# regularizable collinear periodic solutions in the $n$-body problem with
arbitrary masses

Guowei Yu  Chern Institute of Mathematics and LPMC, Nankai University,
Tianjin, China<EMAIL_ADDRESS>
###### Abstract.

For $n$-body problem with arbitrary positive masses, we prove there are
regularizable collinear periodic solutions for any ordering of the masses,
going from a simultaneous binary collision to another in half of a period with
half of the masses moving monotonically to the right and the other half
monotonically to the left. When the masses satisfy certain equality condition,
the solutions have extra symmetry. This gives a new proof of the Schubart
orbit, when $n=3$.

This work is supported by the National Key R&D Program of China
(2020YFA0713303), NSFC (No. 12171253), the Fundamental Research Funds for the
Central Universities and Nankai Zhide Fundation.

##  1\. Introduction

The first solutions ever found in the three body problem were the homothetic
Euler solutions discovered by Euler [7] in 1767. In these solutions, the three
masses come out of a total collision in any ordering, expanding monotonically
along a straight line until their velocities becoming zero simultaneously.
Then they shrink monotonically along the same straight line until reaching
another total collision. Later similar collinear homothetic solutions were
found by Moulton [15] in 1910 for arbitrary $n\geq 3$.

Since collisions with more than two bodies are not regularizable (see [12]),
there is no meaningful way to continue these homothetic solutions before or
after the total collisions. Meanwhile simultaneous binary collisions111A
simultaneous binary collision is a collision contianing only binary
collisions. For simplicity, the same name will be used, even if it only
contains a single binary collision. are known to be regularizable (intuitively
solutions are extended through these collisions by elastic bounces, see [9],
[18], [5] and [6]), so we may ask if there are collinear solutions going from
one simultaneous binary collision to another, and if they can be extended
through these collisions to obtain regularizable periodic solutions.

The first solution of this type was found numerically by Schubart [16] in 1956
for $n=3$ with two equal masses(see Remark 1.2 and Figure 2), and later proven
by Moeckel [14] and Venturelli [19] using different methods(for cases with
unequal masses, see [17] and [20]).

In this paper, we show such solutions existence for any $n\geq 3$ with
arbitrary choice of masses and ordering (see Figure 1 for an illuminating
picture).

######  Theorem 1.1.

Let $\mathcal{S}_{\mathbf{n}}$ be the permutation group of the index set
$\mathbf{n}=\\{1,2,\dots,n\\}$ with $n\geq 3$. For any choice of positive
masses, time $T>0$ and permutation $\sigma\in\mathcal{S}_{\mathbf{n}}$, there
is a regularizable collinear $2T$-periodic solution
$x(t)=(x_{i}(t))_{i=1}^{n}\in\mathbb{R}^{n}$ of the $n$-body problem
satisfying the following.

  1. (a).

For any $t\in\mathbb{R}$, $x(t)=x(t+2T)=x(2T-t)\in\mathbb{R}^{n}$.

  2. (b).

For any $t\in(0,T)$, $x(t)$ is collision-free with $\dot{x}_{\sigma(i)}(t)>0$
for odd $i$, $\dot{x}_{\sigma(i)}(t)<0$ for even $i$.

  3. (c).

When $n$ is odd, $x(0)$ and $x(T)$ each contain $\frac{1}{2}(n-1)$ pairs of
binary collision as

$x_{\sigma(1)}(0)<x_{\sigma(2)}(0)=x_{\sigma(3)}(0)<\cdots<x_{\sigma(2j)}(0)=x_{\sigma(2j+1)}(0)<\cdots<x_{\sigma(n-1)}(0)=x_{\sigma(n)}(0);$
$x_{\sigma(1)}(T)=x_{\sigma(2)}(T)<\cdots<x_{\sigma(2j-1)}(T)=x_{\sigma(2j)}(T)<\cdots<x_{\sigma(n-2)}(T)=x_{\sigma(n-1)}(T)<x_{\sigma(n)}(T).$

  4. (d).

When $n$ is even, $x(0)$ contains $\frac{1}{2}(n-2)$ pairs of binary collision
and $x(T)$ contains $\frac{1}{2}n$ pairs of binary collision as

$\displaystyle x_{\sigma(1)}(0)$
$\displaystyle<x_{\sigma(2)}(0)=x_{\sigma(3)}(0)<\cdots<x_{\sigma(2j)}(0)=x_{\sigma(2j+1)}(0)$
$\displaystyle<\cdots<x_{\sigma(n-2)}(0)=x_{\sigma(n-1)}(0)<x_{\sigma(n)}(0);$
$x_{\sigma(1)}(T)=x_{\sigma(2)}<\cdots<x_{\sigma(2j-1)}(T)=x_{\sigma(2j)}(T)<\cdots<x_{\sigma(n-1)}(T)=x_{\sigma(n)}(T).$

  5. (e).

$\dot{x}_{\sigma(1)}(0)=\dot{x}_{\sigma(n)}(T)=0$, when $n$ is odd,
$\dot{x}_{\sigma(1)}(0)=\dot{x}_{\sigma(n)}(0)=0$, when $n$ is even. Moreover

$\lim_{t\to
0}\frac{m_{\sigma(2j)}\dot{x}_{\sigma(2j)}(t)+m_{\sigma(2j+1)}\dot{x}_{\sigma(2j+1)}(t)}{m_{\sigma(2j)}+m_{\sigma(2j+1)}}=0,\;\forall
1\leq j\leq[(n-1)/2];$ $\lim_{t\to
T}\frac{m_{\sigma(2j-1)}\dot{x}_{\sigma(2j-1)}(t)+m_{\sigma(2j)}\dot{x}_{\sigma(2j)}(t)}{m_{\sigma(2j-1)}+m_{\sigma(2j)}}=0,\;\forall
1\leq j\leq[n/2].$

  6. (f).

$x(t)$ is $C^{1}$-block regularizable in the collinear $n$-body problem, and
$C^{0}$-block regularizable in the planar or spatial $n$-body problem.

######  Remark 1.1.

For the definition of block regularization, see Definition 2.2 and 2.3.

Figure 1.  above $n$ is odd, below $n$ is even; $\sigma=\text{Id}$; black,
$t=0$; gray, $t=T$.

######  Theorem 1.2.

For any $T>0$ and $\sigma\in\mathcal{S}_{\mathbf{n}}$, if the masses satisfy

(1) $m_{\sigma(i)}=m_{\sigma(n+1-i)},\;\;\forall i\in\mathbf{n},$

then there is a regularizable collinear $2T$-periodic solution
$x(t)=(x_{i}(t))_{i\in\mathbf{n}}\in\mathbb{R}^{n}$ satisfying all the
properties listed in Theorem 1.1, and for any $i\in\mathbf{n}$,

  1. (g).

