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: where \(q_{i}\in R^{k}\), \(q=(q_{1},\ldots,q_{N})\), \(0<\alpha<2\), and
$$ m_{i}\ddot{q}_{i}(t)+\frac{\partial U(q)}{\partial q_{i}}=0, $$
(1.1)
$$ U(q)= \sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|q_{i}-q_{j}|^{\alpha}} . $$
(1.2)
Anzeige
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:
(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)
(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 where 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 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
$$ f(q)=\int_{0}^{\tau} \Biggl(\frac{1}{2}\sum _{i=1}^{N} m_{i}\bigl\vert \dot{q}_{i}(t)\bigr\vert ^{2}+U(q) \Biggr)\, dt, $$
(2.1)
$$ q_{i}\in H_{\tau}=\bigl\{ x,\dot{x}\in L^{2}[0, \tau]|x_{i}(0)=0,x_{i}(\tau)=a_{i}\bigr\} , $$
(2.2)
$$ \sum_{j\neq i}\frac{m_{j}m_{i}(a_{j}-a_{i})}{|a_{j}-a_{i}|^{\alpha+2}}=\lambda m_{i}a_{i} . $$
(2.3)
$$ \|q\|_{\tau}= \Biggl(\sum_{i=1}^{N}m_{i} \int_{0}^{\tau}\bigl\vert \dot {q}_{i}(t) \bigr\vert ^{2}\, dt \Biggr)^{1/2}. $$
(2.4)
Lemma 2.1
(Tonelli [16])
Anzeige
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. Since \(q^{n}(t)\rightarrow q(t)\) uniformly we have \(\int_{0}^{\tau} U(q^{n}(t))\, dt\rightarrow+\infty\), and so □
(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)
(ii)
If \(m(S)>0\), then \(\int_{0}^{\tau} U(q)\, dt=+\infty\) and \(f(q)=+\infty\).
$$ \varliminf_{n\rightarrow\infty}f\bigl(q^{n}\bigr)\geq f(q). $$
(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)
\(f(q)\)
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)
Furthermore, when
\(\tau\rightarrow+\infty\)
and
\(\tilde{q}^{\tau}_{i}(t)\rightarrow\tilde{q}_{i}(t)\), \(\tilde{q}_{i}(t)\)
has the following properties:
(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)
(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),
otherwise, we call it a mixed type solution.
(1∘)
if, for all i, we say \(\tilde{q}(t)\) is a parabolic solution;
$$ \bigl\vert \dot{\tilde{q}}_{i}(t)\bigr\vert \rightarrow0,\quad t \rightarrow +\infty $$
(2.10)
(2∘)
if, for all i, we say \(\tilde{q}(t)\) is a hyperbolic solution;
$$ \bigl\vert \dot{\tilde{q}}_{i}(t)\bigr\vert \rightarrow v_{i}>0, \quad t\rightarrow +\infty $$
(2.11)
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 where \((a_{1},a_{2},\ldots,a_{N})\) can be a given central configuration, \(\frac {1}{2}<\beta< \min\{1,\frac{1}{\alpha}\}\), then where and 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 \(0<\theta<1\). □
$$ q_{i}(t)=a_{i}t^{\beta},\quad t\in[0, \tau],a_{i}\in R^{k}, $$
(2.13)
$$\begin{aligned} f\bigl(q(t)\bigr) =&\frac{1}{2}\sum_{i=1}^{N} m_{i}|a_{i}|^{2} \int_{0}^{\tau} \beta ^{2}t^{2(\beta-1)}\, dt +\int_{0}^{\tau} \sum_{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|a_{i}-a_{j}|^{\alpha}}t^{-\alpha \beta}\, dt \\ \leq&\frac{1}{2} \Biggl( \sum_{i=1}^{N}m_{i}|a_{i}|^{2} \Biggr)\frac {\beta^{2}}{2\beta-1}\tau^{2\beta-1} \\ &{}+ \biggl(\sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|a_{i}-a_{j}|^{\alpha }} \biggr)\frac{1}{1-\alpha\beta} \tau^{1-\alpha\beta} \\ \leq& c\tau^{\theta}, \end{aligned}$$
