# On Geodesic Rays of Newtonian Gravitational Systems

Putian Yang  Shiqing Zhang Corresponding author<EMAIL_ADDRESS>
00footnotetext: 2010 Mathematics Subject Classification: 34B15, 34C25, 70F10,
70F16, 70G75

Abstract

In this paper, we focus on the set of geodesics rays of the Newtonian $N$-body
problem. We find that the limits of geodesic rays are also geodesic rays,
hence they are not dense in the space of initial conditions. As a result,
there are many motions whose domain is the half real line has non-negative
total energy, and they are not geodesic rays, the set of such motions has
positive measure, we think this set gives a large space to accommodate many
solutions with “bad” properties. As an application of weak KAM theory for
$N$-body problem, we give a brief proof of the existence of complete parabolic
orbit starting from any given initial position.

Keywords:  Newtonian $N$-body problem, Parabolic motions, Superhyperbolic
motions,
Geodesic rays

###### Contents

  1. 1 Introduction
  2. 2 Variational Setting and Some Results
    1. 2.1 The calibrating curves
    2. 2.2 Superhyperbolic motions
    3. 2.3 Parabolic motions starting from collison set
  3. 3 Proof of Theorem 1.2
  4. 4 Acknowledgement

##  1 Introduction

In 1922, Chazy [4] published a long famous paper on the three body problem, he
claimed that there are seven kinds of motions based on their final evolutions,
among which the parabolic and hyperbolic types attract tremendous interests,
since these types and mixed ones have interesting physical features.

In this paper we suppose that there are $N$ punctual bodies with masses
$m_{1},\dots,m_{N}>0$. They move in the given Euclidean space
$E=\mathbb{R}^{d},d\geq 2$, with $(\cdot,\cdot)$ the standard inner product of
$E$ and $|\cdot|$ is its induced norm. We call $E^{N}$ the configuration space
of the $N$-body problem, it is a linear space with mass scalar product which
is denoted by $g_{m}$ and is defined as

$\langle x,y\rangle=\sum_{i=1}^{N}m_{i}(x_{i},y_{i})$ (1.1)

for $x=(x_{1},\ldots,x_{N}),y=(y_{1},\ldots,y_{N})\in E^{N}$. The moment of
inertia of $x$ is $I(x)=\langle x,x\rangle$ and the norm of $x$ is
$\|x\|=I(x)^{1/2}.$

For $N$ bodies with positions $x_{1}(t),\dots,x_{N}(t)$, the parabolic orbits,
hyperbolic orbits and their mixed types have the definitions simlilar to that
in [4].

######  Definition 1.1.

For $x(t)=(x_{1}(t),\dots,x_{N}(t))\in E^{N}$, as $t\rightarrow+\infty$, the
$N$ bodies system is said to be

  * •

completely parabolic or simply parabolic, if each body $x_{i}$ goes to
infinity and $\dot{x}_{i}\rightarrow 0\in E$;

  * •

hyperbolic, if velocity of each body $\dot{x}_{i}\rightarrow a_{i}\in
E,a_{i}\neq a_{j}$, $i,j=1,\dots,N$;

  * •

hyperbolic-parabolic or partially hyperbolic, if velocity of each body
$\dot{x}_{i}\rightarrow a_{i}\in E$, there are some $a_{i}$ that are equal,
and not all $a_{i}$ be zero.

We say that the motion of a $N$-body system is called expansive if all mutual
distances $r_{ij}(t)=|x_{i}(t)-x_{j}(t)|\rightarrow+\infty$ as
$t\rightarrow+\infty$. On the other hand, a normalized configuration $a\in
E^{N}$ is called the limit shape of a solution $x(t)$, if there are constants
$\alpha,\kappa>0$ such that

$\lim_{t\rightarrow+\infty}\frac{x(t)}{t^{\alpha}}=\kappa a.$

Denote by $\Omega$ the subset of configurations without collision, i.e.,

$\Omega=\left\\{x\in E^{N}\mid x_{i}\neq x_{j},\forall i,j\text{ s.t. }1\leq
i<j\leq N\right\\},$

we also denote by $\Delta=E^{N}\setminus\Omega$ the collision set and by
$TE^{N}\cong E^{N}\times E^{N}$ the space of initial conditions, in particular
an initial position being in $\Delta$ means that its motion is an ejection
emerging from collision.

We equipe $\Omega$ with the Jacobi-Maupertuis metric, then $\Omega$ is an
incomplete Riemannian manifold, hence we can consider geodesic rays starting
from given points in $\Omega$. Apparently any expansive motion has non-
negative total energy and Chazy also proved that for the three body problem,
all the three cases are are expansive. Burgos and Maderna proved in [3] that
geodesic rays of $N$-body problem all belong to the above three types, and all
geodesic rays are expansive, we present this result in Theorem 1.4. We mainly
discuss geodesic rays in this paper.

Chazy [4] had verified that the hyperbolic motions are stable in three-body
problem, in other words, the set of initial conditions leading to hyperbolic
motions is open in $T\Omega$, i.e., a solution that is near a hyperbolic
soluton is also hyperbolic. Besides Chazy’s great work, during the last
century, there are some other early literatures on the initial conditons of
expansive motions, for example, Saari [25], Saari and Hulkower [26]
investigated the manifold structures of some sets of initial conditions of
expanding orbits and their connection to collisions. Another example is from
Saari [28], he found that when the total energy is positive, the growth order
of the minimal distance between particles is either $t^{2/3}$ or $t$. In this
paper, we prove that if both the dimension $d$ of the space $E$ and the number
of the bodies $N$ are greater than $2$, the set of all geodesic rays of
$N$-body problems is closed in the topology of $TE^{N}$. Through out this
paper we denote by $GR$ the set of initial conditions leading to geodesic
rays. We apply the methods from Maderna and Venturelli [14] to get our main
result as the following:

######  Theorem 1.2.

In the Newtonian gravitational $N$-body system, the set of initial conditions
giving rise to geodesic rays is closed.

The existence of expansive motions were unclear in 20th century, for instance,
whether there will be a parabolic eject from a given initial configuration or
to a given limit configuration. It is rather difficult for the calculus method
to solve these existence problems.

Nevertheless, in recent years, weak KAM theory and some related topological
technics have attracted considerable attentions in the community of celestial
mechanics, and by using these tools there have been extensive outstanding
studies in the existence of Newtonian expansive motions, among them we must
emphasize here several researches done by excellent mathematicians in the
field of dynamical systems, and we summarize them as Theorem 1.3.

In 2009, Maderna and Venturelli [13] discovered that given a central
configuration $x_{1}\in E^{N}$ that absolutely minimizes the normalized
potential $\tilde{U}=I^{1/2}U$, and given any $x_{0}$ as a initial point in
$E^{N}$, there exist a motion $x(t)$ such that $x(0)=x_{0}$ and $x_{1}$ is its
limit shape in the sense that

$x(t)=\lambda x_{1}t^{2/3}+o(t^{2/3})$

for some constant $\lambda>0$.

In 2013, Luz and Maderna [7] gives another beautiful proof of the existence of
parabolic motions. They found that the parabolic motions can emerge from any
initial point in $\Omega$, and the motion is asymptotic to the $set$ of
central configuration, in other words, for an arbitrary given position
$x_{0}\in\Omega$, there are velocities of the $N$ particles such that the
motion starting from the chosen initial conditions is fully parabolic.

For the case that the parabolic motions starting from the collision set and
asymptotic to a given central configuration, see [19]. Based on [3], we in our
paper, give another brief proof of the existence of parabolic motions.

In 2020, Maderna and Venturelli published another extrordinary paper [14], he
proved the existence of hyperbolic motions that given any initial
configuration $x_{0}\in E^{N}$, any normalized configuration without collisons
$x_{1}\in\Omega$, and any total energy $h>0$, there exist a hyperbolic motion
$x(t)=\sqrt{2h}x_{1}t+o(t)$ starting from $x_{0}$ and has $x_{1}$ as its limit
shape.

