Noncollision Periodic Solutions for Circular Restricted Planar Newtonian Four-Body Problems

Abstract

We study a class of circular restricted planar Newtonian four-body problems in which three masses are positioned at the vertices of a Lagrange equilateral triangle configuration, each mass revolving around the center of mass in circular orbits. Assuming that the value of the fourth mass is negligibly small (i.e., it does not perturb the motion of the other three masses, though its own motion is influenced by them), we use variational minimization methods to prove the existence of noncollision periodic solutions with some fixed winding numbers. These noncollision solutions exist for both equal and unequal mass values for the three bodies located at the vertices of the Lagrange equilateral configuration.

1. Introduction and Main Results

The Newtonian n-body problem studies the motion of n masses governed by Newton’s second law and the law of universal gravitation:

$$\begin{array}{c}\hfill m_k{\ddot{q}}_k={\displaystyle \frac{{\displaystyle \partial \left(\sum _{1\le k<s\le n}\frac{m_k{m}_s}{|q_k-q_s|}\right)}}{\partial q_k}},\end{array}$$

where $k\in \{1,2,\dots ,n\}$, the mass $m_k>0$, and the corresponding position $q_k\in {\mathbb{R}}^j$ with $j\in \{1,2,3\}$.

Henri Poincaré was one of the earliest scholars to prove that the periodic solution of n-body problem exists by using the variational method. In [1], when he studied the restricted three-body problem, he adopted the method of minimizing the action and proved the existence of a periodic solution. But to simplify analysis, he used a strong force with inverse power greater than 2. n-body problems with Newtonian potentials are very difficult and important problems due to the possible singularites for collisions and higher nonlinear properties. There is a well-established research program that uses variational methods to study noncollision periodic solutions for n-body problems; see [2,3,4,5,6,7,8,9,10,11,12,13]. Especially, Simó [7] used computer explorations to discover many new periodic solutions of n-body problems with Newtonian potentials, and in [2], for a planar Newtonian three-body problem with equal masses, a figure-“eight”-type periodic solution was proven to exist. For the fixed ends (Bolza problems) of Newtonian n-body problems, by developing a few optimization theories, and using the analysis of interference of singularities, Marchal [6] proved that the minimizer for the Lagrangian action has no interior collision. Later, based on [6], Ferrario and Terracini [3] developed an averaging technique by replacing some of the point masses with suitable shapes, and they proved that when the motion has certain symmetry, the noncollision periodic solution exists under some group action.

Another method was developed by Zhang and Zhou [9,10,11]. First, they decomposed the Lagrange actions for both three masses and n masses into sums of relative two-body problems. They then derived the largest possible lower-bound estimate for the Lagrangian action on symmetric generalized periodic solutions and compared it with the Lagrangian action on test orbits. Using this approach, Zhang, Zhou, and Liu [12] proved the existence of new noncollision symmetric periodic solutions of planar Newtonian three-body problems, where the masses of bodies are equal or unequal. In 2015, Ouyang and Xie [14] obtained infinitely many simple choreographic solutions to the Newtonian four-body problem. For more details on this topic, one can refer to [15,16,17].

Restrictive n-body problems have strong practical significance. As early as the 18th century, Euler and Lagrange respectively derived collinear and equilateral triangular solutions for restricted three-body problems. Restricted three-body models have important aerospace applications, e.g., spacecraft motion in the Earth–Moon and Sun–Earth systems. In [18], Álvarez-Ramíırez and Vidal showed the existence of periodic solutions for an equilateral restricted four-body problem, where three bodies were fixed in order to eliminate the time dependence, and the mass of second body and the third body were same. And they characterized the region of the possible motions and the surface of the fixed level for the fourth small-enough mass in the spatial as well as in the planar case. In 2024, Bengochea, Burgos-Garca, and Pérez-Chavela [17] studied symmetric periodic orbits near collision in a nonautonomous restricted planar four-body problem with the three same positive-mass bodies following the figure-“eight” choreography. For more restricted n-body problems, see [19,20,21,22,23,24,25,26,27,28,29] and the references therein.

