Open Access

Connecting orbits for Newtonian-like N-body problems

Journal of Inequalities and Applications20152015:197

https://doi.org/10.1186/s13660-015-0715-3

Received: 16 November 2014

Accepted: 26 May 2015

Published: 16 June 2015

Abstract

Using variational minimizing methods, we prove the existence of a connecting orbit between the center of mass and infinity of Newtonian-like N-body problems with Newtonian-type weak force potentials.

Keywords

N-body problems variational minimizers connecting orbits

MSC

34A34 34C25 34C37 35A15

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 [215]. 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)
 □

Declarations

Acknowledgements

The authors sincerely thank the referees for their many valuable comments which help us improving the paper. This paper was partially supported by NSF of China (No. 11071175 and No. 11426181) and Fundamental Research Funds for the Central Universities (JBK 130401 and JBK 150931).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Economics and Mathematics, Southwestern University of Finance and Economics
(2)
Department of Mathematics, Sichuan University

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