1 Introduction

In the 1989 paper of Rabinowitz [1], we find the first substantial use of variational methods to study heteroclinic orbits for Hamiltonian systems. The perspective of that work appears influential for a number of papers by several authors which followed [2–15]. Especially, we would like to draw attention to Souissi [13], Maderna and Venturelli [14] and Zhang [15] for a study of the parabolic orbits for restricted 3-body problems and complete N-body problems. From those studies, we draw motivation for the present work: namely, we extend the results and methods of Souissi [13] and Zhang [15] to Newtonian-like N-body problems.

Given masses \(m_{1},\ldots,m_{N}>0\) of N bodies, we study the following system of equations with Newtonian-type weak force potentials:

$$ m_{i}\ddot{q}_{i}(t)+\frac{\partial U(q)}{\partial q_{i}}=0, $$
(1.1)

where \(q_{i}\in R^{k}\), \(q=(q_{1},\ldots,q_{N})\), \(0<\alpha<2\), and

$$ U(q)= \sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|q_{i}-q_{j}|^{\alpha}} . $$
(1.2)

We apply the variational minimizing method to prove the following.

Theorem 1.1

For (1.1), there exists one connecting orbit \(\tilde{q}(t)=(\tilde{q}_{1}(t),\ldots,\tilde{q}_{N}(t))\) between the center of mass and infinity such that:

  1. (i)

    For any \(1\leq i\neq j\leq N\),

    $$ \max_{0\leq t\leq+\infty}\bigl\vert \tilde{q}_{i}(t)-\tilde {q}_{j}(t)\bigr\vert =+\infty. $$
    (1.3)
  2. (ii)
    $$ \min_{0\leq t\leq+\infty}\sum_{1}^{N}m_{i} \bigl\vert \dot{\tilde {q}}_{i}(t)\bigr\vert ^{2}=2E \geq0. $$
    (1.4)

2 Variational minimizing critical points

In order to find a connecting orbit of (1.1), we shall first find a solution of the system (1.1) on the open interval \((0,\tau)\) and then consider the limit orbit as \(\tau\rightarrow +\infty\). To find a solution on \((0,\tau)\), we define the functional

$$ f(q)=\int_{0}^{\tau} \Biggl(\frac{1}{2}\sum _{i=1}^{N} m_{i}\bigl\vert \dot{q}_{i}(t)\bigr\vert ^{2}+U(q) \Biggr)\, dt, $$
(2.1)

where

$$ q_{i}\in H_{\tau}=\bigl\{ x,\dot{x}\in L^{2}[0, \tau]|x_{i}(0)=0,x_{i}(\tau)=a_{i}\bigr\} , $$
(2.2)

where \((a_{1},\ldots,a_{i},\ldots,a_{N})\) is a central configuration for the N-body problems which satisfies \(a_{j}\neq a_{i}\), \(1\leq j\neq i\leq N\), and there is \(\lambda \in R\) such that

$$ \sum_{j\neq i}\frac{m_{j}m_{i}(a_{j}-a_{i})}{|a_{j}-a_{i}|^{\alpha+2}}=\lambda m_{i}a_{i} . $$
(2.3)

Since \(\forall q_{i}\in H_{\tau}\), \(q_{i}(0)=0\), for \(q=(q_{1},\ldots,q_{N})\in H_{\tau}\times\cdots\times H_{\tau}\) we have the equivalent norm

$$ \|q\|_{\tau}= \Biggl(\sum_{i=1}^{N}m_{i} \int_{0}^{\tau}\bigl\vert \dot {q}_{i}(t) \bigr\vert ^{2}\, dt \Biggr)^{1/2}. $$
(2.4)

Lemma 2.1

(Tonelli [16])

Let X be a reflexive Banach space and \(f:X\rightarrow R\cup\{+\infty\}\). If f does not always take +∞ and is weakly lower semi-continuous and coercive (\(f(x)\rightarrow +\infty\), as \(\|x\|\rightarrow+\infty\)), then f attains its infimum on X.

Lemma 2.2

The functional \(f(q)\) defined in (2.1) is weakly lower semi-continuous (w.l.s.c.) on \(H_{\tau}\times\cdots\times H_{\tau}\).