$x_{\sigma(i)}(t)=-x_{\sigma(n+1-i)}(T-t)$, when $n$ is odd,
$x_{\sigma(i)}(t)=-x_{\sigma(n+1-i)}(t)$, when $n$ is even.

######  Remark 1.2.

When $n=3$ and $m_{1}=m_{3}$, Theorem 1.2 gives an alternative proof of the
Schubart orbit, as the corresponding solution going from a binary collision
between $m_{1}$ and $m_{2}$ to a binary collision between $m_{2}$ and $m_{3}$
during half a period with the masses forming an Euler configuration at a
quater of a period ($x_{1}(T/2)=-x_{3}(T/2),x_{2}(T/2)=0$, see Figure 2).

Figure 2. Schubart orbit with $m_{1}=m_{3}$; black, t=0; blurred, t=T/2; gray,
t=T.

By Newton’s universal gravitational law, the trajectories
$x(t)=(x_{i}(t))_{i\in\mathbf{n}}\in\mathbb{R}^{nd}$ of $n$ point masses,
$m_{i}>0$, $i\in\mathbf{n}$, moving in $\mathbb{R}^{d}$($d\geq 1$), satisfy
the following equation

(2)
$m_{i}\ddot{x}_{i}=-\sum_{j\in\mathbf{n}\setminus\\{i\\}}\frac{m_{i}m_{j}(x_{i}-x_{j})}{|x_{i}-x_{j}|^{3}},\;\;i\in\mathbf{n}.$

When $d=1$, it is called the collinear $n$-body problem, as all the masses are
always moving in a straight line. Similarly when $d=2$ or $3$, it is called
the planar or spatial $n$-body problem. Since the collinear problem is an
invariant sub-system of the higher dimension problem, knowledges of it will be
crucial in understanding the general problem, as shown by Moeckel in [13].

Our proof is variational in nature. It is well-known solutions of (2) can be
found as critical points of the following action functional

(3)
$\mathcal{A}(x;t_{1},t_{2})=\int_{t_{1}}^{t_{2}}L(x,\dot{x})\,dt=\int_{t_{1}}^{t_{2}}K(\dot{x})+U(x)\,dt,$

where

$K(\dot{x})=\frac{1}{2}\sum_{i\in\mathbf{n}}m_{i}|\dot{x}_{i}|^{2},\;\;U(x)=\sum_{\\{i<j\\}\subset\mathbf{n}}\frac{m_{i}m_{j}}{|x_{i}-x_{j}|}.$

For a long time very few results were obtained in the Newtonian $n$-body
problem using variational methods, because of singularities due to collision.
We denote the set of _collision configurations_ as

$\Delta=\\{x=(x_{i})_{i=1}^{n}\in\mathbb{R}^{nd}:x_{i}=x_{j},\;\text{ for some
}i\neq j\\}.$

The breakthrough is the proof of the famous Figure-Eight solution in the equal
mass three body problem by Chenciner and Montgomery [2]. After that many new
periodic solutions have been found and proved using action minimization
methods. The main difficulty is always to show the corresponding minimizers
are collision-free. It is impossible to give a complete list of the
literature, we refer the readers to [8], [1], [21] and the references within.

So far almost all the periodic solutions were found as collision-free
minimizers under different symmetric constraints. Because of this, certain
equality condition (13) needs to be satisfied by the masses. Up to our
knowledge, the only result that holds for unequal masses is the proof of the
retrograde orbits in the three body problem by Chen [1] (however the masses
still need to satisfy certain conditions, which do not hold for any choice of
masses). From this point of view, Theorem 1.1 is the first result that
establishes the existence of periodic solutions for any $n$ and any choice of
masses using action minimization methods.

The standard approaches to rule out collisions in action minimizers are _local
deformation_ , _Marchal’s average method_ and _level esitimate_. Here we use
none of them, but just the monotonicity of the solution, so the proof is
simple and elementary.

As the solutions we found are collinear, they can not always be collision-
free. However since they are block-regularizable, they can still be used to
explain the behavior of the nearby solutions and understand the global
dynamics of the problem. Compare to collision-free periodic solutions, so far
there are much less results about regularizable periodic solutions based on
action minimization methods. Besides [19], [17] and [20] mentioned above, up
to our knowledge there is also [11].

##  2\. Proof of Theorem 1.1

Since the proofs for any $\sigma\in\mathcal{S}_{\mathbf{n}}$ are the same, we
will only give the details for $\sigma=\text{Id}$.

For simplicity, for any $x\in H^{1}([0,T],\mathbb{R}^{n})$, we set
$\mathcal{A}_{T}(x)=\mathcal{A}(x;0,T)$. To find the desired minimizer, the
following set of admissible set of paths will be considered.

######  Definition 2.1.

$\Gamma=\\{x\in H^{1}([0,T],\mathbb{R}^{n}):x\text{ satisfies the following
conditions}\\}.$

  1. (i).

$x_{1}(0)x_{n}(0)\leq 0$;

  2. (ii).

for any $t\in[0,T]$, $x_{1}(t)\leq x_{2}(t)\leq\cdots\leq x_{n-1}(t)\leq
x_{n}(t)$;

  3. (iii).

when $n$ is odd,

$x_{1}(0)\leq x_{2}(0)=x_{3}(0)\leq\cdots\leq
x_{2j}(0)=x_{2j+1}(0)\leq\cdots\leq x_{n-1}(0)=x_{n}(0),$
$x_{1}(T)=x_{2}(T)\leq\cdots\leq x_{2j-1}(T)=x_{2j}(T)\leq\cdots\leq
x_{n-2}(T)=x_{n-1}(T)\leq x_{n}(T);$

  4. (iv).

when $n$ is even,

$x_{1}(0)\leq x_{2}(0)=x_{3}(0)\leq\cdots\leq
x_{2j}(0)=x_{2j+1}(0)\leq\cdots\leq x_{n-2}(0)=x_{n-1}(0)\leq x_{n}(0),$
$x_{1}(T)=x_{2}(T)\leq\cdots\leq x_{2j-1}(T)=x_{2j}(T)\leq\cdots\leq
x_{n-1}(T)=x_{n}(T).$

######  Lemma 2.1.

There is an $x\in\Gamma$ satisfying
$\mathcal{A}_{T}(x)=\inf\\{\mathcal{A}_{T}(y):y\in\Gamma\\}<\infty$.

###### Proof.

It is not hard to see the infimum of $\mathcal{A}_{T}$ in $\Gamma$ must be
finite.

Since $\Gamma$ is weakly closed in $H^{1}([0,T],\mathbb{R}^{n})$ and
$\mathcal{A}_{T}$ is weakly lower semi-continuous, by a standard result in
calculus of variation, it is enough to show $\mathcal{A}_{T}(x)\to\infty$,
whenever $\|x|_{[0,T]}\|_{H^{1}}\to\infty$.