Recently, Burgos [2] proved the existence of a partially hyperbolic motion.
That is, in an Euclidean space $E=\mathbb{R}^{d}$ with dimension $d\geq 2$,
for any given energy level $h>0$, any configuration $x_{0}\in\Omega$, there
exists a partially hyperbolic motion $x(t)=at+o(t)$ such that
$a\in\Delta-\\{0\\}$. Burgos’s arguement relies on an important work of Chazy
in [4], that the limit shape of hyperbolic motion depends contiunously on the
initial condition. There is a natural question.

######  Question 1.

Given an arbitrary $a\in\Delta\setminus\\{0\\}$, is there a motion that has
$a$ to be its limit shape?

For brevity, we summarize here some of their conclusions as the following
theorem, note that if $x_{0}\in\Delta$, it means that the motion is an
ejection from collision and $x$ is continous at $t=0$, and $x(t)\in\Omega$ is
a maximal solution for $t>0$.

######  Theorem 1.3.

For the Newtonian $N$-body problem, given any intial configuration $x_{0}\in
E^{N}$ and any energy constant $h\geq 0$, there are choices of initial
velocities that determine the motion to be a geodesic rays, they have the form

$x(t)=at+O(t^{2/3})\text{ as }t\rightarrow+\infty,$ (1.2)

and $x(t)$ is hyperbolic if $a\in\Omega$, parabolic if $a=0$, partially
hyperbolic if $a\in\Delta\setminus\\{0\\}$.

If $h=0$, it is completely parabolic, if $h>0$, it is hyperbolic or partially
hyperbolic.

Now we state an important theorem as following that will be used in proof of
our main result. See Burgos and Maderna [3] for the proof of the theorem.

######  Theorem 1.4.

For any geodesic ray $\gamma:[0,+\infty)\rightarrow\Omega$ with energy
constant $h\geq 0$, the motion is expansive, more precisely,
$r_{ij}\thickapprox t^{2/3}$ or $r_{ij}\thickapprox t$.

In this paper we do not directly prove Theorem 1.2, on the contrary we use an
important equivalence between free time minimizers and geodesic rays, please
see Theorem 2.3, and we use the methods concerned with free time minimizers.

Another interesting discussion of our paper is about the non-geodesic motions.
It is known that many solutions without singularity are not geodesic, for
example, the periodic solutions and the superhyperbolic motions. When
considering the case of non-negative total energy, the set of solutions of
Newtonian gravitational system that is not geodesic $may$ includes many
relatively bad solution such as superhyperbolic motions and oscillatory
motions. Based on Theorem 1.2, we have a natural result for that set, that at
least for some small numbers of bodies, such set has positive measure, this
assertion gives us a lot of imagination, we present it as the following
proposition:

######  Proposition 1.5.

For $d=2,3$, $N=2,3,4$, and arbitrary $N$ positive point masses, there are
many initial conditions leading to solutions which do exist in $[0,+\infty)$,
they are not geodesic ray. The set of such initial data near the singularity
set has non-zero measure.

###### Proof.

According to Theorem 1.2, in $GR$ there cannot be a sequence with a limit
leading to motions with singularities. For any point $p$ in the set of initial
conditions $NC$ that leads to the singularity, there must be an open ball
$B(p,r_{p})$ which contains no point in $GR$. On the other hand, Saari [24]
found that $NC$ is of measure zero in case $d=2,3$ and $N=2,3,4$, hence
$\cup_{p\in NC}B(p,r_{p})\setminus NC$ has non-zero measure, it is the desired
set.

∎

Proposition 1.5 implies that even for the positive total energy, the solutions
near the singularity set suffer no singularity and do not minimize the
Lagrangian action in such a strong sense. These solutions are close to the
singularity set so that their behaviors may not look good.

We organize the paper as follows. The second section is devoted to introducing
the background material, some results given by variational method and
notations. In particular, subsection 2.2 introduces superhyperbolic motions,
and subsection 2.3 gives a brief proof of the existence of parabolic motions.
In the third section, our main result is proved, and a question is proposed.

##  2 Variational Setting and Some Results

The dynamics of the $N$ bodies are ruled by the Newton’s universal gravitation
law, that is

$m_{i}\ddot{x}_{i}=\nabla_{x_{i}}U(x),\text{ for }1\leq i\leq N,$ (2.1)

the Newtonian potential

$V(x)=-U(x):=-\sum_{1\leq i<j\leq N}\frac{m_{i}m_{j}}{|x_{i}-x_{j}|},$ (2.2)

where we take suitable units such that the gravitation constant $G=1$.

A configuration that absolutely minimizes $I^{1/2}U$ is also called a
minimizing central configuration.

The simplest hyperbolic motions or parabolic motions are those homothetic
type, in these situations, the limiting configurations are central
configurations. These constructions limit the choices of initial conditions
giving rise to these kinds of solutions, since the central configurations up
to similarity are finite, at least for $N=2,3,4,5$, see Hampton and Moeckel
[10] for the spatial four body problem, Albouy and Kaloshin [1] for the planar
five body case and Hampton and Jensen [9] the spatial five body problem. For
more general $N$-body problem, Painlevé-Wintner conjecture that the number of
central configurations up to similarity is finite. However, recent studies
give us a broad view that the initial conditions leading to our desired orbits
are not necessarily central configurations and they do exist almost
everywhere.

The Lagrangian $L$ on $E^{N}\times E^{N}$ is defined as

$L(x,v)=\frac{1}{2}\|v\|^{2}+U(x).$ (2.3)

According to the famous Hamilton’s principle, a solution of the Newton’s
equation joining two arbitrary given $x,y\in E^{N}$ must be a critical point
of the Lagrangian action, and more precisely a minimizer of Lagrangian action
among space of absolutely continuous curves joining $x$ and $y$ with fixed
time interval. The existence of such minimizers was already confirmed by
Tonelli’s theorem, see F. Clarke [6, p321, Theorem 16.2]. We next give this
approach explicitly.

Define the set of absolutely curves

$AC(x,y,T):=\left\\{\gamma:[0,T]\rightarrow E^{N}\text{ is absolutely
continuous, }\gamma(0)=x,\gamma(T)=y\right\\}$ (2.4)

and

$AC(x,y)=\cup_{T>0}AC(x,y,T),$ (2.5)

since without loss of generality, we can always assume each curve starting at
$0$ ending at some $T>0$ for $T$.

The Hamiltonian $H$ of the Lagrangian is the first integral of the solutions
to the Newton’s equation, and $h$ is the total energy constant. The Hamilton-
Jacobi equation is

$H(x,d_{x}u)=\frac{1}{2}\|d_{x}u\|_{*}-U(x)=h,$ (2.6)

where the norm $\|\cdot\|_{*}$ is the norm of $(E^{*})^{N}$, for
$p\in(E^{*})^{N}$,

$\|p\|_{*}=\sum_{i=1}^{N}m_{i}^{-1}|p_{i}|^{2}.$ (2.7)

The Lagrangian action of $\gamma\in AC(x,y)$ with energy constant $h\geq 0$ is

$A_{h}(\gamma)=\int_{0}^{T}L(\gamma,\dot{\gamma})+hdt=\int_{0}^{T}\frac{1}{2}\|\dot{\gamma}\|^{2}+U(\gamma)+hdt$
(2.8)

when $h=0$, we simply write $A$. It is well defined since $\Omega$ is
connected, and for all $x,y\in\Omega$ there is a smooth $\gamma\in AC(x,y,T)$
and $U$ is bounded in the compact set $\gamma|_{[0,T]}$ hence
$A_{h}(\gamma)<\infty$ and $0\leq A_{h}\not\equiv+\infty$.

However, there was a big obstacle for the calculus of variations that under
Newton potential there will be trajectory with isolated collisions and its
Lagrangian action is finite, this discovery is due to Poincaré [20], leading
to a question that a minimizer may not be a true motion. But mathematicians
did not worry about it long since Marchal [15] have made a great breakthrough,
making sure in case of more than 2 dimensional, the minimization process
confirms the absence of collision in the classical Newtonian $N$-body problem,
see more in Chenciner [5]. In 1-D case, Yu and Zhang [31] has obtained
important results under some conditions.