Proof

(1) It is well known that the norm and its square are w.l.s.c.

(2) \(\forall\{q_{i}^{n}\}\subset H_{\tau}\), if \(q_{i}^{n}\rightharpoonup q_{i}\) weakly, then by the compact embedding theorem, we have the following uniform convergence:

$$ \max_{0\leqslant t\leqslant\tau} \bigl\vert q_{i}^{n}(t)-q_{i}(t) \bigr\vert \rightarrow0,\quad n\rightarrow+\infty. $$
(2.5)

Let \(S=\{\tilde{t}\in[0,\tau]: \exists1\leq i_{0}\neq j_{0}\leq N\mbox{ s.t. } q_{i_{0}}(t_{0})=q_{j_{0}}(t_{0})\}\) and let \(m(S)\) denote the Lebesgue measure of S.

  1. (i)

    If \(m(S)=0\), then \(U(q^{n}(t)) \stackrel{\text{a.e.}}{\rightarrow } U(q(t))\). From Fatou’s lemma we have

    $$ \int_{0}^{\tau} U(q)\, dt\leq \varliminf_{n\rightarrow\infty} \int_{0}^{\tau} U\bigl(q^{n}(t)\bigr)\, dt. $$
    (2.6)
  2. (ii)

    If \(m(S)>0\), then \(\int_{0}^{\tau} U(q)\, dt=+\infty\) and \(f(q)=+\infty\).

Since \(q^{n}(t)\rightarrow q(t)\) uniformly we have \(\int_{0}^{\tau} U(q^{n}(t))\, dt\rightarrow+\infty\), and so

$$ \varliminf_{n\rightarrow\infty}f\bigl(q^{n}\bigr)\geq f(q). $$
(2.7)

 □

The proof of the next lemma is straightforward.

Lemma 2.3

f is coercive on \(H_{\tau}\times\cdots\times H_{\tau}\).

Lemma 2.4

  1. (1)

    \(f(q)\) attains its infimum on \(H_{\tau}\times\cdots\times H_{\tau}\), and the minimizer \(\tilde{q}^{\tau}(t)=(\tilde{q}^{\tau}_{1}(t),\ldots,\tilde {q}_{N}^{\tau}(t))\) is a generalized solution [16].

  2. (2)

    Furthermore, when \(\tau\rightarrow+\infty\) and \(\tilde{q}^{\tau}_{i}(t)\rightarrow\tilde{q}_{i}(t)\), \(\tilde{q}_{i}(t)\) has the following properties:

    1. (i)

      for any \(1\leq i\neq j\leq N\),

      $$ \max_{0\leq t\leq+\infty}\bigl\vert \tilde{q}_{i}(t)-\tilde {q}_{j}(t)\bigr\vert =+\infty, $$
      (2.8)
    2. (ii)
      $$ \min_{0\leq t\leq+\infty}\sum_{1}^{N}m_{i} \bigl\vert \dot{\tilde {q}}_{i}(t)\bigr\vert ^{2}=2E. $$
      (2.9)

Definition 2.5

Concerning the velocities of the solution of (1.1),

(1∘):

if, for all i,

$$ \bigl\vert \dot{\tilde{q}}_{i}(t)\bigr\vert \rightarrow0,\quad t \rightarrow +\infty $$
(2.10)

we say \(\tilde{q}(t)\) is a parabolic solution;

(2∘):

if, for all i,

$$ \bigl\vert \dot{\tilde{q}}_{i}(t)\bigr\vert \rightarrow v_{i}>0, \quad t\rightarrow +\infty $$
(2.11)

we say \(\tilde{q}(t)\) is a hyperbolic solution;

otherwise, we call it a mixed type solution.

The proof of (1) in Lemma 2.4 is obvious using Lemmas 2.1-2.3.

In the following, we will give the proofs of (2.8) and (2.9) of Lemma 2.4.