Motivated by the above work, we consider a special restricted four-body system which is different from others, and our aim in this paper is to use the minimization property to prove the noncollision property, which is small. Assume that three masses, $m_1,m_2$, and $m_3$, move in the circular orbits $q_1\left(t\right),q_2\left(t\right)$, and $q_3\left(t\right)$ in a plane around the center of mass, forming a Lagrange equilateral triangle configuration, and the fourth body has relatively sufficiently small mass compared to the other three bodies; that is, it does not influence the motion of the primary three bodies but moves in the same plane as the three bodies. Its orbit is $q_4\left(t\right)$. More precisely, $q_1\left(t\right),q_2\left(t\right),q_3\left(t\right)$, and $q_4\left(t\right)$ satisfy the following equations:

$$\begin{array}{c}\hfill m_4{\ddot{q}}_4=\sum _{j=1}^3{\displaystyle \frac{m_4{m}_j(q_j-q_4)}{|q_j-q_4{|}^3}},m_4\ll min\{m_1,m_2,m_3\},\end{array}$$

(1)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{m}_k{\ddot{q}}_k={\displaystyle \frac{{\displaystyle \partial \left(\sum _{1\le k<s\le 4}\frac{m_k{m}_s}{|q_k-q_s|}\right)}}{\partial q_k}},k\in \{1,2,3\},\hfill \\ q_k\left(t\right)=r_k{e}^{i(2\pi t+{\theta }_k)},\mathrm{satisfying}{m}_1{q}_1\left(t\right)+m_2{q}_2\left(t\right)+m_3{q}_3\left(t\right)=0,\hfill \end{array}\right.\end{array}$$

(2)

where the angle ${\theta }_k\in [0,2\pi )$, radii $r_k>0$, and $|q_k-q_j|\equiv l>0$ with the integer $1\le j\ne k\le 3$ and constant $l>0$.

In this paper, we use variational methods to study the noncollision periodic solutions of the circular restricted planar Newtonian four-body problems with some fixed winding numbers. For the readers’ convenience, we look back upon the definition of the winding number.

Definition 1  

(Definition 1.1 [12]). Let $[a,b]\ni t\to x\left(t\right)$ be a given oriented continuous closed curve, and let Γ denote its image as a set. Let p be a point in the plane but not on the curve. The mapping $\phi :\Gamma \to S^1$ is given by

$$\begin{array}{c}\hfill \phi \left(x\left(t\right)\right)={\displaystyle \frac{x\left(t\right)-p}{\left|x\right(t)-p|}},t\in [a,b],\end{array}$$

which is defined to be the position mapping of the curve Γ relative to p. When a point goes around the curve Γ once, its image point $\phi \left(x\right(t\left)\right)$ will wind around $S^1$ a number of times. This number is called the winding number of the curve, Γ, relative to p, and we denote it by $deg(\Gamma ,p)$. If p is the origin, it can be written as $deg\Gamma$.

From Definition 1 and (2), we know that

$$\begin{array}{c}\hfill deg(q_2-q_1)=1\mathrm{and}deg(q_3-q_1)=1.\end{array}$$

Let

$$\begin{array}{c}\hfill W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)=\left\{x\left(t\right)|x\left(t\right),\dot{x}\left(t\right)\in L^2(\mathbb{R},{\mathbb{R}}^2),x(t+1)=x\left(t\right)\right\}.\end{array}$$

The norm of $W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)$ is

$$\parallel x\parallel ={\left[{\int }_0^1{\left|x\right|}^{2}dt\right]}^{{\displaystyle \frac{1}{2}}}+{\left[{\int }_0^1{\left|\dot{x}\right|}^{2}dt\right]}^{{\displaystyle \frac{1}{2}}}.$$

(3)

In System (1), we can eliminate $m_4$ from both sides. For simplicity, let $q\left(t\right)=q_4\left(t\right)$ in the subsequent text; then the Lagrangian functional of System (1) is the following:

$$f\left(q\right)={\int }_0^1\left[{\displaystyle \frac{1}{2}}{\left|\dot{q}\right|}^2+\sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}\right]dt$$

(4)

on ${\Lambda }_{\pm }$, where

$${\Lambda }_{-}=\left\{q\in W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)|\begin{array}{c}q(t+{\displaystyle \frac{1}{2}})=-q\left(t\right),q\left(t\right)\ne q_j\left(t\right),\forall t\in [0,1],j=1,2,3,\\ deg(q-q_1)=-1,deg(q-q_2)=-1,\\ deg(q-q_3)=-1\end{array}\right\}$$

and

$${\Lambda }_{+}=\left\{q\in W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)|\begin{array}{c}q(t+{\displaystyle \frac{1}{2}})=-q\left(t\right),q\left(t\right)\ne q_j\left(t\right),\forall t\in [0,1],j=1,2,3,\\ deg(q-q_1)=1,deg(q-q_2)=1,deg(q-q_3)=1\end{array}\right\}.$$

Obviously, the values of the three types of winding numbers in the space ${\Lambda }_{-}$ differ from those in the space ${\Lambda }_{+}$. Here a positive relative winding number means that the relative motion for two bodies is in a counter-clockwise direction, while a negative relative winding number means that the relative motion for two bodies is in a clockwise direction. And it is well known that $f\left(q\right)$ is $C^1$ on ${\Lambda }_{\pm }$. Let the closure of ${\Lambda }_{\pm }$ be $\overline{{\Lambda }_{\pm }}$; then our main results are the following.