######  Theorem 2.1.

(Marchal [15] Chenciner [5]) If $\gamma\in AC(x,y)$ is defined over some
interval $[0,T]\subset\mathbb{R}$ and it minimizes $A$ over $AC(x,y,T)$, then
$\gamma$ experiences no collison for all $t\in(0,T)$.

Therefore a minimizer of the action is a solution of the Newtonian equation
2.1 in $(0,T)$ and a true motion of Newtonian systems.

As investigated by Saari [22, 24], we know that for the planar or three-
dimensional two, three or four body problems under Newtonian potential, in the
sense of Lebesgue, almost all $(x,v)\in T\Omega$ leading to solutions confirm
no singularities and exist for $[0,+\infty)$. We denote $NC\subset TE^{N}$ as
the set of initial coniditions leading to motions that suffer singularities at
some definite times.

Let $x,y\in E^{N}$ be arbitrary, we define the fixed time action potential for
time $T$:

$\phi(x,y,T)=\inf\left\\{A(\gamma)\big{|}\gamma\in AC(x,y,T)\right\\},$

and the free time action potential $\phi_{h}(\cdot,\cdot)$:

$\phi_{h}(x,y)=\inf\\{A_{h}(\gamma)\mid\gamma\in AC(x,y)\\}.$

A curve $\gamma_{0}\in AC(x,y,T)$ is called a fixed-time minimizer of $A$ if
it minimizes $A$ in $AC(x,y,T)$. Similarly, a curve $\gamma_{0}\in AC(x,y)$ is
called a free time minimizer of $A_{h}$ if it minimizes $A_{h}$ in $AC(x,y)$,
the existence of free time minimizers is proved, see [7, Theorem 3.1] and [14,
Lemma 4.2]. It is obvious that a free time minimizer is a fixed time minimizer
hence experiences no collision in the middle of the interval.

######  Theorem 2.2.

In a Newtonian $N$-bodies system with energy constant $h\geq 0$, free time
minimizing curve of its action potential $\phi_{h}(x,y)$ does exist for any
distinct $x,y\in E^{N}$.

A curve $\gamma$ defined in $[0,+\infty)$ is called a free time minimizer if
it is a free time minimizer of the Lagrangian action when restricted in any
compact subinterval $[a,b]\subset[0,+\infty)$, that is, $\gamma_{[a,b]}$
minimizes Lagrangian over $AC(\gamma(a),\gamma(b))$. Following the idea of
Marchal, if $\gamma\in AC(x,y,T)$ minimizes $A_{h}$ over $AC(x,y)$, then it
must minimize $A_{h}$ over $AC(x,y,T)$, hence minimizes $A$ over $AC(x,y,T)$
since $A_{h}=A+hT$. Therefore, if $\gamma\in AC(x,y)$ minimizes $A_{h}$, then
$\gamma|_{(0,T)}\subset\Omega$.

We have to mention that the methods of free time action minimizing is not the
only way to find those expansive motions. For instance, Maderna and Venturelli
[13] did not assert the solution $x(t)$ approximating to a given central
configuration to be free time minimizing. There are some other literatures as
examples, see Duignan, Moeckel, Montgomery and Yu [8] for bi-hyperbolic
motions, and Zhang [32] for restricted three body problem. On the other hand,
Hu, Ou and Yu [11] suggested that the results in [11] could be useful in the
application of non-action minimization methods in the Newtonian $N$-body
problem. Therefore it is a natural question that whether a hyperbolic, a
parabolic or a partial hyperbolic motion is necessarily geodesic, this is an
unsolved problem.

The Jacobi-Maupertuis metric is defined as $j_{h}=2(h+U)g_{m}$. A geodesic ray
of the $N$-body system is an arclength parameterized geodesic
$\gamma:[0,+\infty)\rightarrow E^{N}$ such that all of its restrictions to
compact subintervals are minimizing geodesics.

The total energy constant of a given motion $x(t)$ is
$h=\frac{1}{2}\|\dot{x}(t)\|-U(x(t))$, which implies that the motion must be
confined to the Hill’s region:

$\Omega_{h}=\\{x\in\Omega\mid U(x)\geqq-h\\}.$ (2.9)

We see that $\Omega_{h}=\Omega$ if and only if $h\geqq 0$, the geodesics of
the Jacobi-Maupertuis metric and free time minimizers of $A_{h}$ are
equivalent, roughly speaking, this is probably because for any $\gamma\in
AC(x,y)$,

$\displaystyle\int_{s}^{t}\|\dot{\gamma}\|_{j_{h}}d\tau$
$\displaystyle=\int_{s}^{t}\sqrt{2(h+U)}\|\dot{\gamma}\|d\tau=\int_{s}^{t}\sqrt{2}\left(\frac{1}{2}\|\dot{\gamma}\|^{2}-U+U\right)^{1/2}\|\dot{\gamma}\|d\tau$
$\displaystyle=\int_{s}^{t}\|\dot{\gamma}\|^{2}d\tau=\int_{s}^{t}\frac{1}{2}\|\dot{\gamma}\|^{2}+U+\frac{1}{2}\|\dot{\gamma}\|^{2}-Ud\tau$
$\displaystyle=\int_{s}^{t}L(\gamma,\dot{\gamma})+hd\tau=A_{h}\left(\gamma|_{[s,t]}\right).$

The rigorous proof is given by Burgos and Maderna [3].

######  Theorem 2.3.

After a suitable reparameterization, a geodesic ray of the metric $j_{h}$ is a
free time minimizer of $A_{h}$, and a free time minimizer of $A_{h}$ is a
geodesic ray of the metric.

Further more, we denote by $d_{h}$ the Riemannian distance induced by the
metric $j_{h}$, Maderna and Venturelli [14] have pointed out that the
completion of $(\Omega,d_{h})$ is exactly $(E^{N},\phi_{h})$. $\phi_{h}$ is a
distance, it is verified by Maderna [12] for $h=0$, but it also holds for
cases of $h\geq 0$. We owe the proof of the following proposition to Maderna.

######  Proposition 2.4.

(Maderna [12]) For any $h\geq 0$, $\phi_{h}(\cdot,\cdot)$ is a distance in
$E^{N}$.

###### Proof.

We first prove that $\phi_{h}$ satiesfies trianglular inequality. For any
$x,y,z\in E^{N}$, any $\gamma_{1}\in AC(x,y)$ and $\gamma_{2}\in AC(y,x)$
defined in $[0,T_{1}]$ and $[0,T_{2}]$ respectively, we set

$\gamma(t):=\begin{cases}\gamma_{1}(t)&0\leq t\leq T_{1}\\\
\gamma_{2}(t-T_{1})&T_{1}\leq t\leq T_{1}+T_{2}\end{cases},$ (2.10)

thus

$\phi_{h}(x,z)=\inf\left\\{A_{h}(\eta)\mid\eta\in AC(x,z)\right\\}\leq
A_{h}(\gamma)\leq A_{h}(\gamma_{1})+A_{h}(\gamma_{2}).$ (2.11)

Since $\gamma_{1},\gamma_{2}$ are arbitrary, we have
$\phi_{h}(x,z)\leq\phi_{h}(x,y)+\phi_{h}(y,z)$.

Second, we verify that $\phi_{h}(x,y)=0$ makes $x=y$. For any
$x=(x_{1},\ldots,x_{N}),y=(y_{1},\ldots,y_{N})\in\Omega$ and
$\gamma=(\gamma_{1},\ldots,\gamma_{N})\in AC(x,y)$ defined over $[0,T]$. For
any $1\leq i\leq N$,

$|x_{i}-y_{i}|=|\gamma_{i}(T)-\gamma_{i}(0)|\leq\int_{0}^{T}|\dot{\gamma}_{i}|dt\leq\sqrt{T}\left(\int_{0}^{T}|\dot{\gamma}_{i}|^{2}\right)^{1/2}.$

Therefore

$A_{h}(\gamma)\geq\int_{0}^{T}\frac{1}{2}\|\dot{\gamma}\|^{2}dt=\frac{1}{2}\sum
m_{i}\int_{0}^{T}|\dot{\gamma}_{i}|^{2}dt\geq\frac{1}{2T}\sum
m_{i}|x_{i}-y_{i}|^{2}.$

So eventually $\phi_{h}(x,y)=0$ indicates $x=y$.