By Definition 2.1, for any $x\in\Gamma$ we can define a new path $z\in
H^{1}([0,nT],\mathbb{R}^{n})$ as

$z(t)=\begin{cases}x_{2j-1}(t-2jT+2T),&\text{ if
}t\in[(2j-2)T,(2j-1)T],\;\forall j=1,2,\dots,[\frac{n+1}{2}];\\\
x_{2j}(2jT-t),&\text{ if }t\in[(2j-1)T,2jT],\;\forall
j=1,2,\dots,[\frac{n}{2}].\end{cases}$

Notice that $\|x|_{[0,T]}\|_{H^{1}}=\|z|_{[0,nT]}\|_{H^{1}}$, as

$\int_{0}^{nT}|z|^{2}\,dt=\int_{0}^{T}\sum_{i=1}^{n}|x_{i}|^{2}\,dt,\;\;\int_{0}^{nT}|\dot{z}|^{2}\,dt=\int_{0}^{T}\sum_{i=1}^{n}|\dot{x}_{i}|^{2}\,dt.$

Meanwhile by condition (i) in Definition 2.1, there is a $t_{0}\in[0,nT]$
satisfying $z(t_{0})=0$. Then

$|z(t)|=|z(t)-z(t_{0})|\leq\int_{0}^{nT}|\dot{z}|\,dt\leq(nT)^{\frac{1}{2}}\left(\int_{0}^{nT}|\dot{z}|^{2}\,dt\right)^{\frac{1}{2}}$

As a result, $\int_{0}^{nT}|z|^{2}\,dt\leq
n^{2}T^{2}\int_{0}^{nT}|\dot{z}|^{2}\,dt$. Let
$\mu=\min\\{m_{1},m_{2},\dots,m_{n}\\}$. Then

$\displaystyle\|x|_{[0,T]}\|^{2}_{H^{1}}$
$\displaystyle=\|z|_{[0,nT]}\|^{2}_{H^{1}}\leq(1+n^{2}T^{2})\int_{0}^{nT}|\dot{z}|^{2}\,dt\leq(1+n^{2}T^{2})\int_{0}^{T}\sum_{i=1}^{n}|\dot{x}_{i}|^{2}\,dt$
$\displaystyle\leq\frac{2(1+n^{2}T^{2})}{\mu}\int\frac{1}{2}\sum_{i=1}^{n}m_{i}|\dot{x}_{i}|^{2}\,dt\leq\frac{2(1+n^{2}T^{2})}{\mu}\mathcal{A}_{T}(x).$

∎

Let $x|_{[0,T]}$ be a minimizer of $\mathcal{A}_{T}$ in $\Gamma$ given by the
above lemma for the rest of this section. To obtain property (a) in Theorem
1.1, we can extend it to a $2T$-period Sobolev loop by

(4) $x(t)=x(2T-t),\;x(t)=x(t+2T),\;\forall t\in\mathbb{R}.$

######  Lemma 2.2.

For any $0\leq t_{1}<t_{2}\leq T$, $x_{i}(t_{1})<x_{i}(t_{2})$, if $i$ is odd;
$x_{i}(t_{1})>x_{i}(t_{2})$, if $i$ is even.

###### Proof.

We give the detail for $i$ being odd, the case for $i$ being even is similar
and will be omitted.

First we prove a weaker result stated as: for any $0\leq t_{1}<t_{2}\leq T$,
$x_{i}(t_{1})\leq x_{i}(t_{2})$, if $i$ is odd. By a contradiction argument,
let’s assume there is an odd $i_{0}$ and $0\leq t_{1}<t_{2}\leq T$ with
$x_{i_{0}}(t)$ strictly decreasing, $\forall t\in[t_{1},t_{2}]$. This allows
us to define a new path $x^{*}\in\Gamma$ as:

$x^{*}_{i_{0}}(t)=\begin{cases}x_{i_{0}}(t)&\text{ when }t\in[0,t_{1}],\\\
2x_{i_{0}}(t_{1})-x_{i_{0}}(t)&\text{ when }t\in[t_{1},t_{2}],\\\
x_{i_{0}}(t)+2(x_{i_{0}}(t_{1})-x_{i_{0}}(t_{2}))&\text{ when
}t\in[t_{2},T];\end{cases}$ $x^{*}_{i}(t)=\begin{cases}x_{i}(t)&\text{ when
}t\in[0,T],\;\text{ if }i<i_{0},\\\
x_{i}(t)+2(x_{i_{0}}(t_{1})-x_{i_{0}}(t_{2}))&\text{ when }t\in[0,T],\;\text{
if }i>i_{0}.\end{cases}$

As a result,

$\int_{0}^{T}\frac{1}{2}\sum_{i=1}^{n}m_{i}|\dot{x}^{*}_{i}|^{2}\,dt=\int_{0}^{T}\frac{1}{2}\sum_{i=1}^{n}m_{i}|\dot{x}_{i}|^{2}\,dt,\;\;\int_{0}^{T}U(x^{*})\,dt<\int_{0}^{T}U(x)\,dt.$

This implies $\mathcal{A}_{T}(x^{*})<\mathcal{A}_{T}(x)$, which is absurd.
This proves the weaker result stated above.

Now let’s further assume there is an odd $i_{0}<n$ and $0\leq t_{1}<t_{2}\leq
T$ with

$x_{i_{0}}(t)=x_{i_{0}}(t_{1}),\;\forall t\in[t_{1},t_{2}].$

Without lose of generality, we may even assume $t_{1}>0$ (if not, just replace
$t_{1}$ by $t_{2}/2$). Then for any $\varepsilon>0$ small enough, we can
define a new path $\hat{x}\in\Gamma$ as following:

$\hat{x}_{i_{0}}(t)=\begin{cases}x_{i_{0}}(t)&\text{ when }t\in[0,t_{1}],\\\
x_{i_{0}}(t_{1})+\frac{t-t_{1}}{t_{2}-t_{1}}\varepsilon&\text{ when
}t\in[t_{1},t_{2}],\\\ x_{i_{0}}(t)+\varepsilon&\text{ when
}t\in[t_{2},T];\end{cases}$ $\hat{x}_{i}(t)=\begin{cases}x_{i}(t)&\text{ when
}t\in[0,T],\text{ if }i<i_{0},\\\ x_{i}(t)+\varepsilon&\text{ when
}t\in[0,T],\text{ if }i>i_{0}.\end{cases}$

As a result,

(5) $|\hat{x}_{i}(t)-\hat{x}_{j}(t)|\geq|x_{i}(t)-x_{j}(t)|,\;\forall
t\in[0,T],\;\forall i\neq j;$ (6)
$\hat{x}_{n}(t)-\hat{x}_{1}(t)=x_{n}(t)-x_{1}(t)+\varepsilon,\;\;\forall
t\in[0,t_{1}].$