Theorem 1.  

The minimizer of $f\left(q\right)$ on $\overline{{\Lambda }_{-}}$ exists and there exists a noncollision periodic solution of System (1) for some three unequal $m_1,m_2$, and $m_3$. Moreover, there also exists a noncollision periodic solution of System (1) for three equal $m_1$, $m_2$, and $m_3$, too.

Theorem 2.  

The minimizer of $f\left(q\right)$ on $\overline{{\Lambda }_{+}}$ exists and there exists a noncollision periodic solution of System (1) for some three unequal $m_1,m_2$, and $m_3$. Moreover, there also exists a noncollision periodic solution of System (1) for three equal $m_1$, $m_2$, and $m_3$, too.

2. Preliminaries

In this section, we list some basic lemmas and useful inequalities for our proofs of Theorems 1 and 2.

Lemma 1  

(Page 4 [30]). Suppose that X is a reflexive Banach space with the norm $\parallel ·\parallel$, and let $S\subset X$ be a weakly closed subset of X. Suppose that $f:S\to \mathbb{R}\cup \{+\infty \}$, and if $f\not\equiv +\infty$ is weakly lower semi-continuous and coercive, $\left(f\right(x)\to +\infty$ as $\parallel x\parallel \to +\infty )$, then f attains its infimum on S.

Lemma 2  

(Proposition 1.3 (Wirtinger inequality) [31]). Let $x\in W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)$ and ${\int }_0^{1}x\left(t\right)dt=0$; then

$$\begin{array}{c}\hfill {\int }_0^1{\left|x\left(t\right)\right|}^{2}dt\le {\displaystyle \frac{1}{4{\pi }^2}}{\int }_0^1{\left|\dot{x}\left(t\right)\right|}^{2}dt.\end{array}$$

Lemma 3  

(Page 22 [32]). Let σ be an orthogonal representation of a finite or compact group, G, and $f:H\to R$ satisfies $f(\sigma ·x)=f\left(x\right),\forall \sigma \in G,\forall x\in H$, where H is a real Hilbert space. Set $F=\{x\in H|\sigma ·x=x,\forall \sigma \in G\}$. Then the critical point of f in F is also a critical point of f in H.

Remark 1.  

For $q\in \overline{{\Lambda }_{\pm }}$, $t\in [0,1]$, one has $q(t+{\displaystyle \frac{1}{2}})=-q\left(t\right)$. And notice that the action functional is invariant under the time translation and rotational transformation for the positions; then by Lemma 3 and the perturbation invariance for winding numbers, we can obtain that the critical point of $f\left(q\right)$ in ${\Lambda }_{\pm }$ is a noncollision periodic solution of System (1).

Lemma 4  

(Lemma 2.1 [5]). Let $x\in W^{1,2}([t_1,t_2],{\mathbb{R}}^2)$, and $x\left(t_1\right)=x\left(t_2\right)=0$. Then for any $s>0$, we have

$$\begin{array}{c}\hfill {\int }_{t_1}^{t_2}({\displaystyle \frac{1}{2}}|\dot{x}{|}^2+{\displaystyle \frac{s}{\left|x\right|}})dt\ge {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}{s}^{2/3}{(t_2-t_1)}^{1/3}.\end{array}$$

Lemma 5  

(Lemma 2.1 [11]). Let $x\in W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2),{\int }_0^{1}xdt=0$; then for any $s>0$, we have

$$\begin{array}{c}\hfill {\int }_0^1({\displaystyle \frac{1}{2}}|\dot{x}{|}^2+{\displaystyle \frac{s}{\left|x\right|}})dt\ge {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}{s}^{2/3}.\end{array}$$

Lemma 6  

(Lemmas 4.1 and 4.2 [13]). The noncollision periodic solution $(q_1,q_2,q_3)$ of System (2) exists. This is the equilateral triangle solution of Lagrange. Moreover, we have the following conclusions:

(1) 

$l=\sqrt[3]{{\displaystyle \frac{M}{4{\pi }^2}}}$, where l is defined in System (2) and $M=m_1+m_2+m_3$.