It is not difficult to see $\phi_{h}(x,x)=0$ and
$\phi_{h}(x,y)=\phi_{h}(y,x)$. ∎

We must mention an important estimate for $\phi_{h}(\cdot,\cdot)$ such that we
can apply Ascoli’s theorem to get our desired solution. This estimate is given
by Maderna [12].

######  Theorem 2.5.

(Maderna [12]) For a Newtonian $N$-body system in a configuration space
$E^{N}$, the masses of the particles and the number $N$ determine two positive
constants $C_{1},C_{2}$ such that

$\phi(x,y,t)\leq C_{1}\frac{l^{2}}{t}+C_{2}\frac{t}{l}$ (2.12)

for any $x,y\in E^{N}$, any $l>\|x-y\|$ and $t>0$.

The following theorem is about a uniform bound for a sequence of action
potential, it is a tiny extension of [14, Theorem 2.11] .

######  Theorem 2.6.

If $\\{h_{n}\\}$ is a sequence of bounded non-negative energy constants. There
exist constants $\alpha>0$ and $\beta>0$ such that,

$\phi_{h_{n}}(x,y)\leq\mu(\|x-y\|),$ (2.13)

where $\mu(\|x-y\|)=\left(\alpha\|x-y\|+\beta\|x-y\|^{2}\right)^{1/2}$ and
$\alpha,\beta$ only depend on the bound of $h_{n}$, the number of bodies $N$
and their masses.

###### Proof.

Suppose $h_{n}$ are bounded by a positive number $M$, since
$\phi_{h_{n}}(x,y)=\inf_{t>0}\left\\{\phi(x,y,t)+h_{n}t\right\\}$, and follows
from Theorem 2.5 we first estimate the upper bound for $\phi+h_{n}t$:

$\phi(x,y,t)+h_{n}t\leq\frac{C_{1}l^{2}}{t}+(\frac{C_{2}}{l}+h_{n})t,$

for all $t>0,l>\|x-y\|$, now consider the right handside of the above
inequality as a function of $t$, we calculate its minimal value to be
$2\sqrt{C_{1}l^{2}\left(\frac{C_{2}}{l}+h_{n}\right)}=\left(4C_{1}C_{2}l+4C_{1}h_{n}l^{2}\right)^{1/2}$,
thus if we let $\alpha=4C_{1}C_{2},\beta=4C_{1}M$, then by $h_{n}\leq M$ we
have

$\phi_{h_{n}}(x,y)\leq\phi(x,y,t)+h_{n}t\leq\left(\alpha l+\beta
l^{2}\right)^{1/2}$

holds for all $l>\|x-y\|$, then let $l\rightarrow\|x-y\|$ to justify our
claim. ∎

###  2.1 The calibrating curves

The purpose of this subsection is to present certain minimizers which are
called calibrating curves, they relate the action potentials and the viscosity
subsolutions of the corresponding Hamilton-Jacobi equation. Proposition 2.9
claim the existence of such curves, it was proved by Maderna and Venturelli
[14, Theorem 3.2, p525] for $h>0$, the paper [14] implies that the result also
holds in case of zero energy. For convenience, we present here the proof of
Proposition 2.9 and owe it to Maderna and Venturelli. Calibrating curves is
quite essential through out of our paper.

######  Definition 2.7.

A absolutely continuous curve $\gamma:I\rightarrow E^{N}$ in some interval $I$
is called a calibrating curve of a function $u\in C(E^{N})$ if

$u\left(\gamma(t_{2})\right)-u\left(\gamma(t_{1})\right)=A_{h}\left(\gamma\big{|}_{[t_{1},t_{2}]}\right)$
(2.14)

for any closed compact subinterval $[t_{1},t_{2}]\subset I$. We also call
$\gamma$ $h$-calibrates $u$ in $I$.

Before proving the proposition, we need von Zeipel’s theorem, see von Zeipel
[30] and McGehee [17].

######  Theorem 2.8.

If the solution $x(t)$ has a singularity at $t^{*}$, then $\lim_{t\rightarrow
t^{*}}I(x(t))=I^{*}$ exists and lies in $[0,+\infty]$. In particular, in the
case of $\lim_{t\rightarrow t^{*}}I(x(t))=I^{*}<+\infty$, there exists a
$x^{*}\in\Delta$ such that $\lim_{t\rightarrow t^{*}}x(t)=x^{*}$, i.e., $x(t)$
suffers a collision.

In the case of $I^{*}=+\infty$, the particles of $x(t)$ is called experiencing
a pseudo-collison.

######  Proposition 2.9.

For $h_{n}\geq 0$ and $h_{n}\rightarrow h\geq 0$, if $u\in C^{0}(E^{N})$ and
$u(x)=\lim_{n\rightarrow\infty}\left(\phi_{h_{n}}(0,p_{n})-\phi_{h_{n}}(x,p_{n})\right)$
for some sequence $p_{n}\in E^{N}$ with $\|p_{n}\|\rightarrow+\infty$. Then
for any initial configuration $x\in E^{N}$, there exist a calibrating curve
$\gamma$ defined over $[0,+\infty)$ with $\gamma(0)=x$. i.e.,

$u(\gamma(t))-u(x)=A_{h}\left(\gamma\big{|}_{[0,t]}\right)$

for all $t>0$.

###### Proof.

We first show that for any $r>0$ there exist some $y_{r}\in E^{N}$ and a curve
$\gamma_{r}\in AC(x,y_{r})$ defined in a finite interval, such that
$\|x-y_{r}\|=r$ and $u(y_{r})-u(x)=A_{h}(\gamma_{r})$.

Denote $\phi_{h_{n}}(\cdot,p_{n})$ by $v_{n}(\cdot)$ as a function in $E^{N}$,
then it is continuous since $\phi_{h_{n}}$ is a distance. According to Theorem
2.2, the action $\phi_{h_{n}}(x,p_{n})$ attains its minimum in $AC(x,p_{n})$
at some $\gamma_{n}$ defined over $[0,t_{n}]$ with $\gamma_{n}(t_{n})=p_{n}$,
then for any $t\in[0,t_{n}]$,

$v_{n}(\gamma(t))-v_{n}(p_{n})=\phi_{h_{n}}(\gamma(t),p_{n})=A_{h_{n}}\left(\gamma_{n}|_{[t,t_{n}]}\right),$

which implies that for any subinterval $[t_{1},t_{2}]\subset[0,t_{n})$,

$\displaystyle v_{n}(\gamma(t_{1}))-v_{n}(\gamma(t_{2}))$ $\displaystyle=$
$\displaystyle\big{(}v_{n}(p_{n})-v_{n}(\gamma(t_{2}))\big{)}-\big{(}v_{n}(p_{n})-v_{n}(\gamma(t_{1}))\big{)}$
$\displaystyle=$ $\displaystyle-
A_{h_{n}}\left(\gamma|_{[t_{2},t_{n}]}\right)+A_{h_{n}}\left(\gamma|_{[t_{1},t_{n}]}\right)$
$\displaystyle=$ $\displaystyle
A_{h_{n}}\left(\gamma|_{[t_{1},t_{2}]}\right).$

Now because $\|p_{n}\|\rightarrow+\infty$, suppose $\|p_{n}-x\|>r$, then for
each $n$, there is a $y_{n}=\gamma(\tau_{n})$ for some $\tau_{n}$ such that
$\|y_{n}-x\|=r$, thus we assume $y_{n}\rightarrow y_{r}$ for some
$y_{r}\in\partial B(x,r)$. Since $\gamma_{n}$ is a calibrating curve, the
following equation holds,

$v_{n}(x)-v_{n}(y_{n})=A_{h_{n}}\left(\gamma_{n}\big{|}_{[0,\tau_{n}]}\right)=\phi_{h_{n}}(x,y_{n}).$

On the otherhand, since $\phi_{h_{n}}$ is distance, then
$\phi_{h_{n}}(y_{n},p_{n})\leq\phi_{h_{n}}(y_{n},y_{r})+\phi_{h_{n}}(y_{r},p_{n})$
and
$\phi_{h_{n}}(y_{r},p_{n})\leq\phi_{h_{n}}(p_{n},y_{n})+\phi_{h_{n}}(y_{n},y_{r})$
imlplies $|v_{n}(y_{n})-v_{n}(y_{r})|\leq\phi_{h_{n}}(y_{r},y_{n})$, then we
apply Theorem 2.6 to have
$|v_{p_{n}}(y_{n})-v_{p_{n}}(y_{r})|\leq\mu(\|y_{n}-y_{r}\|)\rightarrow 0$ as
$n\rightarrow\infty$.