Meanwhile notice that

$C_{1}=\sup\\{x_{n}(t)-x_{1}(t):t\in[0,t_{1}]\\}>0,$

as otherwise $U(x(t))=\infty$, $\forall t\in[0,t_{1}]$. This implies
$\mathcal{A}_{T}(x)=\infty$, which is absurd. Then

$\displaystyle\int_{0}^{T}U(\hat{x})\,dt-\int_{0}^{T}U(x)\,dt$
$\displaystyle\leq\int_{0}^{t_{1}}\frac{m_{1}m_{n}}{\hat{x}_{n}(t)-\hat{x}_{1}(t)}-\frac{m_{1}m_{n}}{x_{n}(t)-x_{1}(t)}\,dt$
$\displaystyle=-\int_{0}^{t_{1}}\frac{m_{1}m_{n}\varepsilon}{(x_{n}(t)-x_{1}(t))^{2}+\varepsilon(x_{n}(t)-x_{1}(t))}\,dt\leq-\frac{m_{1}m_{n}t_{1}\varepsilon}{C_{1}^{2}+C_{1}\varepsilon}.$

At the same time a direct computation shows

$\int_{0}^{T}K(\dot{\hat{x}})\,dt-\int_{0}^{T}K(\dot{x})\,dt=\frac{m_{i_{0}}}{2}\int_{t_{1}}^{t_{2}}\dot{\hat{x}}_{i_{0}}^{2}-\dot{x}_{i_{0}}^{2}\,dt=\frac{m_{i_{0}}}{2(t_{2}-t_{1})}\varepsilon^{2}.$

As a result, for $\varepsilon\in(0,C_{1})$ small enough, we have the following
inequality, which is absurd.

$\mathcal{A}_{T}(\hat{x})-\mathcal{A}_{T}(x)\leq-\frac{m_{1}m_{n}t_{1}}{2C_{1}^{2}}\varepsilon+\frac{m_{i_{0}}}{2(t_{2}-t_{1})}\varepsilon^{2}<0.$

Now the only case that is not covered by the above argument is when $i_{0}=n$
with $n$ being odd. In this case, assume $t_{2}<T$ (if not, just replace
$t_{2}$ by $(t_{1}+t_{2})/2$) and like above, we have

$C_{2}=\sup\\{x_{n}(t)-x_{1}(t):t\in[t_{2},T]\\}>0.$

Then the same $\hat{x}$ defined as above satisfies (5) and

$\hat{x}_{n}(t)-\hat{x}_{1}(t)=x_{n}(t)-x_{1}(t)+\varepsilon,\;\;\forall
t\in[t_{1},t_{2}].$

A contradiction now can be reached by a similar estimate as above. ∎

The above lemma first implies $x(t)$ is collision-free, $\forall t\in(0,T)$;
second it implies if we make small enough perturbations near $x(t)$, the new
path will still belong to $\Gamma$. As a result, $x|_{[t-\delta,t+\delta]}$ is
a fix-end local minimizer of $\mathcal{A}$, for $\delta>0$ small enough, which
implies the following proposition.

######  Proposition 2.1.

$x(t)$, $t\in(0,T)$, is a collision-free smooth solution of (2).

###### Proof of (b), (c) & (d).

For property (b), we will give the details for $i$ being odd, while the other
case can be proven similarly.

By a contradiction argument, assume there is an odd $i_{0}<n$ and a
$t_{0}\in(0,T)$ with $\dot{x}_{i_{0}}(t_{0})=0$. Then by Lemma 2.2, there is a
constant $C_{1}>0$, such that for any $\varepsilon>0$ small enough,

(7) $0\leq\dot{x}_{i_{0}}(t)\leq C_{1}(t-t_{0}),\;\forall
t\in[t_{0},t_{0}+\varepsilon].$

Now define a new path $\hat{x}\in H^{1}([0,T],\mathbb{R}^{n})$ as following:

$\hat{x}_{i_{0}}(t)=\begin{cases}x_{i_{0}}(t)&\text{ when }t\in[0,t_{0}],\\\
x_{i_{0}}(t)+(t-t_{0})(2\varepsilon-(t-t_{0}))&\text{ when
}t\in[t_{0},t_{0}+\varepsilon],\\\ x_{i_{0}}(t)+\varepsilon^{2}&\text{ when
}t\in[t_{0}+\varepsilon,T];\end{cases}$
$\hat{x}_{i}(t)=\begin{cases}x_{i}(t)&\text{ when }t\in[0,T]\text{ and
}i<i_{0},\\\ x_{i}(t)+\varepsilon^{2}&\text{ when }t\in[0,T]\text{ and
}i>i_{0}.\end{cases}$

As a result,

$|\hat{x}_{i}(t)-\hat{x}_{j}(t)|\geq|x_{i}(t)-x_{j}(t)|,\;\forall
t\in[0,T],\;\forall i\neq j;$
$\hat{x}_{n}(t)-\hat{x}_{1}(t)=x_{n}(t)-x_{1}(t)+\varepsilon^{2},\;\;\forall
t\in[0,t_{0}].$

Meanwhile as $C_{2}=\sup\\{x_{n}(t)-x_{1}(t):t\in[0,t_{0}]\\}$ is a positive
constant, we have

$\displaystyle\int_{0}^{T}U(\hat{x})\,dt-\int_{0}^{T}U(x)\,dt$
$\displaystyle\leq\int_{0}^{t_{0}}\frac{m_{1}m_{n}}{\hat{x}_{n}(t)-\hat{x}_{1}(t)}-\frac{m_{1}m_{n}}{x_{n}(t)-x_{1}(t)}\,dt$
$\displaystyle\leq-\int_{0}^{t_{0}}\frac{m_{1}m_{n}\varepsilon^{2}}{(x_{n}(t)-x_{1}(t))^{2}+\varepsilon^{2}(x_{n}(t)-x_{1}(t))}\,dt\leq-\frac{m_{1}m_{n}t_{0}\varepsilon^{2}}{C_{2}^{2}+C_{2}\varepsilon^{2}}.$

The definition of $\hat{x}$ and (7) then imply

$\displaystyle\int_{0}^{T}K(\dot{\hat{x}})-K(\dot{x})\,dt$
$\displaystyle=\frac{m_{i_{0}}}{2}\int_{t_{0}}^{t_{0}+\varepsilon}\dot{\hat{x}}^{2}_{i_{0}}-\dot{x}^{2}_{i_{0}}\,dt=2m_{i_{0}}\int_{t_{0}}^{t_{0}+\varepsilon}\dot{x}_{i_{0}}(t)(\varepsilon+t_{0}-t)+(\varepsilon+t_{0}-t)^{2}\,dt$
$\displaystyle\leq\int_{t_{0}}^{t_{0}+\varepsilon}C_{1}(t-t_{0})(\varepsilon+t_{0}-t)+(\varepsilon+t_{0}-t)^{2}\,dt\leq\frac{(2+C_{1})m_{i_{0}}}{3}\varepsilon^{3}.$

By the above estimates, for $\varepsilon\in(0,\sqrt{C_{2}})$ small enough, we
have the following, which is absurd.