Lemma 2.6

There exist constants \(c>0\) and \(0<\theta<1\) independent of τ such that

$$ f\bigl(\tilde{q}^{\tau}\bigr)\leq c\tau^{\theta}. $$
(2.12)

Proof

We choose a special orbit defined by

$$ q_{i}(t)=a_{i}t^{\beta},\quad t\in[0, \tau],a_{i}\in R^{k}, $$
(2.13)

where \((a_{1},a_{2},\ldots,a_{N})\) can be a given central configuration, \(\frac {1}{2}<\beta< \min\{1,\frac{1}{\alpha}\}\), then

$$\begin{aligned} f\bigl(q(t)\bigr) =&\frac{1}{2}\sum_{i=1}^{N} m_{i}|a_{i}|^{2} \int_{0}^{\tau} \beta ^{2}t^{2(\beta-1)}\, dt +\int_{0}^{\tau} \sum_{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|a_{i}-a_{j}|^{\alpha}}t^{-\alpha \beta}\, dt \\ \leq&\frac{1}{2} \Biggl( \sum_{i=1}^{N}m_{i}|a_{i}|^{2} \Biggr)\frac {\beta^{2}}{2\beta-1}\tau^{2\beta-1} \\ &{}+ \biggl(\sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|a_{i}-a_{j}|^{\alpha }} \biggr)\frac{1}{1-\alpha\beta} \tau^{1-\alpha\beta} \\ \leq& c\tau^{\theta}, \end{aligned}$$
(2.14)

where

$$ \theta= \max(2\beta-1,1-\alpha\beta) $$
(2.15)

and

$$ c=\frac{1}{2}\sum_{1}^{N}m_{i}|a_{i}|^{2} \frac{\beta^{2}}{2\beta-1}+ \sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|a_{i}-a_{j}|^{\alpha}} \frac{1}{1-\alpha\beta}>0. $$
(2.16)

When \(0<\alpha<2\), we have \(\frac{1}{\alpha}>\frac{1}{2}\). We can choose \(\frac{1}{2}<\beta<\frac{1}{\alpha}\), then \(2\beta-1>0\), \(1-\alpha\beta>0\), and hence \(\theta>0\). When \(\beta<1\), \(2\beta-1<1\), then \(0<\theta<1\). □

Lemma 2.7

Let \(\tilde{q}^{n}(t)=(\tilde{q}^{n}_{1}(t),\ldots,\tilde{q}_{N}^{n}(t))\) be critical points corresponding to the minimizing critical values \(\min_{H_{n}}f(q)\), where \(H_{n}\) was defined in (2.2) when \(\tau =n\). Then the maximum distance between \(\tilde{q}^{n}_{i}\) and \(\tilde{q}^{n}_{j}\) on \(R^{+}\) satisfies

$$ \bigl\Vert \tilde{q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t) \bigr\Vert _{\infty}\rightarrow +\infty, \quad \textit{when } n\rightarrow+ \infty. $$
(2.17)

Proof

By the definition of \(f(\tilde{q}^{n})\) and Lemma 2.6, we have the inequalities

$$ cn^{\theta}\geq f\bigl(\tilde{q}^{n}\bigr)\geq\int _{0}^{n} \sum _{1\leq i< j\leq N} \frac{m_{i}m_{j}}{|\tilde{q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t)|^{\alpha}}\, dt. $$
(2.18)

Hence

$$ \sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{\|\tilde{q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t)\|^{\alpha}_{\infty}}\leq c n^{\theta-1} \rightarrow0, $$
(2.19)

from which it follows that \(\forall1\leq i< j\leq N\), \(\|\tilde {q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t)\|_{\infty}\rightarrow +\infty\), \(n\rightarrow+\infty\). □

Lemma 2.8

\(\{\tilde{q}^{n}(t)\}\) is equi-continuous and uniformly bounded on any compact interval.