On the otherhand, since $\phi_{h_{n}},\phi_{h}$ are distances in $E^{N}$
induced by comformal metrics and $h_{n}\rightarrow h$, thus
$\lim_{n}\phi_{h_{n}}(x,y_{r})=\phi_{h}(x,y_{r})$, and by the Theorem 2.6, we
have

$|\phi_{h_{n}}(x,y_{r})-\phi_{h_{n}}(x,y_{n})|\leq\phi_{h_{n}}(y_{r},y_{n})\leq\mu\|y_{r}-y_{n}\|\rightarrow
0,$ (2.15)

and then

$|\phi_{h}(x,y_{r})-\phi_{h_{n}}(x,y_{n})|\leq|\phi_{h}(x,y_{r})-\phi_{h_{n}}(x,y_{r})|+|\phi_{h_{n}}(x,y_{r})-\phi_{h_{n}}(x,y_{n})|\rightarrow
0,$ (2.16)

i.e., $\lim_{n}\phi_{h_{n}}(x,y_{n})=\phi_{h}(x,y_{r})$. Therefore

$\displaystyle u(y_{r})-u(x)$
$\displaystyle=\lim_{n}\bigl{(}v_{n}(x)-v_{n}(y_{r})\bigr{)}$
$\displaystyle=\lim_{n}\bigl{(}v_{n}(x)-v_{n}(y_{n})+v_{n}(y_{n})-v_{n}(y_{r})\bigr{)}$
$\displaystyle=\lim_{n}\bigl{(}\phi_{h_{n}}(x,y_{n})+v_{n}(y_{n})-v_{n}(y_{r})\bigr{)}$
$\displaystyle=\phi_{h}(x,y_{r}),$

we again apply Lemma 2.2 to take a minimizer $\gamma_{r}\in AC(x,y_{r})$ such
that $A_{h}(\gamma_{r})=\phi_{h}(x,y_{r})=u(y_{r})-u(x)$.

Now we apply Zorn’s Lemma to have a maximal calibrating curve
$\gamma:[0.t^{*})\rightarrow E^{N}$, then we try to prove $t^{*}=+\infty$. We
argue by contradiction. Let $t^{*}\neq+\infty$, $\gamma$ is a free time
minimizer thus it is a true motion in $[0,t^{*})$, the maximal property of
$t^{*}$ implies it is a singularity. Applying von Zeipel’s theorem 2.8 gives
$I(\gamma(t))\rightarrow I^{*}\in[0,+\infty]$, If $I^{*}<+\infty$, Zeipel’s
theorem tells us $\lim_{t\rightarrow t^{*}}\gamma(t)$ exists in finite
position, then we pick another calibrating curve $\eta$ defined in some
$[t^{*},t^{*}+\sigma)$ $(\sigma>0)$ with $\eta(t^{*})=\lim_{t\rightarrow
t^{*}}\gamma(t)$, concatenating $\gamma,\eta$ produces a new calibrating curve
defined in $[0,t^{*}+\sigma)$, a contradiction. If $I^{*}=+\infty$, we choose
a sequence $x_{n}=\gamma(t_{n}),(t_{n}\rightarrow t^{*-})$ such that
$\|x_{n}-x\|\rightarrow+\infty$ and denote
$A_{n}=A_{h}\left(\gamma\big{|}_{[0,t_{n}]}\right)$, then we have

$\|x_{n}-x\|^{2}=\left\|\int_{0}^{t_{n}}\dot{\gamma}dt\right\|^{2}=\left\|\int_{0}^{t_{n}}1\cdot\dot{\gamma}dt\right\|^{2}\leq
t_{n}\int_{0}^{t_{n}}\|\dot{\gamma}\|^{2}dt\leq 2t_{n}A_{n},$ (2.17)

and since $\gamma$ minimizes $A_{h}$, we have

$\displaystyle\frac{\|x_{n}-x\|^{2}}{2t_{n}}+ht_{n}$ $\displaystyle\leq
A_{n}+ht_{n}=A_{h}(\gamma\big{|}_{[0,t_{n}]})=\phi(x,x_{n})$
$\displaystyle\leq\left(\alpha\|x-x_{n}\|+\beta\|x-x_{n}\|^{2}\right)^{1/2},$

this is a contradiction since when $\|x_{n}-x\|$ approaching infinity, the
left handside has order $O(\|x_{n}-x\|^{2})$ but is governed by the right
handside whose order is $O(\|x_{n}-x\|)$.

∎

######  Remark 2.10.

The function $u$ defined in Theorem 2.9 is called Buseman function, denote by
$\mathcal{B}_{h}$ the space of these functions, like Maderna and Venturelli
[14]. They are also verified in [14] to be viscosity subsolutions of the
Hamilton-Jacobi equations 2.6. We will show that it is non-empty in the proof
of Theorem 1.2. Then we can say that there exists a calibrating curve starting
with any prescribed initial position $x_{0}\in E^{N}$ and energy $h\geq 0$.

###  2.2 Superhyperbolic motions

We need to introduce an important classification of motions of $N$-body
problem made by Marchal and Saari [16], which helps us to achieve our main
result. We wirte $R(t)=\max_{i<j}r_{ij}(t)$ and $r(t)=\min_{i<j}r_{ij}(t)$
which represent respectively the maximal and minimal distances between the $N$
bodies.

######  Theorem 2.11.

Suppose $\gamma:[0,+\infty)\rightarrow\Omega$ is an arbitrary solution of the
Newton’s equation, then its motion, no matter the sign of its energy, can only
be one of the two cases:

  1. 1.

$R(t)/t\rightarrow+\infty$ and $r(t)\rightarrow 0$ as $t\rightarrow+\infty$.

  2. 2.

$\gamma(t)=at+O(t^{2/3})$ for some $a\in E^{N}$. Here $O(t^{2/3})$ represents
a term bounded by $Ct^{2/3}$ for some constant $C$.

We need to mention that Pollard [21] proved $R(t)/t\rightarrow+\infty$ if and
only if $r(t)\rightarrow 0$ as $t\rightarrow+\infty$, therefore the first case
of the above theorem is valid and is called a superhyperbolic motion.

The following proposition asserts that $\gamma$ is not superhyperbolic in some
case, we owe it to [14].

######  Proposition 2.12.

For $h\geq 0$, a free time minimizer $\gamma$ of $A_{h}$ is not
superhyperbolic

###### Proof.

First we give a claim, that for any $x,y\in E^{N},t>0$ and any $\gamma\in
AC(x,y,t)$ defined over $[a,b]$ with length $t$, we have

$\frac{\|x-y\|^{2}}{2t}\leq A(\gamma),$

which is true since

$\|x-y\|^{2}\leq\left(\int_{a}^{b}\|\dot{\gamma}\|dt\right)^{2}\leq
t\int_{a}^{b}\|\dot{\gamma}\|^{2}dt,$

then

$\frac{\|x-y\|^{2}}{2t}\leq\frac{1}{2}\int_{a}^{b}\|\dot{\gamma}\|^{2}dt\leq
A(\gamma).$