$\mathcal{A}_{T}(\hat{x})-\mathcal{A}_{T}(x)\leq-\frac{m_{1}m_{n}t_{0}}{2C_{2}^{2}}\varepsilon^{2}+\frac{(2+C_{1})m_{i_{0}}}{3}\varepsilon^{3}<0.$

What’s left now is when $i_{0}=n$ (with $n$ being odd) and
$\dot{x}_{i_{0}}(t_{0})=0$, for some $t_{0}\in(0,T)$. In this case, a
contradiction can be reached by a similar argument as above, as

$\hat{x}_{n}(t)-\hat{x}_{1}(t)=x_{n}(t)-x_{1}(t)+\varepsilon^{2},\;\forall
t\in[t_{0},T],$

and $C_{3}=\sup\\{x_{n}(t)-x_{1}(t):t\in[t_{0},T]\\}$ is a positive constant
$C_{3}$.

For property (c) and (d), as $x\in\Gamma$, it must satisfy Condition (iii) and
(iv) in Definition 2.1, and Lemma 2.2 implies the inequalities in these
conditions must be strict. ∎

Recall that the velocity of the center of mass of a smooth solutions must be a
constant.

######  Lemma 2.3.

$v_{0}=m_{0}^{-1}\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(t)=0,\;\forall
t\in(0,T)$, where $m_{0}=\sum_{i\in\mathbf{n}}m_{i}$.

###### Proof.

Notice the path $\hat{x}|_{[0,T]}$ with $\hat{x}_{i}(t)=x_{i}(t)-v_{0}t$,
$\forall i\in\mathbf{n}$, still belongs to $\Gamma$, and $\forall t\in[0,t]$,

$2K(\dot{x})-2K(\dot{\hat{x}})=\sum_{i\in\mathbf{n}}m_{i}\big{(}\dot{x}_{i}^{2}-(\dot{x}_{i}-v_{0})^{2}\big{)}=\sum_{i\in\mathbf{n}}m_{i}\big{(}2v_{0}\dot{x}_{i}-v_{0}^{2}\big{)}=m_{0}v_{0}^{2}.$

As $U(\hat{x}(t))=U(x(t))$, $v_{0}$ must be zero. Otherwise the following
inequality holds, which is absurd.

$\mathcal{A}_{T}(x)-\mathcal{A}_{T}(\hat{x})=\int_{0}^{T}K(\dot{x})-K(\dot{\hat{x}})\,dt=\frac{1}{2}m_{0}v_{0}^{2}T>0.$

∎

###### Proof of (e).

By the definition of $\Gamma$, for any $\xi\in
C^{\infty}([0,T],\mathbb{R}^{n})\cap\Gamma$, $x+\varepsilon\xi\in\Gamma$, when
$\varepsilon>0$ small enough. Then the minimizing property of $x(t)$ implies

$\frac{d}{d\varepsilon}\Big{|}_{\varepsilon=0}\mathcal{A}_{T}(x+\varepsilon\xi)=\int_{0}^{T}\sum_{i\in\mathbf{n}}(m_{i}\dot{x}_{i}\dot{\xi}_{i}+\partial_{x_{i}}U(x)\xi_{i})\,dt=0.$

By (2) and integration by parts,

(8) $\lim_{t\to
T^{-}}\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(t)\xi_{i}(t)-\lim_{t\to
0^{+}}\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(t)\xi_{i}(t)=0.$

Since the velocity of the center of mass of each pair of masses approaching a
binary collision has a finite limit, when the masses go to a binary collision
(see [8, Section 4]), the following one-sided limits are well-defined and
finite for the corresponding choices of $j$,

$\lim_{t\to 0^{+}}m_{2j}\dot{x}_{2j}(t)+m_{2j+1}\dot{x}_{2j+1}(t),\;\lim_{t\to
T^{-}}m_{2j-1}\dot{x}_{2j-1}(t)+m_{2j}\dot{x}_{2j}(t).$

As $x(t)=x(t+2T)=x(2T-t)$, $\forall t\in\mathbb{R}$, the following limits
exist and we may set

$m_{2j}\dot{x}_{2j}(0)+m_{2j+1}\dot{x}_{2j+1}(0)=\lim_{t\to
0}m_{2j}\dot{x}_{2j}(t)+m_{2j+1}\dot{x}_{2j+1}(t),\;\forall 1\leq
j\leq[(n-1)/2];$ $m_{2j-1}\dot{x}_{2j-1}(T)+m_{2j}\dot{x}_{2j}(T)=\lim_{t\to
T}m_{2j-1}\dot{x}_{2j-1}(t)+m_{2j}\dot{x}_{2j}(t),\;\forall 1\leq j\leq[n/2].$

As $\xi\in\Gamma$, (8) can be written as

(9)
$\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(0)\xi_{i}(0)-\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(T)\xi_{i}(T)=0,$

and by Lemma 2.3, we may write
$v_{0}=\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(t)=0$, for all $t\in[0,T]$.

We divide the proof to four different cases and assume $\xi\in
C^{\infty}([0,T],\mathbb{R}^{n})\cap\Gamma$ always satisfying

(10) $\sum_{i\in\mathbf{n}}m_{i}\xi_{i}(t)=0,\;\;\forall t\in[0,T].$

_Case 1_. $n$ is odd and $t=0$. First let’s choose a $\xi$ with
$\xi_{i}(T)=0$, $\forall i\in\mathbf{n}$; $\xi_{i}(0)=\xi_{2}(0)\neq 0$,
$\forall i\in\mathbf{n}\setminus\\{1\\}$. By (10),
$m_{1}\xi_{1}(0)=-(m_{0}-m_{1})\xi_{2}(0)$. Using this and (9), we have

$\displaystyle\sum_{i\in\mathbf{n}}m_{i}$
$\displaystyle\dot{x}_{i}(0)\xi_{i}(0)=0\Longrightarrow
m_{1}\dot{x}_{1}(0)\xi_{1}(0)+\xi_{2}(0)\sum_{i=2}^{n}m_{i}\dot{x}_{i}(0)=0$
$\displaystyle\stackrel{{\scriptstyle
v_{0}=0}}{{\Longrightarrow}}m_{1}\dot{x}_{1}(0)\big{(}\xi_{1}(0)-\xi_{2}(0)\big{)}=0\Longrightarrow-
m_{0}\xi_{2}(0)\dot{x}_{1}(0)=0\stackrel{{\scriptstyle\xi_{2}(0)\neq
0}}{{\Longrightarrow}}\dot{x}_{1}(0)=0.$

Now fix an arbitrary integer $j$ with $1\leq j\leq[n/2]$, and choose a $\xi$
satisfying

