i+1}(t)=q_{i}(t+\dfrac{1}{N}),i=1,\ldots,N-1,\,\,q_{1}(t)=q_{N}(t+\dfrac{1}{N});$
(3.6) $\displaystyle
q_{N+2}(t)=q_{N+1}(t+\dfrac{1}{3}),\,q_{N+3}(t)=q_{N+2}(t+\dfrac{1}{3}),\,q_{N+1}(t)=q_{N+3}(t+\dfrac{1}{3});$
(3.7) $\displaystyle q_{i}(t+\dfrac{1}{3})=q_{i}(t),\,i=1,\ldots,N;$ (3.8)
$\displaystyle q_{j}(t+\dfrac{1}{N})=q_{j}(t),\,j=N+1,N+2,N+3.$ (3.9)

Case $1$: $q_{1},q_{2}$ collide at $t=0$.

By $(3.5)$, we can deduce $q_{1},q_{2}$ collide at
$t=\dfrac{i}{r},\,i=0,\ldots,r-1.$

Furthermore, by $(3.8)$, we can deduce $q_{1},q_{2}$ collide at

$t=\dfrac{i}{r},\,\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3}\,(mod\,1).$
$None$

From $(3.6)$ and $(3.10)$, we have

$\displaystyle q_{2},q_{3}\,\text{collide
at}\,\dfrac{i}{r}+\dfrac{N-1}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{N-1}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{N-1}{N}(mod\,1),\,\,i=0,\ldots,r-1,$
$\displaystyle q_{3},q_{4}\,\text{collide
at}\,\dfrac{i}{r}+\dfrac{N-2}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{N-2}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{N-2}{N}(mod\,1),\,\,i=0,\ldots,r-1,$
$\displaystyle\vdots$ $\displaystyle q_{N-1},q_{N}\,\text{collide
at}\,\dfrac{i}{r}+\dfrac{2}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{2}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{2}{N}(mod\,1),\,i=0,\ldots,r-1,$
$\displaystyle q_{N},q_{1}\,\text{collide
at}\,\dfrac{i}{r}+\dfrac{1}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{1}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{1}{N}(mod\,1),\,i=0,\ldots,r-1.$

######  Lemma 3.3.

$\forall\,\,0\leq i,\,j\leq r-1,\,0\leq k\leq 2,\,\,(i-j)^{2}+k^{2}\neq 0$, we
have

$\dfrac{i}{r}\neq\dfrac{j}{r}+\dfrac{k}{3}(mod\,1)$ $None$

###### Proof.

If there exist $0\leq i_{0},j_{0}\leq r-1,0\leq k_{0}\leq
2,(i_{0}-j_{0})^{2}+k_{0}^{2}\neq 0$ such that

$\dfrac{i_{0}}{r}=\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}(mod\,1).$

Then we have

$1|(\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}).$

Since

$\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}\geq-\dfrac{r-1}{r}=-1+\dfrac{1}{r}>-1,$

and

$\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}\leq\dfrac{r-1}{r}+\dfrac{2}{3}<2,$

we can deduce

$\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=0$ or
$\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=1.$

If $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=0$, then
$3(i_{0}-j_{0})=k_{0}r$. When $k_{0}=0$, we get $i_{0}=j_{0}$, which is a
contradiction with our assumptions on the $i_{0},\,j_{0},\,k_{0}$; when
$k_{0}\neq 0$, notice $0<k_{0}\leq 2$, we can deduce $3|r$, which is a
contradiction since $(r,3)=1.$

If $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=1$, then
$3(j_{0}-i_{0})=(3-k_{0})r$. When $k_{0}=0$, we get $r=j_{0}-i_{0}$, which is
a contradiction since $-r+1\leq j_{0}-i_{0}\leq r-1$; when $k_{0}\neq 0$,
notice $1\leq 3-k_{0}<3$, we can deduce $3|r$, which is also a contradiction
since $(r,3)=1.$ ∎

By $(3.10)$ and Lemma $3.3$, we know that $q_{1},q_{2}$ collide at

$t_{i}=\dfrac{i}{3r},\,\,i=0,\ldots,\,3r-1.$ $None$

Then by Lemma $2.5$, $(3.12)$, we have

$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{1}(t)-\dot{q}_{2}(t)|^{2}+\dfrac{N+3}{|q_{1}(t)-q_{2}(t)|})dt$
$\displaystyle=\sum_{i=0}^{3r-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}|\dot{q}_{1}(t)-\dot{q}_{2}(t)|^{2}+\dfrac{N+3}{|q_{1}(t)-q_{2}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}3r(\dfrac{1}{3r})^{\frac{1}{3}}.$
(3.13)