Now suppose $\gamma$ is superhyperbolic and we take $t_{n}\rightarrow+\infty$
such that $R(t_{n})/t_{n}\rightarrow+\infty$. Denote
$\lambda_{n}=\|\gamma(t_{n})-\gamma(0)\|$, first notice that
$R(t)\thickapprox\|\gamma(t)\|$ as $t\rightarrow+\infty$ implies
$\lambda_{n}/t_{n}\rightarrow+\infty$. Then since $\gamma$ minimizes the
action $A_{h}$, and
$A(\gamma|_{[0,t_{n}]})+ht_{n}=A_{h}(\gamma|_{[0,t_{n}]})=\phi_{h}(\gamma(0),\gamma(t_{n}))$,
therefore by the previous assertion and Theorem 2.6 we have the following
inequality about $\phi_{h}(\gamma(0),\gamma(t_{n}))$ for any $n$:

$\frac{\lambda_{n}^{2}}{2t_{n}}+ht_{n}\leq
A(\gamma|_{[0,t_{n}]})+ht_{n}=\phi_{h}(\gamma(0),\gamma(t_{n}))\leq\left(\alpha\lambda_{n}+\beta\lambda_{n}^{2}\right)^{1/2}$
(2.18)

for some constants $\alpha,\beta$, but this is incompatible for
$\lambda_{n}\rightarrow+\infty$ as $t_{n}\rightarrow+\infty$. ∎

We have to explain here that Saari [23] had found that there is no
superhyperbolic motion in the three body problem. In detail, Saari gave
important estimates that in three body problem, either $R(t)\approx At$ or
$R(t)$ is bounded by $Ct^{2/3}$ as $t\rightarrow+\infty$, where $A$ and $C$
are constants.

For the $N$-body problem with $N\geq 4$, whether there is a superhyperbolic
motion still needs a complete answer. Saari and Xia [27] gave an example that
for the collinear Newtonian four body problem, the superhyperbolic motions
exist in a weak sense, i.e., a motion with eslatic collision. For the high-
dimensional and collisionless case, there is no example at present.

We have already known that geodesic rays are the second type, but the converse
is not true, oscillatory motion is an example, it is known that it exists in
three body problem, see Sitnikov [29]. According to Saari [23], their mutual
distances are bounded by $Ct^{2/3}$ for some constant $C$ hence it belongs to
the second case of the above theorem, but its complex and weird behavior
exclude the possibility of being a geodesic ray.

###  2.3 Parabolic motions starting from collison set

In 2014, Luz and Maderna [7] proved the existence of parabolic solutions for
any given initial position $x_{0}\in\Omega$, the case of starting from
collision set was omitted in that paper, physically speaking, when two bodies
$x_{i},x_{j}$ have the same initial position $x_{i}(0)=x_{j}(0)\in E$, their
motions are ejections from their shared starting point. In this subsection we
use the calibrating curves to give a brief proof for all $x_{0}\in E^{N}$.
Before that we need a lemma that is needed not only in this subsection, but
also in the proof of Theorem 1.2.

######  Lemma 2.13.

For $h\geq 0$, $w\in\mathcal{B}_{h}$ is dominated by $L+h$. For any function
$u$ dominated by $L+h$, if $\gamma$ $h$-calibrates $u$, then $\gamma$ is a
free time minimizer of the action potential $A_{h}$.

###### Proof.

Since $\phi_{h}(\cdot,\cdot)$ is a distance, for any $x,y$ in the domain of
$w$, suppose

$w_{n}=\phi(0,p_{n})-\phi(x,p_{n}),\quad w=\lim_{n}w_{n},$

then

$w_{p_{n}}(y)-w_{p_{n}}(x)=\phi_{h}(x,p_{n})-\phi_{h}(y,p_{n})\leq\phi_{h}\left(x,y\right),$
(2.19)

and since $w\in\mathcal{B}_{h}$, let $n\rightarrow+\infty$ to have
$w(y)-w(x)\leq\phi_{h}\left(x,y\right)$.

The second claim arise from
$A_{h}(\gamma\big{|}_{[a,b]})=u(\gamma(b))-u(\gamma(a))\leq\phi_{h}(\gamma(b),\gamma(a))\leq
A_{h}(\gamma\big{|}_{[a,b]})$. ∎

Now we formally give the statement of the result and its proof.

######  Theorem 2.14.

Given arbitrary $x_{0}\in E^{N}$, there exists a parabolic solution of
Newtonian $N$-body problem having $x_{0}$ as its initial position.

###### Proof.

According to Proposition 2.9, there is a calibrating curve $x(t)$ starting
from $x_{0}\in E^{N}$ with energy $h=0$, by Lemma 2.13, it is a free time
minimizer, and by Proposition 2.12, it is not superhyperbolic, hence by
Theorem 2.11, we have $x(t)=at+O(t^{2/3})$ for some $a\in E^{N}$. On the other
hand Theorem 1.4 states that all mutual distances of a free time minimizer
goes to infinity, hence
$\lim_{t\rightarrow+\infty}\frac{1}{2}\|\dot{x}(t)\|^{2}-U\big{(}x(t)\big{)}=\frac{1}{2}\|a\|^{2}$.
On the other hand, by the conservation of total energy,
$\frac{1}{2}\|\dot{x}(t)\|^{2}-U\big{(}x(t)\big{)}\equiv 0$, hence
$\frac{1}{2}\|a\|^{2}=0$, which imlplies $a=0$ and then $x(t)$ is parabolic. ∎

######  Remark 2.15.

If the parabolic orbit $\gamma$ in the proof of the above proposition has an
asymptotic configuration or limit shape, it is determined by the function $u$
which is calibrated by $\gamma$.

A free time minimizer with zero total energy is a parabolic motion, and
conversely it is true in some sense. Moeckel and Montgomery [18] have found
out in the three body problem, that every parabolic solution which is
asymptotic to a Lagrange configuration is eventually geodesic. They also found
that in some range of mass ratios, all geodesic rays is asymptotic to
Lagrange’s equilateral configuration, which is the absolute minimum of
$\tilde{U}$. But the case of Eulerian collinear configuration is unclear. The
above arguement combined with Maderna’s main theorem in [13] left us a
question.

######  Question 2.

Is a parabolic motion necessarily a geodesic ray?

##  3 Proof of Theorem 1.2

We give the proof of our main result in this section, before that, we need an
vital technics in the following lemma.

######  Lemma 3.1.

For sequence $a_{n}\in E^{N}$, suppose $\left\|a_{n}\right\|$ has a limit
$\lambda\geq 0$, there is a time sequence $t_{n}$ that goes to infinity such
that the $\|\cdot\|$ norm of

$p_{n}:=a_{n}t_{n}+O\left(t_{n}^{2/3}\right),$

goes to infinity. Here $O\left(t^{2/3}\right)$ means
$O\left(t^{2/3}\right)\approx t^{2/3}$ as $t\rightarrow+\infty$.

###### Proof.

  * •

If $\lambda=0$ and there is a $N^{*}$ such that $\left\\{a_{n}\right\\}$
vanishes for $n>N^{*}$, thus for any positive sequence
$\left\\{t_{k}\right\\}$ that goes to infinity, we have
$\left\|p_{n_{k}}\right\|\rightarrow+\infty$.

  * •

If $\lambda=0$ and there is a $N^{*}$ such that $\left\|a_{n}\right\|>0$ for
all $n>N$, we take $t_{n}=\left\|a_{n}\right\|^{-1}$, which also makes
$\left\|p_{n}\right\|$ and $t_{n}$ go to infinity as $t\rightarrow+\infty$.

  * •

If $\lambda>0$, then there exist a positive integer $N^{*}$ and a positive
real number $L$ such that $\left\|a_{n}\right\|>L$ for all $n>N^{*}$, hence
for any positive sequence $t_{n}$ that goes to infinity,
$\left\|p_{n}\right\|=$ $\|a_{n}t_{n}+O(t_{n}^{2/3})\|\geq
Lt_{n}+O(t_{n}^{2/3})\rightarrow+\infty$.

∎

Now we can conduct the proof of the main Theorem 1.2

###### Proof.

Recall that we have denoted by $GR\subset TE^{N}$ the set of initial data that
lead to geodesic rays. Take an arbitrary sequence
$\\{(x_{0}^{(n)},v_{0}^{(n)})\\}\subset GR$ satiesfying
$x_{0}^{(n)}\rightarrow x_{0}\in E^{N},v_{0}^{(n)}\rightarrow v_{0}\in E^{N}$
in the norm $\|\cdot\|$ as $n\rightarrow\infty$. We prove the motion $\gamma$
determined by $x_{0}$ and $v_{0}$ is also a geodesic ray, then any initial
condition in $TE^{N}\setminus GR$ must have an open neighberhood in $TE^{N}$
such that it contains no point of $GR$, and hence $TE^{N}\setminus GR$ is open
and $GR$ is closed.