(2.14)
$$ \theta= \max(2\beta-1,1-\alpha\beta) $$
(2.15)
$$ c=\frac{1}{2}\sum_{1}^{N}m_{i}|a_{i}|^{2} \frac{\beta^{2}}{2\beta-1}+ \sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{|a_{i}-a_{j}|^{\alpha}} \frac{1}{1-\alpha\beta}>0. $$
(2.16)
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 Hence from which it follows that \(\forall1\leq i< j\leq N\), \(\|\tilde {q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t)\|_{\infty}\rightarrow +\infty\), \(n\rightarrow+\infty\). □
$$ cn^{\theta}\geq f\bigl(\tilde{q}^{n}\bigr)\geq\int _{0}^{n} \sum _{1\leq i< j\leq N} \frac{m_{i}m_{j}}{|\tilde{q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t)|^{\alpha}}\, dt. $$
(2.18)
$$ \sum _{1\leq i< j\leq N}\frac{m_{i}m_{j}}{\|\tilde{q}_{i}^{n}(t) - \tilde{q}_{j}^{n}(t)\|^{\alpha}_{\infty}}\leq c n^{\theta-1} \rightarrow0, $$
(2.19)
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 \(\forall T>0\), Then, for any \(0\leq s,r\leq T\), we have
$$ \sum_{i=1}^{N}m_{i}\int _{0}^{T}\bigl\vert \dot{\tilde{q}}_{i}^{n}(t) \bigr\vert ^{2}\, dt \leq cT^{\theta}. $$
(2.20)
$$\begin{aligned} \bigl|\tilde{q}_{i}^{n}(s)-\tilde{q}_{i}^{n}(r)\bigr| &\leq\int_{r}^{s}\bigl|\dot{\tilde{q}}_{i}^{n}(t)\bigr| \, dt \\ &\leq|s-r|^{1/2} \biggl(\int_{r}^{s} \bigl\vert \dot{\tilde{q}}_{i}^{n}(t)\bigr\vert ^{2}\, dt \biggr)^{1/2} \\ &\leq \biggl(\frac{cT^{\theta}}{m_{i}} \biggr)^{1/2}|s-r|^{1/2}. \end{aligned}$$
(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 rewritten as
$$ E=\sum_{i=1}^{N}\frac{1}{2}m_{i}| \dot{\tilde{q}}_{i}|^{2}- \sum _{1\leq i< j\leq N} \frac{m_{i}m_{j}}{|\tilde{q}_{i}-\tilde{q}_{j}|^{\alpha}}\geq0, $$
(2.23)
$$ \sum _{i=1}^{N}\frac{1}{2}m_{i}| \dot{\tilde{q}}_{i}|^{2}= \sum _{1\leq i< j\leq N} \frac{m_{i}m_{j}}{|\tilde{q}_{i}-\tilde{q}_{j}|^{\alpha}}+E. $$
(2.24)
Now we claim:
(i) for any \(1\leq i\neq j\leq N\), suppose there exist \(1\leq i_{0}< j_{0}\leq N\) and \(d>0\) such that
$$ \max_{t\in R^{+}}\bigl\vert \tilde{q}_{i}(t)- \tilde{q}_{j}(t)\bigr\vert = +\infty $$
(2.25)
$$ \bigl\vert \tilde{q}_{i_{0}}(t)-\tilde{q}_{j_{0}}(t)\bigr\vert < d,\quad \forall t\in R^{+}. $$
(2.26)
By (2.24), there exist \(1\leq k_{0}\leq N\) and \(e>0\) such that then we have
$$ \vert \dot{\tilde{q}}_{k_{0}}\vert >e, \quad \forall t\in R^{+}, $$
(2.27)
$$ ct^{\theta}\geq\frac{1}{2}\int_{0}^{t} \sum_{i=1}^{N} m_{i}\vert \dot{ \tilde{q}}_{i}\vert ^{2}\, dt\geq\frac{1}{2}\int _{0}^{t} m_{k_{0}}\vert \dot{ \tilde{q}}_{k_{0}}\vert ^{2}\, dt\geq\frac{1}{2}m_{k_{0}}e^{2}t. $$
(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)
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The research and writing of this manuscript was a collaborative effort from all the authors. All authors read and approved the final manuscript.