From $(3.6)$ and $(3.12)$, we have

$\displaystyle q_{2},q_{3}\,\text{collide
at}\,\dfrac{i}{3r}+\dfrac{N-1}{N}(mod\,1),\,\,i=0,\ldots,3r-1,$ $\displaystyle
q_{3},q_{4}\,\text{collide
at}\,\dfrac{i}{3r}+\dfrac{N-2}{N}(mod\,1),\,\,i=0,\ldots,3r-1,$
$\displaystyle\vdots$ $\displaystyle q_{N-1},q_{N}\,\text{collide
at}\,\dfrac{i}{3r}+\dfrac{2}{N}(mod\,1),\,\,i=0,\ldots,3r-1,$ (3.14)
$\displaystyle q_{N},q_{1}\,\text{collide
at}\,\dfrac{i}{3r}+\dfrac{1}{N}(mod\,1),\,\,i=0,\ldots,3r-1.$ (3.15)

######  Lemma 3.4.

$\forall\,0\leq i,i^{\prime}\leq 3r-1,1\leq j,j^{\prime}\leq
N-1,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}\neq 0$, we have

$\dfrac{i}{3r}+\dfrac{j}{N}\neq\dfrac{i^{\prime}}{3r}+\dfrac{j^{\prime}}{N}(mod\,1).$
$None$

The proof is similar to Lemma $3.3$.

Remark 3.1 From Lemma $3.4$, $\forall\,0\leq i,i^{\prime}\leq r-1,\,\,1\leq
j,j^{\prime}\leq N-1,\,\,0\leq k,k^{\prime}\leq
2,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}+(k-k^{\prime})^{2}\neq 0$, we have

$\displaystyle\dfrac{i}{r}+\dfrac{j}{N}+\dfrac{k}{3}\neq\dfrac{i^{\prime}}{r}+\dfrac{j^{\prime}}{N}+\dfrac{k^{\prime}}{3}(mod\,1).$

By Lemma $2.5$, Lemma $3.4$ and $(3.15)$, we have

$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{j+1}(t)-\dot{q}_{j+2}(t)|^{2}+\dfrac{N+3}{|q_{j+1}(t)-q_{j+2}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}3r(\dfrac{1}{3r})^{\frac{1}{3}},\,\,\,(j=1,\ldots,N-2),$
(3.17)
$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{N}(t)-\dot{q}_{1}(t)|^{2}+\dfrac{N+3}{|q_{N}(t)-q_{1}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}3r(\dfrac{1}{3r})^{\frac{1}{3}}.$
(3.18)

Let

$\displaystyle M_{1}=\sum_{j=0}^{N-2}$
$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{j+1}(t)-\dot{q}_{j+2}(t)|^{2}+\dfrac{N+3}{|q_{j+1}(t)-q_{j+2}(t)|})dt+$
$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{N}(t)-\dot{q}_{1}(t)|^{2}+\dfrac{N+3}{|q_{N}(t)-q_{1}(t)|})dt.$

Then by $(3.13),(3.17),(3.18)$, Lemma $2.5$, and notice that $\forall\,1\leq
i\leq N,\,N+1\leq j\leq
N+3,\,\,\int_{0}^{\frac{1}{3}}q_{i}(t)dt=0,\,\,\int_{0}^{\frac{1}{N}}q_{j}(t)dt=0$,
so we have

$\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt$
$\displaystyle=\dfrac{1}{N+3}\\{\,\,M_{1}+[\sum_{1\leq i<j\leq
N}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt-
M_{1}\,]+$ $\displaystyle\qquad\qquad\sum_{1\leq i\leq N,1\leq j\leq
3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{N+j}(t)|})dt+$
$\displaystyle\qquad\qquad\sum_{N+1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt\,\,\\}$
$\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,N\times
3r(\frac{1}{3r})^{\frac{1}{3}}+3\times(\frac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-N)+3N+3N(\frac{1}{N})^{\frac{1}{3}}\,]$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}A.$ (3.19)

In the following cases, we firstly study the cases under $N$ is even.

Case $2$: $q_{1},q_{k+2}(k=1,\ldots,\dfrac{N}{2}-2)$ collide at $t=0$.