Proof

By the proof of Lemma 2.6, we can see \(\forall T>0\),

$$ \sum_{i=1}^{N}m_{i}\int _{0}^{T}\bigl\vert \dot{\tilde{q}}_{i}^{n}(t) \bigr\vert ^{2}\, dt \leq cT^{\theta}. $$
(2.20)

Then, for any \(0\leq s,r\leq T\), we have

$$\begin{aligned} \bigl|\tilde{q}_{i}^{n}(s)-\tilde{q}_{i}^{n}(r)\bigr| &\leq\int_{r}^{s}\bigl|\dot{\tilde{q}}_{i}^{n}(t)\bigr| \, dt \\ &\leq|s-r|^{1/2} \biggl(\int_{r}^{s} \bigl\vert \dot{\tilde{q}}_{i}^{n}(t)\bigr\vert ^{2}\, dt \biggr)^{1/2} \\ &\leq \biggl(\frac{cT^{\theta}}{m_{i}} \biggr)^{1/2}|s-r|^{1/2}. \end{aligned}$$
(2.21)

By \(q^{n}(0)=0\) and the above inequality, for \(0< s< T\), we have

$$ \bigl\vert \tilde{q}_{i}^{n}(s)\bigr\vert \leq \biggl( \frac{cT^{\theta}}{m_{i}} \biggr)^{1/2}|s|^{1/2}\leq \biggl( \frac{cT^{\theta}}{m_{i}} \biggr)^{1/2}T^{1/2}. $$
(2.22)

 □

Now we can prove Theorem 1.1.

Proof of Theorem 1.1

For any compact interval \([a,b]\) of \(R^{+}\), Marchal’s theorem [17] implies that \(\tilde{q}^{n}(t)\) has no collision on \((a,b)\), so, by the Ascoli-Arzelà theorem, we know \(\{\tilde {q}^{n}\}\) has a sub-sequence converging uniformly to a limit \(\tilde{q}(t)\) on any compact set \([c,d]\subset(a,b)\), and \(\tilde{q}(t)\in C^{2}(R^{+},R^{k})\) is a solution of (1.1). By the energy conservation law and (2.17), we have

$$ E=\sum_{i=1}^{N}\frac{1}{2}m_{i}| \dot{\tilde{q}}_{i}|^{2}- \sum _{1\leq i< j\leq N} \frac{m_{i}m_{j}}{|\tilde{q}_{i}-\tilde{q}_{j}|^{\alpha}}\geq0, $$
(2.23)

rewritten as

$$ \sum _{i=1}^{N}\frac{1}{2}m_{i}| \dot{\tilde{q}}_{i}|^{2}= \sum _{1\leq i< j\leq N} \frac{m_{i}m_{j}}{|\tilde{q}_{i}-\tilde{q}_{j}|^{\alpha}}+E. $$
(2.24)

Now we claim:

(i) for any \(1\leq i\neq j\leq N\),

$$ \max_{t\in R^{+}}\bigl\vert \tilde{q}_{i}(t)- \tilde{q}_{j}(t)\bigr\vert = +\infty $$
(2.25)

suppose there exist \(1\leq i_{0}< j_{0}\leq N\) and \(d>0\) such that

$$ \bigl\vert \tilde{q}_{i_{0}}(t)-\tilde{q}_{j_{0}}(t)\bigr\vert < d,\quad \forall t\in R^{+}. $$
(2.26)

By (2.24), there exist \(1\leq k_{0}\leq N\) and \(e>0\) such that

$$ \vert \dot{\tilde{q}}_{k_{0}}\vert >e, \quad \forall t\in R^{+}, $$
(2.27)

then we have

$$ ct^{\theta}\geq\frac{1}{2}\int_{0}^{t} \sum_{i=1}^{N} m_{i}\vert \dot{ \tilde{q}}_{i}\vert ^{2}\, dt\geq\frac{1}{2}\int _{0}^{t} m_{k_{0}}\vert \dot{ \tilde{q}}_{k_{0}}\vert ^{2}\, dt\geq\frac{1}{2}m_{k_{0}}e^{2}t. $$
(2.28)

This is a contradiction, since \(0<\theta<1\) and \(t\in R^{+}\).

Now by (2.24), we have:

$$ (\mathrm{ii})\quad \min_{t\in R^{+}}\sum_{i=1}^{N}m_{i} \bigl\vert \dot{\tilde{q}}_{i}(t)\bigr\vert ^{2}=2E\geq0. $$
(2.29)

 □