Based on Theorem 1.3, There is a sequence of free time minimizers
$\gamma_{n}(t)=a_{n}t+O(t^{2/3})$ defined in $[0,+\infty)$ with intial data
$\gamma_{n}(0)=x_{0}^{(n)}\in E^{N},\dot{\gamma}_{n}(0)=v_{0}^{(n)}\in E^{N}$,
here $a_{n}\in E^{N}$. Theorem 1.4 implies $O(t^{2/3})\approx t^{2/3}$. The
energy $h_{n}$ of $\gamma_{n}$ is a constant regardless of time,
$h_{n}=\frac{1}{2}\|\dot{\gamma}_{n}(t)\|^{2}-U(\gamma(t))=\frac{1}{2}\|v_{0}^{(n)}\|^{2}-U(x_{0}^{(n)})=\frac{1}{2}\|a_{n}\|^{2}\geq
0$, therefore we suppose $h_{n}\rightarrow h\geq
0,\|a_{n}\|\rightarrow\lambda\geq 0$ as $x_{0}^{(n)},v_{0}^{(n)}$ approaching
their limits.

Let

$p_{n}=\gamma_{n}(t_{n})=a_{n}t_{n}+O(t_{n}^{2/3}),$

By Lemma 3.1 we can assume a sequence $t_{n}$ tending to infinity such that
$\left\|p_{n}\right\|\rightarrow+\infty$. Define
$u_{n}(x)=\phi_{h_{n}}(0,p_{n})-\phi_{h_{n}}(x,p_{n})$, and by Theorem 2.6, we
have

$u_{n}(x)-u_{n}(y)=\phi_{h_{n}}(y,p_{n})-\phi_{h_{n}}(x,p_{n})\leq\phi_{h_{n}}(x,y)\leq\mu(\|x-y\|),$
(3.1)

similarly $u_{n}(y)-u_{n}(x)\leq\phi_{h_{n}}(x,y)\leq\mu(\|x-y\|)$, thus we
have the inequality

$|u_{n}(y)-u_{n}(x)|\leq\phi_{h_{n}}(x,y)\leq\mu(\|x-y\|),$

which eventually means $\\{u_{n}\\}$ is equicontinuous and uniformly bounded
in any compact set of $E^{N}$ since
$|u_{n}(x)|\leq\left(A\|x\|+B\|x\|^{2}\right)^{1/2}\leq M$, $M$ is a bound
depending on the compact set. Therefore according to Ascoli’s Theorem, we can
take a subsequence of $u_{n}$ which is also denoted by $u_{n}$ for convenience
to define a function on $E^{N}$:

$u(x)=\lim_{n\rightarrow+\infty}u_{n}(x),$ (3.2)

we therefore verified that $\mathcal{B}_{h}$ is non empty. $u(x)$ is
continuous since

$|u(x)-u(y)|=\lim_{n}\left(u_{n}(x)-u_{n}(y)\right)\leq\mu(\|x-y\|).$

Because $\gamma_{n}$ is a free time minimizer of $A_{h_{n}}$, so for arbitrary
$t\geq 0$ and $t_{n}>t$, we have

$\displaystyle u_{n}(\gamma_{n}(t_{n}))-u_{n}(\gamma_{n}(t))$
$\displaystyle=u_{n}(p_{n})-u_{n}(\gamma_{n}(t))=\phi_{h_{n}}\left(\gamma_{n}(t),p_{n}\right)$
(3.3) $\displaystyle=A_{h_{n}}\left(\gamma_{n}\big{|}_{[t,t_{n}]}\right).$
(3.4)

Therefore $\gamma_{n}$ $h_{n}$-calibrates $u_{n}$, since for any
$t,t^{\prime},t_{n}$ such that $t_{n}>t^{\prime}>t\geq 0$,

$\displaystyle u_{n}(\gamma_{n}(t^{\prime}))-u_{n}(\gamma_{n}(t))=$
$\displaystyle\big{(}u_{n}(\gamma_{n}(t_{n}))-u_{n}(\gamma_{n}(t))\big{)}$
$\displaystyle-\big{(}u_{n}(\gamma_{n}(t_{n}))-u_{n}(\gamma_{n}(t^{\prime}))\big{)}$
$\displaystyle=$ $\displaystyle
A_{h_{n}}\left(\gamma_{n}\big{|}_{[t,t_{n}]}\right)-A_{h_{n}}\left(\gamma_{n}\big{|}_{[t^{\prime},t_{n}]}\right)$
$\displaystyle=$ $\displaystyle
A_{h_{n}}\left(\gamma_{n}\big{|}_{[t,t^{\prime}]}\right).$

For any $t>0$ we have

$u_{n}(\gamma_{n}(t))-u_{n}(x_{0}^{(n)})=A_{h_{n}}\left(\gamma_{n}\big{|}_{[0,t]}\right).$
(3.5)

Let $\gamma:[0,t^{*})\rightarrow\Omega$ be a solution with the maximal $t^{*}$
such that $\gamma(0)=x_{0},\dot{\gamma}(0)=v_{0}$, $\gamma$ has energy
constant

$\frac{1}{2}\|v_{0}\|^{2}-U\big{(}(x_{0})\big{)}=\lim_{n}\bigl{(}\frac{1}{2}\|v_{0}^{(n)}\|^{2}-U(x_{0}^{(n)})\bigr{)}=\lim_{n}h_{n}=h.$

Then we try to verify that $t^{*}=+\infty$ and $\gamma$ $h-$calibrates $u$,
thus by Lemma 2.13, $\gamma$ is an free time minimizer hence a geodesic ray,
and in particular by Lemma 2.12, it is not superhyperbolic.

We now prove that $\gamma$ $h$-calibrates $u$. By the continuity with respect
to the initial conditions, we have $\gamma_{n}\rightarrow\gamma$ and
$\dot{\gamma}_{n}\rightarrow\dot{\gamma}$ uniformly over any compact subset of
$[0,t^{*})$, then for every $t\in[0,t^{*})$, let $n\rightarrow+\infty$, we
have

$\displaystyle\lim_{n\rightarrow+\infty}A_{h_{n}}\left(\gamma_{n}|_{[0,t]}\right)=$
$\displaystyle\lim_{n}\left[A_{h_{n}}(\gamma_{n}|_{[0,t]})-A_{h}(\gamma|_{[0,t]})\right]+A_{h}(\gamma|_{[0,t]})$
$\displaystyle=$
$\displaystyle\lim_{n}\left[A_{h_{n}}(\gamma_{n}|_{[0,t]})-A_{h}(\gamma_{n}|_{[0,t]})+A_{h}(\gamma_{n}|_{[0,t]})-A_{h}(\gamma|_{[0,t]})\right]$
$\displaystyle+A_{h}(\gamma|_{[0,t]})$ $\displaystyle=$
$\displaystyle\lim_{n}(h_{n}-h)t+\lim_{n}\left[A_{h}(\gamma_{n}|_{[0,t]})-A_{h}(\gamma|_{[0,t]})\right]+A_{h}(\gamma|_{[0,t]})$
$\displaystyle=$ $\displaystyle A_{h}(\gamma|_{[0,t]})$

On the other hand, the sequence $h_{n}$ is bounded since it has a limit $h\geq
0$, hence according to the inequality 3.1, when $n\rightarrow+\infty$, we have

$|u_{n}(x_{0}^{(n)})-u_{p_{n}}(x_{0})|\leq\phi_{h_{n}}(x_{0}^{(n)},x_{0})\leq\mu(\|x_{0}^{(n)}-x_{0}\|)\rightarrow
0,$

and thus

$\lim_{n\rightarrow\infty}u_{n}(x_{0}^{(n)})=\lim_{n}\left(u_{n}(x_{0}^{(n)})-u_{n}(x_{0})+u_{n}(x_{0})\right)=u(x_{0}),$
(3.6)

similarly

$\lim_{n}u_{n}(\gamma_{n}(t))=u(\gamma(t)).$ (3.7)

Let $n$ in the equality 3.5 goes to infinity, we have

$u(\gamma(t))-u(x_{0})=A_{h}\left(\gamma\big{|}_{[0,t]}\right)$ (3.8)

for $t\in[0,t^{*})$, hence $\gamma$ is an $h$-calibrating curve of $u$ in its
domain.