By $(3.5)$, we can deduce $q_{1},q_{k+2}(k=1,\ldots,\dfrac{N}{2}-2)$ collide
at $t=\dfrac{i}{r},\,i=0,\ldots,r-1.$

Then by $(3.8)$ , $q_{1},q_{k+2}$ collide at

$t=\dfrac{i}{r},\,\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3}\,(mod\,1),\,\,i=0,\cdots,r-1.$
$None$

From Lemma $3.3$, we get $q_{1},q_{k+2}$ collide at

$t=\dfrac{i}{3r},i=0,\ldots,3r-1.$ $None$

Then by $(3.8)$, we have

$\displaystyle q_{2},q_{k+3}\,\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{N-1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle
q_{3},q_{k+4}\,\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{N-2}{N}(mod\,1),\,i=0,\ldots,3r-1,$
$\displaystyle\vdots$ $\displaystyle q_{N-k-1},q_{N}\,\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{k+2}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle
q_{N-k},q_{1},\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{k+1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle
q_{N-k+1},q_{2}\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{k}{N}(mod\,1),\,i=0,\ldots,3r-1,$
$\displaystyle\vdots$ $\displaystyle q_{N},q_{k+1}\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{1}{N}(mod\,1),\,i=0,\ldots,3r-1.$ (3.22)

Then by Lemma $2.5$, Lemma $3.3$, Lemma $3.4$, $(3.21)-(3.22)$, we have

$\displaystyle f(q)$
$\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,N\times
3r(\frac{1}{3r})^{\frac{1}{3}}+3\times(\frac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-N)+3N+3N(\frac{1}{N})^{\frac{1}{3}}\,]$
$\displaystyle=A.$ (3.23)

Case 3: $q_{1},q_{\frac{N}{2}+1}$ collide at $t=0$.

By $(3.5),(3.6),\,\,(3.8)$, $q_{1},q_{\frac{N}{2}+1}$ collide at

$\displaystyle t=$
$\displaystyle\dfrac{i}{r},\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3},\,$
$\displaystyle\dfrac{i}{r}+\dfrac{\frac{N}{2}}{N},\,\frac{i}{r}+\dfrac{1}{3}+\dfrac{\frac{N}{2}}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{\frac{N}{2}}{N}(mod\,1),i=0,\ldots,r-1.$
(3.24)

Simplify $(3.24)$ , we get $q_{1},q_{\frac{N}{2}+1}$ collide at

$\displaystyle t=\dfrac{i}{r}+\dfrac{j}{6},\,i=0,\ldots,r-1,\,j=0,\ldots,5$
(3.25)

######  Lemma 3.5.

$\forall\,0\leq i,i^{\prime}\leq r-1,0\leq j,j^{\prime}\leq
5,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}\neq 0$, we have

$\dfrac{i}{r}+\dfrac{j}{6}\neq\dfrac{i^{\prime}}{r}+\dfrac{j^{\prime}}{6}(mod\,1)$
$None$

###### Proof.

If there exist $0\leq i_{0},i_{1}\leq r-1,\,0\leq j_{0},j_{1}\leq
5,(i_{0}-i_{1})^{2}+(j_{0}-j_{1})^{2}\neq 0$ such that

$\displaystyle\dfrac{i_{0}}{r}+\dfrac{j_{0}}{6}=\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}(mod1)$
(3.27)

Since

$\displaystyle\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\geq-\dfrac{r-1}{r}-\dfrac{5}{6}>-2,$
$\displaystyle\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\leq\dfrac{r-1}{r}+\dfrac{5}{6}<2,$

then we deduce

$\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=-1$ , or
$\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=0$, or
$\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=1$.

If $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=-1$,
we have $r(6+j_{1}-j_{0})=6(i_{0}-i_{1})$. When $i_{0}=i_{1}$, which is a
contradiction since $r(6+j_{1}-j_{0})\neq 0$ ; when $i_{0}\neq i_{1}$ and
$j_{0}=j_{1}$ , we can deduce $r=i_{0}-i_{1}$, which is a contradiction since
$-r+1\leq i_{0}-i_{1}\leq r-1$; when $i_{0}\neq i_{1}$ and $j_{0}\neq j_{1}$,
we can deduce $6|r$, which is a contradiction since $(r,3)=1$.