$\xi_{i}(T)=0,\forall i\in\mathbf{n};\xi_{2j}(0)=\xi_{2j+1}(0)\neq
0,\;\xi_{i}(0)=0,\;\forall i\in\mathbf{n}\setminus\\{1,2j,2j+1\\}.$

Using (9) and $\dot{x}_{1}(0)=0$, we get

$\displaystyle\sum_{i\in\\{1,2j,2j+1\\}}m_{i}\dot{x}_{i}(0)\xi_{i}(0)=0$
$\displaystyle\stackrel{{\scriptstyle\dot{x}_{1}(0)=0}}{{\Longrightarrow}}(m_{2j}\dot{x}_{2j}(0)+m_{2j+1}\dot{x}_{2j+1}(0))\xi_{2j}(0)=0$
$\displaystyle\stackrel{{\scriptstyle\xi_{2j}(0)\neq
0}}{{\Longrightarrow}}m_{2j}\dot{x}_{2j}(0)+m_{2j+1}\dot{x}_{2j+1}(0)=0$

_Case 2_. $n$ is odd and $t=T$. First choose a $\xi$ with $\xi_{i}(0)=0$,
$\forall i\in\mathbf{n}$; $\xi_{i}(T)=\xi_{1}(T)\neq 0$, $\forall
i\in\mathbf{n}\setminus\\{n\\}$. By (10),
$m_{n}\xi_{n}(T)=-(m_{0}-m_{n})\xi_{1}(T)$. Using this and (9), we get

$\displaystyle\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(T)$
$\displaystyle\xi_{i}(T)=0\Longrightarrow\xi_{1}(0)\sum_{i=1}^{n-1}m_{i}\dot{x}_{i}(T)+m_{n}\dot{x}_{n}(T)\xi_{n}(T)=0$
$\displaystyle\stackrel{{\scriptstyle
v_{0}=0}}{{\Longrightarrow}}m_{n}\dot{x}_{n}(T)\Big{(}\xi_{n}(T)-\xi_{1}(T))=0\Longrightarrow-
m_{0}\xi_{1}(T)\dot{x}_{n}(T)=0\stackrel{{\scriptstyle\xi_{1}(T)\neq
0}}{{\Longrightarrow}}\dot{x}_{n}(T)=0.$

Now fix an arbitrary integer $j$ with $1\leq j\leq[n/2]$ and choose a
$\xi\in\Gamma$ satisfying

$\xi_{i}(0)=0,\;\forall i\in\mathbf{n};\xi_{2j-1}(T)=\xi_{2j}(T)\neq
0,\;\xi_{i}(T)=0,\;\forall i\in\mathbf{n}\setminus\\{2j-1,2j,n\\}.$

Using (9) and $\dot{x}_{n}(T)=0$, we get

$\displaystyle\sum_{i\in\\{2j-1,2j,n\\}}m_{i}\dot{x}_{i}(T)\xi_{i}(T)=0$
$\displaystyle\stackrel{{\scriptstyle\dot{x}_{n}(T)=0}}{{\Longrightarrow}}\Big{(}m_{2j-1}\dot{x}_{2j-1}(T)+m_{2j}\dot{x}_{2j}(T)\Big{)}\xi_{2j}(T)=0$
$\displaystyle\stackrel{{\scriptstyle\xi_{2j}(T)\neq
0}}{{\Longrightarrow}}m_{2j-1}\dot{x}_{2j-1}(T)+m_{2j}\dot{x}_{2j}(T)=0.$

_Case 3_. $n$ is even and $t=0$. First choose a $\xi$ with $\xi_{i}(T)=0$,
$\forall i\in\mathbf{n}$; $\xi_{1}(0)\neq 0$ and $\xi_{i}(0)=0$, $\forall
i\in\mathbf{n}\setminus\\{1,n\\}$. By (10),
$m_{n}\xi_{n}(0)=-m_{1}\xi_{1}(0)$. Using this and (9), we get

$\displaystyle\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(0)\xi_{i}(0)=0$
$\displaystyle\Longrightarrow
m_{1}\dot{x}_{1}(0)\xi_{1}(0)+m_{n}\dot{x}_{n}(0)\xi_{n}(0)=0$
$\displaystyle\Longrightarrow
m_{1}\xi_{1}(0)\Big{(}\dot{x}_{1}(0)-\dot{x}_{n}(0)\Big{)}=0\stackrel{{\scriptstyle\xi_{1}(0)\neq
0}}{{\Longrightarrow}}\dot{x}_{1}(0)=\dot{x}_{n}(0).$

Next choose a $\xi$ with $\xi_{i}(T)=0$, $\forall i\in\mathbf{n}$;
$\xi_{i}(0)=\xi_{2}(0)\neq 0$, $\forall i\in\mathbf{n}\setminus\\{1,n\\}$.
Then

$\displaystyle\sum_{i\in\mathbf{n}}m_{i}\dot{x}_{i}(0)\xi_{i}(0)=0$
$\displaystyle\stackrel{{\scriptstyle
v_{0}=0}}{{\Longrightarrow}}m_{1}\dot{x}_{1}(0)\xi_{1}(0)+m_{n}\dot{x}_{n}(0)\xi_{n}(0)-\Big{(}m_{1}\dot{x}_{1}(0)+m_{n}\dot{x}_{n}(0)\Big{)}\xi_{2}(0)=0$
$\displaystyle\stackrel{{\scriptstyle\dot{x}_{1}(0)=\dot{x}_{n}(0)}}{{\Longrightarrow}}\Big{(}m_{1}\xi_{1}(0)+m_{n}\xi_{n}(0)\Big{)}\dot{x}_{1}(0)-(m_{1}+m_{n})\xi_{2}(0)\dot{x}_{1}(0)=0$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq;
xi=0}}}{{\Longrightarrow}}-m_{0}\xi_{2}(0)\dot{x}_{1}(0)=0\stackrel{{\scriptstyle\xi_{2}(0)\neq
0}}{{\Longrightarrow}}\dot{x}_{1}(0)=0.$

Now fix an arbitrary integer $j$ with $1\leq j\leq\frac{n}{2}-1$ and choose a
$\xi$ satisfying

$\xi_{i}(T)=0,\;\forall i\in\mathbf{n};\xi_{2j}(0)=\xi_{2j+1}(0)\neq
0,\;\xi_{i}(0)=0,\;\forall i\in\mathbf{n}\setminus\\{1,2j,2j+1,n\\}.$

Using (10) and $\dot{x}_{1}(0)=\dot{x}_{n}(0)=0$, we get

$\displaystyle\sum_{i\in\\{1,2j,2j+1,n\\}}m_{i}\dot{x}_{i}(0)\xi_{i}(0)=0$
$\displaystyle\stackrel{{\scriptstyle\dot{x}_{1}(0)=0}}{{\Longrightarrow}}\Big{(}m_{2j}\dot{x}_{2j}(0)+m_{2j+1}\dot{x}_{2j}(0)\Big{)}\xi_{2j}(0)=0$
$\displaystyle\stackrel{{\scriptstyle\xi_{2j}(0)\neq
0}}{{\Longrightarrow}}m_{2j}\dot{x}_{2j}(0)+m_{2j+1}\dot{x}_{2j+1}(0)=0.$