If $t^{*}<+\infty$, we assume $t_{n}>t^{*}$ for any $n$, Since
$u\in\mathcal{B}_{h}$, we apply Maderna’s Theorem 2.9, take
$0<t^{\prime}<t^{*}$, there is an $h$-calibrating curve
$\rho:[0,+\infty)\rightarrow E^{N}$ of $u$ such that
$\rho(0)=\gamma(t^{\prime})$. The concatenation of
$\gamma\big{|}_{[0,t^{\prime}]}$ with $\rho$ is also an $h$-calibrating curve,
we denote the concatenation as $\tilde{\gamma}$. By Lemma 2.13,
$\tilde{\gamma}$ is a global free time minimizer. We notice that Theorem 2.1
implies that the concatenation $\tilde{\gamma}$ defined over $[0,+\infty)$ is
the true motion and $\dot{\gamma}(t^{\prime})=\dot{\rho}(0)$. This is a
contradiction since $t^{*}$ is finite. Hence $\gamma$ is defined in
$[0,+\infty)$ and calibrates $u$.

On the otherhand, for $u_{n}(x)-u_{n}(y)\leq\phi_{h_{n}}(x,y)$ from the
inequality 3.1, we let $n\rightarrow+\infty$ to have

$u(x)-u(y)\leq\phi_{h}(x,y),$ (3.9)

thus by Lemma 2.13 it is a free time minimizer, and by Theorem 2.3 it is a
geodesic ray.

∎

In the above proof, the sequence $\\{\|a_{n}\|\\}$ has a limit, hence
$\\{a_{n}\\}$ is bounded. We conjecture that limit shape of $\gamma$ in the
above proof is an accumulation point of the set $\\{a_{n}\\}$ in $E^{N}$.
Therefore we have another question.

######  Question 3.

Determine the limit shape of $\gamma$, or give its location range.

##  4 Acknowledgement

This work is partially supported by National Natural Science Foundation of
China (Grant No.12071316).

## References

  * [1] A. Albouy and V. Kaloshin.  Finiteness of central configurations of five bodies in the plane.  Annals of Mathematics, 176(1):535–588, 2012.
  * [2] J. Burgos.  Existence of partially hyperbolic motions in the N-body problem.  Proceedings of the American Mathematical Society, 150(04):1729–1733, 2022.
  * [3] J. Burgos and E. Maderna.  Geodesic rays of the N-body problem.  Archive for Rational Mechanics and Analysis, 243(2):807–827, Feb 2022.
  * [4] Jean Chazy.  Sur l’allure du mouvement dans le problème des trois corps quand le temps croît indéfiniment.  Annales scientifiques de l’École Normale Supérieure, 3e série, 39:29–130, 1922.
  * [5] A. Chenciner.  Action minimizing solutions of the Newtonian $n$-body problem: From homology to symmetry.  In Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. III: Invited lectures, pages 279–294, Beijing, 2002. Beijing: Higher Education Press.
  * [6] F. Clarke.  Functional Analysis, Calculus of Variations and Optimal Control.  Graduate Texts in Mathematics. Springer London, London, 1 edition, 2013\.
  * [7] A. D. Luz and E. Maderna.  On the free time minimizers of the Newtonian N-body problem.  Mathematical Proceedings of the Cambridge Philosophical Society, 156(2):209–227, 2014.
  * [8] N. Duignan, R. Moeckel, R. Montgomery, and G. Yu.  Chazy-type asymptotics and hyperbolic scattering for the $n$-body problem.  Archive for Rational Mechanics and Analysis, 238(1):255–297, Oct 2020.
  * [9] M. Hampton and A. Jensen.  Finiteness of spatial central configurations in the five-body problem.  Celestial Mechanics and Dynamical Astronomy, 109(4):321–332, Apr 2011.
  * [10] M. Hampton and R. Moeckel.  Finiteness of relative equilibria of the four-body problem.  Inventiones mathematicae, 163(2):289–312, Feb 2006.
  * [11] X. Hu, Y. Ou, and G. Yu.  An index theory for collision, parabolic and hyperbolic solutions of the Newtonian $n$-body problem.  Archive for Rational Mechanics and Analysis, 240(1):565–603, Apr 2021.
  * [12] E. Maderna.  On weak KAM theory for N-body problems.  Ergodic Theory and Dynamical Systems, 32(3):1019–1041, 2012.
  * [13] E. Maderna and A. Venturelli.  Globally minimizing parabolic motions in the Newtonian N-body problem.  Archive for Rational Mechanics and Analysis, 194(1):283–313, Oct 2009.
  * [14] E. Maderna and A. Venturelli.  Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem.  Annals of Mathematics, 192(2):499 – 550, 2020.
  * [15] C. Marchal.  How the method of minimization of action avoids singularities.  Celestial Mechanics and Dynamical Astronomy, 83(1):325–353, May 2002.
  * [16] C. Marchal and D. G. Saari.  On the final evolution of the n-body problem.  Journal of Differential Equations, 20(1):150–186, 1976.
  * [17] R. McGehee.  Von Zeipel’s theorem on singularities in celestial mechanics.  Expo. Math., 4:335–345, 1986.
  * [18] R. Moeckel, R. Montgomery, and H. S. Morgado.  Free time minimizers for the three-body problem.  Celestial Mechanics and Dynamical Astronomy, 130(3):28, Mar 2018\.
  * [19] Boris Percino and Héctor Sánchez-Morgado.  Busemann functions for the n-body problem.  Archive for Rational Mechanics and Analysis, 213(3):981–991, Sep 2014.
  * [20] H. Poincaré.  Sur les solutions périodiques et le principe de la moindre action.  C. R. Acad. Sci., Paris, 123:915–918, 1896.
  * [21] H. Pollard.  The behavior of gravitational systems.  Journal of Mathematics and Mechanics, 17(6):601–611, 1967.
  * [22] D. G. Saari.  Improbability of collisions in Newtonian gravitational systems.  Transactions of the American Mathematical Society, 162:267–271, 1971.
  * [23] D. G. Saari.  On oscillatory motion in gravitational systems.  Journal of Differential Equations, 14(2):275–292, 1973.
  * [24] D. G. Saari.  A global existence theorem for the four-body problem of Newtonian mechanics.  Journal of Differential Equations, 26(1):80–111, 1977.
  * [25] D. G. Saari.  The manifold structure for collision and for hyperbolic-parabolic orbits in the n-body problem.  Journal of Differential Equations, 55(3):300–329, 1984.
  * [26] D. G. Saari and N. D. Hulkower.  On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem.  Journal of Differential Equations, 41(1):27–43, 1981.
  * [27] D. G Saari and Z. Xia.  The existence of oscillatory and superhyperbolic motion in Newtonian systems.  Journal of Differential Equations, 82(2):342–355, 1989.
  * [28] Donald G. Saari.  Expanding gravitational systems.  Transactions of the American Mathematical Society, 156:219–240, 1971.
  * [29] K. A. Sitnikov.  The existence of oscillatory motions in the three-body problem.  Soviet physics. Doklady, 5:647, 1961.
  * [30] H. von Zeipel.  Sur les singularités du probleme des n corps.  Ark. Math. Astr. Fys., (4):1–4, 1908.
  * [31] X. Yu and S. Zhang.  Action-minimizing solutions of the one-dimensional N-body problem.  Celestial Mechanics and Dynamical Astronomy, 130(37), 2018.
  * [32] S. Zhang.  Variational minimizing parabolic and hyperbolic orbits for the restricted 3-body problems.  Science China Mathematics, 55(4):721–725, Apr 2012.