We can use similar arguments to prove
$\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\neq 0$
and $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\neq
1$. ∎

From $(3.25)$ and $(3.26)$, we can deduce $q_{1},q_{\frac{N}{2}+1}$ collide at

$t_{i}=\dfrac{i}{6r},\,\,r=0,\ldots,6r-1.$ $None$

Then by Lemma $2.5$ and $(3.28)$, we have

$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{1}(t)-\dot{q}_{\frac{N}{2}+1}(t)|^{2}+\dfrac{N+3}{|q_{1}(t)-q_{\frac{N}{2}+1}(t)|})dt$
$\displaystyle=\sum_{i=0}^{6r-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}|\dot{q}_{1}(t)-\dot{q}_{\frac{N}{2}+1}(t)|^{2}+\dfrac{N+3}{|q_{1}(t)-q_{\frac{N}{2}+1}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}6r(\dfrac{1}{6r})^{\frac{1}{3}}.$
(3.29)

By $(3.6)$, $(3.28)$, we have

$\displaystyle q_{2},q_{\frac{N}{2}+2},\,\text{collide
at}\,t=\dfrac{i}{6r}+\dfrac{\frac{N}{2}-1}{N},\,i=0,\ldots,6r-1,$
$\displaystyle q_{3},q_{\frac{N}{2}+3},\,\text{collide
at}\,t=\dfrac{i}{6r}+\dfrac{\frac{N}{2}-2}{N},\,i=0,\ldots,6r-1,$
$\displaystyle\vdots$ $\displaystyle q_{\frac{N}{2}},q_{N}\,\text{collide
at}\,t=\dfrac{i}{6r}+\dfrac{1}{N},\,i=0,\ldots,6r-1.$ (3.30)

######  Lemma 3.6.

$\forall\,0\leq i,i^{\prime}\leq 6r-1,1\leq
j,j^{\prime}\leq\dfrac{N}{2}-1,\,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}\neq 0$,
we have

$\dfrac{i}{6r}+\dfrac{j}{N}\neq\dfrac{i^{\prime}}{6r}+\dfrac{j^{\prime}}{N}.$
$None$

The proof is similar to Lemma $3.5$.

By Lemma $2.5$, Lemma $3.6$, $(3.30)-(3.31)$, we have

$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{j+1}(t)-\dot{q}_{\frac{N}{2}+j+1}(t)|^{2}+\dfrac{N+3}{|q_{j+1}(t)-q_{\frac{N}{2}+j+1}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}6r(\dfrac{1}{6r})^{\frac{1}{3}}\quad(j=1,\ldots,\frac{N}{2}-1).$
(3.32)

Let

$\displaystyle
M_{2}=\sum_{j=0}^{\frac{N}{2}-1}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{j+1}(t)-\dot{q}_{\frac{N}{2}+j+1}(t)|^{2}+\frac{N+3}{|q_{j+1}(t)-q_{\frac{N}{2}+j+1}(t)|})dt$

Then from Lemma $2.5$, Lemma $3.6$, $(3.29)$ and $(3.32)$, we obtain

$\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt$
$\displaystyle=\dfrac{1}{N+3}\\{\,\,M_{2}+[\sum_{1\leq i<j\leq
N}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt-
M_{2}]+$ $\displaystyle\quad\qquad\sum_{1\leq i\leq N,1\leq j\leq
3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{N+j}(t)|})dt+$
$\displaystyle\quad\qquad\sum_{N+1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt\,\,\\}$
$\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,\dfrac{N}{2}\times
6r(\frac{1}{6r})^{\frac{1}{3}}+3\times(\frac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-\dfrac{N}{2})+3N+3N(\frac{1}{N})^{\frac{1}{3}}\,]$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}B.$ (3.33)

Finally, we study the cases under $N$ is odd.

Case $2^{\prime}$: $q_{1},q_{k+2}(k=1,\ldots,{\frac{N+1}{2}}-2)$ collide at
$t=0$.

By $(3.5),(3.8)$, $q_{1},q_{k+2}(k=1,\ldots,\frac{N+1}{2}-2)$ collide at

$t=\dfrac{i}{r},\,\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3}(mod\,1),\,i=0,\ldots,r-1,$
$None$

from Lemma $3.3$, we get $q_{1},q_{k+2}(k=1,\ldots,\frac{N+1}{2}-2)$ collide
at

$t=\dfrac{i}{3r},\,\,i=0,\ldots,3r-1,$ $None$

then by $(3.6)$, we have

$\displaystyle q_{2},q_{k+3}\,\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{N-1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle
q_{3},q_{k+4}\,\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{N-2}{N}(mod\,1),\,i=0,\ldots,3r-1,$
$\displaystyle\vdots$ $\displaystyle q_{N-k-1},q_{N}\,\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{k+2}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle
q_{N-k},q_{1},\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{k+1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle
q_{N-k+1},q_{2}\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{k}{N}(mod\,1),\,i=0,\ldots,3r-1,$
$\displaystyle\vdots$ $\displaystyle q_{N},q_{k+1}\text{collide
at}\,t=\dfrac{i}{3r}+\dfrac{1}{N}(mod\,1),\,i=0,\ldots,3r-1.$ (3.36)