_Case 4_. $n$ is even and $t=T$. Fix an arbitrary integer $j$ with $1\leq
j\leq n/2$ and choose a $\xi$ satisfying

$\xi_{i}(0)=0,\;\forall
i\in\mathbf{n};\xi_{2j-1}(T)=\xi_{2j}(T),\;\xi_{i}(T)=\xi_{0}\neq 0,\;\forall
i\in\mathbf{n}\setminus\\{2j-1,2j\\}.$

By (10),
$(m_{2j-1}+m_{2j})\xi_{2j}(T)=-\xi_{0}\sum_{i\in\mathbf{n}\setminus\\{2j-1,2j\\}}m_{i}$.
With this and (9), we get

$\displaystyle\sum_{i\in\mathbf{n}}m_{i}$
$\displaystyle\dot{x}_{i}(T)\xi_{i}(T)=0\stackrel{{\scriptstyle
v_{0}=0}}{{\Longrightarrow}}\big{(}m_{2j-1}\dot{x}_{2j-1}(T)+m_{2j}\dot{x}_{2j}(T)\big{)}(\xi_{2j}(T)-\xi_{0})=0$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq;
xi=0}}}{{\Longrightarrow}}-\frac{m_{0}\xi_{0}\big{(}m_{2j-1}\dot{x}_{2j-1}(T)+m_{2j}\dot{x}_{2j}(T)\big{)}}{m_{2j-1}+m_{2j}}=0\stackrel{{\scriptstyle\xi_{0}\neq
0}}{{\Longrightarrow}}m_{2j-1}\dot{x}_{2j-1}(T)+m_{2j}\dot{x}_{2j}(T)=0.$

∎

Let $M=\text{diag}(m_{1},\dots,m_{1},\dots,m_{n},\dots,m_{n})$ be the mass
matrix. Then (2) can be written as the following vector fild, whose
corresponding flow will be denoted as $\phi_{t}$.

(11) $\dot{y}=v;\;\;\dot{v}=M^{-1}\nabla U(y).$

######  Definition 2.2.

We say a collision singularity $y^{*}\in\Delta$ is $C^{k}$ block-
regularizable, if for any collision solution $y^{-}|_{[t_{0}-\delta,t_{0}]}$
with $y^{-}(t_{0})=y^{*}$ and $y(t)\notin\Delta$, $\forall t\neq t_{0}$, one
can associate a unique ejection solution, $y^{+}|_{[t_{0},t_{0}+\delta]}$ with
$y^{+}(t_{0})=y^{*}$ and $y(t)\notin\Delta$, $\forall t\neq t_{0}$, such that
there are two co-dimension one cross sections $\Sigma^{\pm}$ (with
$(y^{\pm},\dot{y}^{\pm})(t_{0}\pm\delta)\in\Sigma^{\pm}$ correspondingly)
transversal to the vector field (11), and a $C^{k}$-homeomorphism
$\psi:\Sigma^{-}\to\Sigma^{+}$ obtained as an extension of the Poincaré map
from $\Sigma_{-}$ to $\Sigma_{+}$ induced from the flow $\phi_{t}$ by sending
$(y^{-},\dot{y}^{-})(t_{0}-\delta)$ to $(y^{+},\dot{y}^{+})(t_{0}+\delta)$.

######  Proposition 2.2.

[5, Theorem A], [6, Corollary G] A simultaneous binary collision singularity
is $C^{1}$ block-regularizable, when $d=1$ and $C^{0}$ block-regularizable,
when $d=2$ and $3$.

######  Remark 2.1.

Block regularization was introduced by Easton [4]. When $n=4$, the
simultaneous binary collisions are $C^{\frac{8}{3}}$ block-regularizable, when
$d=1$ see [10] and when $d=2$ see [3].

######  Definition 2.3.

Given a collision solution $y\in H^{1}(\mathbb{R},\mathbb{R}^{dn})$ of (2)
with the set of collision moments $\\{t\in\mathbb{R}:y(t)\in\Delta\\}$
isolated in $\mathbb{R}$, we say it is a $C^{k}$ block-regularizable solution
of (2), if for any collision moment $t_{0}$, there is a $\delta>0$ small
enough, such that $y|_{[t_{0},t_{0}+\delta]}$ is the unique ejection solution
associated with the collision solution $y|_{[t_{0}-\delta,t_{0}]}$ given in
Definition 2.2 .

###### Proof of (f).

Since $x(t)$ is $2T$ periodic, it is enough to show
$x|_{[t_{0}-\delta,t_{0}+\delta]}$ is block-regularizable, when $t_{0}=0$ and
$T$. As the argument is almost identical, we will only give the details for
$t_{0}=0$.

Since at $t_{0}=0$, $m_{2j}$ and $m_{2j+1}$ forming the $j$-th binary
collision, let

$\xi_{j}(t)=x_{2j+1}(t)-x_{2j}(t),\;s_{j}(t)=\xi_{j}(t)/|\xi_{j}(t)|,\;\;\forall
j=1,2,\dots,[(n-1)/2].$

The intrinsic energy of the $j$-th pair then is

$\alpha_{j}(t)=\frac{1}{2}\frac{m_{2j}m_{2j+1}}{m_{2j}+m_{2j+1}}|\dot{\xi}_{j}(t)|^{2}-\sqrt{\frac{m_{2j}m_{2j+1}}{m_{2j}+m_{2j+1}}}\frac{m_{2j}m_{2j+1}}{|\xi_{j}(t)|}.$

Despite of the binary collisions at $t=0$, the following limits exist and are
finite (see [6] or [8]),

$\lim_{t\to 0^{\pm}}s_{j}(t)=s^{\pm}_{j},\;\lim_{t\to
0^{\pm}}\alpha_{j}(t)=\alpha_{j}^{\pm}.$

By results from [5] and [6], $x|_{[-\delta,\delta]}$ is $C^{k}$-block
regularizable if

$s^{-}_{j}=s^{+}_{j}\;\text{ and }\;\alpha_{j}^{-}=\alpha_{j}^{+},\;\;\forall
j=1,2,\dots,[(n-1)/2].$

The first identities follow from the fact that $x(t)$ is collinear with
ordering of the mass preserving all the time. Meanwhile by (4),
$x_{i}(t)=x_{i}(-t)$, $\dot{x}_{i}(t)=-\dot{x}_{i}(-t)$, $\forall
t\in(0,\delta)$ and $\forall i\in\mathbf{n}$. Hence

$|\xi_{j}(t)|=|\xi_{j}(-t)|,\;\dot{\xi}_{j}(t)=-\dot{\xi}_{j}(-t),\;\;\forall
j=1,2,\dots,[(n-1)/2].$

This then implies $\alpha^{-}_{j}=\alpha^{+}_{j}$ for all corresponding $j$’s.
∎

##  3\. Proof of Theorem 1.2

We will only give the details for $\sigma=\text{Id}$. Notice that when
$\sigma=\text{Id}$, (1) becomes

(12) $m_{i}=m_{n+1-i},\;\;\forall i\in\mathbf{n}.$

The solutions will be found as minimizers under the proper symmetric
constraints. We recall some basic notations and results following [8]. Let
$\Lambda=H^{1}(\mathbb{R}/2T,\mathbb{R}^{nd})$ be the space of $2T$-period
Sobolev loops. For any finite group $G$, we may define its action on $\Lambda$
as follows:

$g\big{(}y(t)\big{)}=\big{(}\uprho(g)y_{\upsigma(g^{-1})(1)},\uprho(g)y_{\upsigma(g^{-1})(2)},\dots,\uprho(g)y_{\upsigma(g^{-1})(n)}\big{)}\big{(}\uptau(g^{-1})t\big{)},\;\;\forall
g\in G,$

where

  1. (a).

$\uprho:G\to O(d)$ represents the action of $G$ on $\mathbb{R}^{d}$;

  2. (b).

$\upsigma:G\to\mathcal{S}_{\mathbf{n}}$ represents the action of $G$ on
$\mathbf{n}$;

  3. (c).

$\uptau:G\to O(2)$ represents the action of $G$ on the time circle
$\mathbb{R}/2T\mathbb{Z}$.

Then $\Lambda^{G}=\\{y\in\Lambda|\;g(y(t))=y(t),\;\forall g\in G\\}$ is the
space of _$G$ -equivariant loops_. By Palais’ symmetric principle, a critical
point of $\mathcal{A}$ in $\Lambda^{G}$ is also a critical point of
$\mathcal{A}$ in $\Lambda$, when

(13) $m_{i}=m_{j},\;\text{ if }\exists g\in G,\text{ such that
}\upsigma(g)(i)=j$

For us, the dihedral group $D_{2}:=\langle
f,h|\;f^{2}=h^{2}=(fh)^{2}=1\rangle$ with following action will be used

$\uprho(f)=\text{Id},\quad\upsigma(f)=\text{Id},\quad\uptau(f)t=2T-t;$
$\uprho(h)=-\text{Id},\quad\upsigma(h)=\prod_{i=1}^{n}(i,n+1-i),\quad\uptau(h)t=\begin{cases}T-t,&\text{
when }n\text{ is odd},\\\ t,&\text{ when }n\text{ is even}.\end{cases}$

For such a group action, (12) implies (13) and each $x\in\Lambda^{D_{2}}$
satisfies

$x_{i}(t)=x_{i}(2T-t)\;\text{ and
}\;x_{i}(t)=\begin{cases}-x_{n+1-i}(T-t),\;&\text{ when }n\text{ is odd},\\\
-x_{n+1-i}(t),\;&\text{ when }n\text{ is even},\end{cases}.$

Let $\Omega=\\{x|_{[0,T]}:x\in\Lambda^{D_{2}}\\}$. Theorem 1.2 now can be
proven by similar arguments as in the proof of Theorem 1.1 with the desired
solutions found as minimizers of $\mathcal{A}$ in $\Omega\cap\Gamma$.

## References

  * [1] K.-C. Chen.  Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses.  Ann. of Math. (2), 167(2):325–348, 2008.
  * [2] A. Chenciner and R. Montgomery.  A remarkable periodic solution of the three-body problem in the case of equal masses.  Ann. of Math. (2), 152(3):881–901, 2000.
  * [3] N. Duignan and H. R. Dullin.  On the $C^{8/3}$-regularisation of simultaneous binary collisions in the planar four-body problem.  Nonlinearity, 34(7):4944–4982, 2021.
  * [4] R. Easton.  Regularization of vector fields by surgery.  J. Differential Equations, 10:92–99, 1971.
  * [5] M. S. ElBialy.  Simultaneous binary collisions in the collinear $N$-body problem.  J. Differential Equations, 102(2):209–235, 1993.
  * [6] M. S. ElBialy.  The flow of the $N$-body problem near a simultaneous-binary-collision singularity and integrals of motion on the collision manifold.  Arch. Rational Mech. Anal., 134(4):303–340, 1996.
  * [7] L. Euler.  De motu rectilineo trium corporum se mutuo attrahentium.  Novi Comm. Acad. Sci. Imp. Petrop., 11:144–151, 1767.
  * [8] D. L. Ferrario and S. Terracini.  On the existence of collisionless equivariant minimizers for the classical $n$-body problem.  Invent. Math., 155(2):305–362, 2004.
  * [9] T. Levi-Civita.  Sur la régularisation du problème des trois corps.  Acta Math., 42(1):99–144, 1920.
  * [10] R. Martí nez and C. Simó.  The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems.  Nonlinearity, 13(6):2107–2130, 2000.
  * [11] E. Mateus, A. Venturelli, and C. Vidal.  Quasiperiodic collision solutions in the spatial isosceles three-body problem with rotating axis of symmetry.  Arch. Ration. Mech. Anal., 210(1):165–176, 2013.
  * [12] R. McGehee.  Triple collision in the collinear three-body problem.  Invent. Math., 27:191–227, 1974.
  * [13] R. Moeckel.  Symbolic dynamics in the planar three-body problem.  Regul. Chaotic Dyn., 12(5):449–475, 2007.
  * [14] R. Moeckel.  A topological existence proof for the Schubart orbits in the collinear three-body problem.  Discrete Contin. Dyn. Syst. Ser. B, 10(2-3):609–620, 2008.
  * [15] F. R. Moulton.  The straight line solutions of the problem of $n$ bodies.  Ann. of Math. (2), 12(1):1–17, 1910.
  * [16] J. Schubart.  Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem.  Astronom. Nachr., 283:17–22, 1956.
  * [17] M. Shibayama.  Minimizing periodic orbits with regularizable collisions in the $n$-body problem.  Arch. Ration. Mech. Anal., 199(3):821–841, 2011.
  * [18] C. Simó and E. A. Lacomba.  Regularization of simultaneous binary collisions in the $n$-body problem.  J. Differential Equations, 98(2):241–259, 1992.
  * [19] A. Venturelli.  A variational proof of the existence of von Schubart’s orbit.  Discrete Contin. Dyn. Syst. Ser. B, 10(2-3):699–717, 2008.
  * [20] D. Yan.  A simple existence proof of Schubart periodic orbit with arbitrary masses.  Front. Math. China, 7(1):145–160, 2012.
  * [21] G. Yu.  Simple choreographies of the planar Newtonian $N$-body problem.  Arch. Ration. Mech. Anal., 225(2):901–935, 2017.