Then by Lemma $2.5$, Lemma $3.4$, $(3.35),(3.36)$, we have

$\displaystyle f(q)\geq$
$\displaystyle\dfrac{3}{2}\times(\dfrac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,N\times
3r(\dfrac{1}{3r})^{\frac{1}{3}}+3\times(\dfrac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-N)+3N+3N(\dfrac{1}{N})^{\frac{1}{3}}\,]$
$\displaystyle=A.$ (3.37)

Case $4$: $q_{N+1},q_{1}$ collide at $t=0$.

By $(3.5)$, we have

$q_{N+1},q_{1}$ collide at

$t=\dfrac{i}{r},\quad i=0,\ldots,r-1.$ $None$

Then by Lemma $2.5$, $(3.37)$, we have

$\displaystyle\int_{0}^{1}$
$\displaystyle(\dfrac{1}{2}|\dot{q}_{1}(t)-\dot{q}_{N+1}(t)|^{2}+\dfrac{N+3}{|q_{1}(t)-q_{N+1}(t)|})dt$
$\displaystyle=\sum_{i=0}^{r-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}|\dot{q}_{1}(t)-\dot{q}_{N+1}(t)|^{2}+\dfrac{N+3}{|q_{1}(t)-q_{N+1}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}r(\dfrac{1}{r})^{\frac{1}{3}}.$
(3.39)

From $(3.38),(3.5)-(3.9)$, we can obtain

$q_{N+2},q_{1},$ collide at $t=\dfrac{i}{r}+\dfrac{2}{3}(mod\,1)$,
$q_{N+3},q_{1}$ collide at
$t=\dfrac{i}{r}+\dfrac{1}{3}(mod\,1),\,\,i=0,\ldots,r-1,$

$q_{N+1},q_{2}$ collide at $\dfrac{i}{r}+\dfrac{N-1}{N}(mod\,1)$,
$q_{N+2},q_{2}$ collide at $\dfrac{i}{r}+\dfrac{N-1}{N}+\dfrac{2}{3}(mod\,1)$,
$q_{N+3},q_{2}$ collide at
$\dfrac{i}{r}+\dfrac{N-1}{N}+\dfrac{1}{3}(mod\,1),\,i=0,\ldots,r-1,$

⋮

$q_{N+1},q_{N-1}$ collide at $\dfrac{i}{r}+\dfrac{2}{N}(mod\,1)$,
$q_{N+2},q_{N-1}$ collide at $\dfrac{i}{r}+\dfrac{2}{N}+\dfrac{2}{3}(mod\,1)$,
$q_{N+3},q_{N-1}$ collide at
$\dfrac{i}{r}+\dfrac{2}{N}+\dfrac{1}{3}(mod\,1),\,i=0,\ldots,r-1,$

$q_{N+1},q_{N}$ collide at $\dfrac{i}{r}+\dfrac{1}{N}(mod\,1)$,
$q_{N+2},q_{N}$ collide at $\dfrac{i}{r}+\dfrac{1}{N}+\dfrac{2}{3}(mod\,1)$,
$q_{N+3},q_{N}$ collide at
$\dfrac{i}{r}+\dfrac{1}{N}+\dfrac{1}{3}(mod\,1),\,i=0,\ldots,r-1.$

Then by Lemma $2.5$, Lemma $3.3$, Remark $3.1$, we have $\forall\,0\leq i\leq
r-1,1\leq j\leq 3,$

$\displaystyle\int_{0}^{1}$
$\displaystyle(\dfrac{1}{2}|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)|^{2}+\dfrac{N+3}{|q_{i}(t)-q_{N+j}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}r(\dfrac{1}{r})^{\frac{1}{3}}.$
(3.40)

So we get

$\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt$
$\displaystyle=\dfrac{1}{N+3}(\,\,\sum_{\stackrel{{\scriptstyle 1\leq i\leq
N}}{{1\leq j\leq
3}}}\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)|^{2}+\dfrac{N+3}{|q_{i}(t)-q_{N+j}(t)|})dt+$
$\displaystyle\qquad\qquad\sum_{1\leq i<j\leq
N}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt+$
$\displaystyle\qquad\qquad\sum_{N+1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt\,\,)$
$\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,3N\times
r(\dfrac{1}{r})^{\frac{1}{3}}+3\times(\dfrac{1}{3})^{\frac{1}{3}}C_{N}^{2}+3N(\dfrac{1}{N})^{\frac{1}{3}}\,]$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}C.$ (3.41)

Case $5$: $q_{N+1},q_{N+2}$ collide at $t=0$.

Then by $(3.5),(3.9)$, we deduce

$q_{N+1},q_{N+2}$ collide at

$\displaystyle t=\dfrac{i}{r}+\dfrac{j}{N}(mod\,1),\,i=0,\ldots
r-1,\,j=0,\ldots,N-1.$ (3.42)

From Remark $3.1$, and $(3.42)$, we can deduce $q_{N+1},q_{N+2}$ collide at

$t_{i}=\dfrac{i}{Nr},\quad i=0,\ldots,Nr-1.$ $None$

Then we have

$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{N+1}(t)-\dot{q}_{N+2}(t)|^{2}+\dfrac{N+3}{|q_{N+1}(t)-q_{N+2}(t)|})dt$
$\displaystyle=\sum_{i=0}^{Nr-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}|\dot{q}_{N+1}(t)-\dot{q}_{N+2}(t)|^{2}+\dfrac{N+3}{|q_{N+1}(t)-q_{N+2}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}Nr(\dfrac{1}{Nr})^{\frac{1}{3}}.$
(3.44)

By$(3.7)$, we deduce $q_{N+2},q_{N+3}$, collide at

$\displaystyle t=\dfrac{i}{Nr}+\dfrac{2}{3},\quad i=0,\ldots,Nr-1,$ (3.45)

$q_{N+3},q_{N+1}$ collide at

$\displaystyle t=\dfrac{i}{Nr}+\dfrac{1}{3},\quad i=0,\ldots,Nr-1.$ (3.46)

Then by Lemma $2.5$, Remark $3.1$, $(3.45)$, and $(3.46)$, we have

$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{N+2}(t)-\dot{q}_{N+3}(t)|^{2}+\dfrac{N+3}{|q_{N+2}(t)-q_{N+3}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}Nr(\dfrac{1}{Nr})^{\frac{1}{3}}$
(3.47)
$\displaystyle\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{N+3}(t)-\dot{q}_{N+1}(t)|^{2}+\dfrac{N+3}{|q_{N+3}(t)-q_{N+1}(t)|})dt$
$\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}Nr(\dfrac{1}{Nr})^{\frac{1}{3}}.$
(3.48)

So, we obtain

$\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt$
$\displaystyle=\dfrac{1}{N+3}(\sum_{N+1\leq i<j\leq
N+3}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt+$
$\displaystyle\qquad\qquad\qquad\sum_{\stackrel{{\scriptstyle 1\leq i\leq
N}}{{1\leq j\leq
3}}}\int_{0}^{1}(\dfrac{1}{2}|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)|^{2}\dfrac{N+3}{|q_{i}(t)-q_{N+j}(t)|})dt+$
$\displaystyle\qquad\qquad\qquad\sum_{1\leq i<j\leq
N}\int_{0}^{1}(\frac{1}{2}\,|\dot{q}_{i}(t)-\dot{q}_{j}(t)|^{2}+\frac{N+3}{|q_{i}(t)-q_{j}(t)|})dt)$
$\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,3\times
Nr(\dfrac{1}{Nr})^{\frac{1}{3}}+3\times(\dfrac{1}{3})^{\frac{1}{3}}C_{N}^{2}+3N\,]$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}D.$ (3.49)

When $N$ is odd, let $\tilde{A}=\inf{\\{A,\,C,\,D\\}}$, then on the collision
set, the action functional $f\geq\tilde{A}$.

When $N$ is even, let $\tilde{B}=\inf{\\{A,\,B,\,C,\,D\\}}$, then on the
collision set, the action functional $f\geq\tilde{B}$.

(1)Take $N=4,d=3,r=7,k_{1}=3,k_{2}=-4$.

We choose the following function as the test function:

Let $a>0,\,\,b>0$, and

$\displaystyle q_{i}=a(\cos{(6\pi t+\dfrac{2\pi(i-1)}{4})},\sin{(6\pi
t+\dfrac{2\pi(i-1)}{4})}\,),\,\,i=1,\ldots,4,$ $\displaystyle
q_{j}=b(\cos{(-8\pi t+\dfrac{2\pi(j-5)}{3})},\sin{(-8\pi
t+\dfrac{2\pi(j-5)}{3})}),\,\,j=5,6,7.$

We choose $a=0.2300,\,\,b=0.0880$,then

$\displaystyle A\approx 144.6215,\,B\approx 138.9586,\,$ $\displaystyle
C\approx 170.7479,\,\,D\approx 139.2196,\,\,\tilde{B}=138.9586,$
$\displaystyle f(q)\approx 135.5123<\tilde{B}.$

This proves that the minimizer of $f(q)$ on the closure $\bar{\Lambda}_{2}$ is
a non-collision solution of the seven-body problem.

(2)Take $N=5,d=3,r=8,k_{1}=3,k_{2}=-5$.

We choose the following function as the test function:

Let $a>0,\,\,b>0$, and

$\displaystyle q_{i}=a(\cos{(6\pi t+\dfrac{2\pi(i-1)}{5})},\,\sin{(6\pi
t+\dfrac{2\pi(i-1)}{5})}\,),\,\,i=1,\ldots,5,$ $\displaystyle
q_{j}=b(\cos{(-10\pi t+\dfrac{2\pi(j-6)}{3})},\,\sin{(-10\pi
t+\dfrac{2\pi(j-6)}{3})}),\,\,j=6,7,8.$

We choose $a=0.2450,\,\,b=0.0760$, then

$\displaystyle A\approx 193.5057,\,\,$ $\displaystyle C\approx
181.0305,\,\,D\approx 228.7437,\,\,\tilde{A}=181.0305,$ $\displaystyle
f(q)\approx 175.2312<\tilde{A}.$

This proves that the minimizer of $f(q)$ on the closure $\bar{\Lambda}_{2}$ is
a non-collision solution of the eight-body problem.

(3)Take $N=7,d=3,r=10,k_{1}=3,k_{2}=-7$.

We choose the following function as the test function:

Let $a>0,\,b>0$, and

$\displaystyle q_{i}=a(\cos{(6\pi t+\dfrac{2\pi(i-1)}{7})},\sin{(6\pi
t+\dfrac{2\pi(i-1)}{7})}\,),\,\,i=1,\ldots,7,$ $\displaystyle
q_{j}=b(\cos{(-14\pi t+\dfrac{2\pi(j-8)}{3})},\sin{(-14\pi
t+\dfrac{2\pi(j-8)}{3})}),\,\,j=8,9,10.$

We choose $a=0.2500,\,\,b=0.0640$, then

$\displaystyle A\approx 305.0645,\,\,$ $\displaystyle C\approx
274.1354,\,\,D\approx 360.6557,\,\,\tilde{A}=274.1354,$ $\displaystyle
f(q)\approx 266.6297<\tilde{A}.$

This proves that the minimizer of $f(q)$ on the closure $\bar{\Lambda}_{2}$ is
a non-collision solution of the ten-body problem.

## References

  * [1] G.Arioli1, V. Barutello, S.Terracini,A new branch of mountain pass solutions for the choreographical 3-Body problem, Commun. Math. Phys. 268(2006), 439-463.
  * [2] G. Arioli, F.Gazzola and S.Terracini, Minimization properties of Hill’s orbits and application to some N-body problems, Ann.Inst.Henri Poincaré Anal. Nonlineairé 17(2000), 617-650.
  * [3] A.Bahari and P.Rabinowitz, Periodic solutions of Hamiltonian systems of three body type, Ann.Inst.Henri Poincaré Anal. Nonlinéaire 8(1991),561-649.
  * [4] V.Barutello and S.Terracini, Action minimizing orbits in the N-body problem with simple choreography constraint, Nonlinearity 17(2004), 2015-2039.
  * [5] V.Barutello, D.Ferrario, and S. Terracini, Symmetry groups of the planar three-body problem and action-minimizing trajectories, Arch.Rational Mech.Anal. 190(2008), 189-226.
  * [6] U.Bessi and V.Coti Zelati, Symmetries and noncollision closed orbits for planar N-body-type problems, Nonlinear Anal. 16(1991), 587-598.
  * [7] G.Buttazzo and M.Giaquinta and S.Hildebrandt, One-dimensional variational problems, Oxford University Press, 1998.
  * [8] K.C.Chen, Action minimizing orbits in the parellelogram four-body problem with equal masses, Arch.Rational Mech.Anal. 158(2001),293-318.
  * [9] K.C.Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions, Arch.Rational Mech.Anal. 170(2003),247-276.
  * [10] K.C.Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problems, Ergodic Theory and Dynamical Systems, 23 (2003), 1691-1715.
  * [11] K.C.Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math, 167(2008), 325-348.
  * [12] A.Chenciner, Action minimizing solutions of the Newtonian n-body problem, From homology to symmetry, ICM 2002, Vol.3, 279-294, Vol.1, 641-643.
  * [13] A.Chenciner, Collisions totales, Mouvements complètement paraboliques et réduction des homothéties dans le problème des n corps, Regular and chaotic dynamics V.3, 3(1998), 93-106.
  * [14] A. Chenciner, Simple non-planar periodic solutions of the n-body problem, In Proceedings of the NDDS Conference, Kyoto, 2002.
  * [15] A.Chenciner and N.Desolneux, Minima de l’intégrale d’action et équilibres relatifs de n corps, C.R.Acad.Sci.Paris, 327 séries I(1998), 193.
  * [16] A.Chenciner and R.Montgomery, A remarkable periodic solutions of the three-body problem in the case of equal masses, Annals of Math, 152(2000), 881-901.
  * [17] A.Chenciner and A.Venturelli, Minima de l’intégrale d’action du problème newtonien de $4$ corps de masses égales dans $\mathbb{R}^{3}$ : orbites “ hip-hop”, Celestial Mechanics, 77(2000), 139-152.
  * [18] V.Coti Zelati, The periodic solutions of N-body type problems, Ann. Inst. H.Poincaré Anal. Nonlineairé 7(1990), 477-492.
  * [19] M.Degiovanni and F.Giannoni, Dynamical systems with Newtonian type potentials, Ann. Sc. Norm. Sup. Pisa 15(1989), 467-494.
  * [20] C.H.Deng, S.Q.Zhang, New periodic solutions for $N+2$ body problem,Journal of Geometry and Physics, 61(2011), 2369-2377.
  * [21] C.H.Deng, S.Q.Zhang and Q.Zhou, Rose solutions with three petals for planar 4-body problems, Sci.China Math, 53(2010), 3085-3094.
  * [22] D.Ferrario and S.Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invention Math. 155(2004),305-362.
  * [23] D.Ferrario, Transitive decomposition of symmetry groups for the n-body problem, Advances in Mathematics 213(2007), 763-784.
  * [24] W.B.Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99(1977), 961-971.
  * [25] W.B.Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204(1975), 113-135.
  * [26] Y.M.Long and S.Q.Zhang, Geometric characterizations for variational minimization solutions of the 3-body problems, Act Math. Sinica 16(2000), 579-592.
  * [27] C.Marchal, How the method of minimization of action avoids singularities, Cel.Mech.Dyn.Astr.83(2002),325-353.
  * [28] R.Montgomery, The N-body problem , the braid group, and action-minimizing periodic solutions, Nonlinearity 11(1998), 363-376.
  * [29] C.Moore, Braids in classical gravity, Phys, Rev.Lett.70(1993), 3675-3679.
  * [30] R.Palais, The principle of symmetric criticality, Comm. Math. Phys. 69(1979),19-30.
  * [31] C.Simó, Dynamical properties of the figure eight solution of the three-body problem, Contemp.Math. 292 AMS.Providence,RI(2002), 209-228.
  * [32] C.Simó,New families of solutions in N-body problems, Progress Math. 21(2001),101-115.
  * [33] X.Su, S.Q.Zhang, New periodic solutions for planar five-body and seven-body problems, Reports on Mathematical Physics 70(2012), 27-38.
  * [34] S. Terracini and A.Venturelli, Symmetric trajectories for the 2N-body problem with equal masses, Arch. Rational Mech. Anal. 184 (2007), 465-493.
  * [35] A.Venturelli, Une caractérisation variationnelle des solutions de Lagrange du problème plan des trois corps, C.R. Acad. Sci. Paris 332(2001), 641-644.
  * [36] S.Q.Zhang, periodic soluitons of N-body problems, in Progress in Nonlinear Analysis ed. by K.C.Chang and Y.M.Long, World Scientific, 2000, 423-443.
  * [37] S.Q.Zhang and Q.Zhou, A minimizing property of Lagrangian solutions, Acta Math. Sinica 17(2001), 497-500.
  * [38] S.Q.Zhang and Q.Zhou, Variational methods for the choregraphy solution to the three-body problem, Sci.China 45(2002), 594-597.
  * [39] S.Q.Zhang and Q.Zhou, Nonplanar and noncollision periodic solutions for N-body problems, Disc. Cont. Dyn.Syst. 10(2004),679-685.
  * [40] S.Q.Zhang and Q.Zhou and Y.Liu, New periodic solutions for 3-body problems, Cel.Mech.Dyn.Astr. 88(2004), 365-378.
  * [41] W.P.Ziemer, Weakly differentiable functions, Springer, 1989.