(2) 

The radii $r_1,r_2,$ and $r_3$ of the planar circular orbits are

$$\begin{array}{c}\hfill r_1={\displaystyle \frac{\sqrt{m_2^2+m_2{m}_3+m_3^2}}{M}}l,r_2={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_3+m_3^2}}{M}}l,r_3={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_2+m_2^2}}{M}}l,\end{array}$$

and

$$\begin{array}{ccc}\hfill sin{\theta }_1& =& {\displaystyle \frac{-m_2+m_3}{2\sqrt{m_2^2+m_2{m}_3+m_3^2}}},cos{\theta }_1={\displaystyle \frac{\sqrt{3}(m_2+m_3)}{2\sqrt{m_2^2+m_2{m}_3+m_3^2}}},\hfill \\ \hfill sin{\theta }_2& =& {\displaystyle \frac{m_1+2m_3}{2\sqrt{m_1^2+m_1{m}_3+m_3^2}}},cos{\theta }_2=-{\displaystyle \frac{\sqrt{3}{m}_1}{2\sqrt{m_1^2+m_1{m}_3+m_3^2}}},\hfill \\ \hfill sin{\theta }_3& =& -{\displaystyle \frac{m_1+2m_2}{2\sqrt{m_1^2+m_1{m}_2+m_2^2}}},cos{\theta }_3=-{\displaystyle \frac{\sqrt{3}{m}_1}{2\sqrt{m_1^2+m_1{m}_2+m_2^2}}}.\hfill \end{array}$$

Lemma 7.  

The infimum of functional $f\left(q\right)$ in (4) can be attained on $\overline{{\Lambda }_{\pm }}$.

Proof. 

With the aid of Lemma 2, we observe that for all $q\in \overline{{\Lambda }_{-}}$, an equivalent norm of (3) in $\overline{{\Lambda }_{-}}$ is

$$\begin{array}{c}\hfill \parallel q\parallel \cong {\left[{\int }_0^1{\left|\dot{q}\right|}^{2}dt\right]}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

(5)

Using the definition of $f\left(q\right)$ in (4) and (5), we know that

$$\begin{array}{ccc}\hfill f\left(q\right)& =& {\int }_0^1\left[{\displaystyle \frac{1}{2}}{\left|\dot{q}\right|}^2+\sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}\right]dt\hfill \\ & \ge & {\displaystyle \frac{1}{2}}{\int }_0^1{\left|\dot{q}\right|}^{2}dt\hfill \\ & =& {\displaystyle \frac{1}{2}}{\parallel q\parallel }^2,\hfill \end{array}$$

which implies that f is coercive on $\overline{{\Lambda }_{-}}$.

Firstly, we prove that f is weakly lower semi-continuous on $\overline{{\Lambda }_{-}}$. The proof is similar to ([Lemma 2.6] [12]). In fact, for $\forall q_{\left(k\right)}\in \overline{{\Lambda }_{-}}$, suppose that $q_{\left(k\right)}\rightharpoonup q$; then, by the compact embedding theorem, we obtain the uniform convergence:

$$\underset{0\le t\le 1}{max}|q_{\left(k\right)}\left(t\right)-q\left(t\right)|\to 0,k\to +\infty .$$

(6)

On the one hand, if $q\in {\Lambda }_{-}$, it follows from (6) that

$${\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q_{\left(k\right)}-q_j|}}dt\to {\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}dt,k\to +\infty .$$

(7)

Since the norm and its square are weakly lower semi-continuous, by (7), we get

$$\begin{array}{c}\hfill \underset{k\to +\infty }{\underline{lim}}f\left(q_{\left(k\right)}\right)\ge f\left(q\right).\end{array}$$

(8)

Then we need to prove that when $q\in \partial {\Lambda }_{-}$, (8) also holds.

Let $E=\{t\in [0,1]|\exists 1\le j_0\le 3\mathrm{such}\mathrm{that}{q}_{j_0}\left(t\right)=q\left(t\right)\}$ and $\mu \left(E\right)$ denote the Lebesgue measure. Next, we will discuss this issue in different situations.

(i) If $\mu \left(E\right)=0$, by (6), we have the almost-everywhere convergence:

$$\begin{array}{c}\hfill \sum _{j=1}^3{\displaystyle \frac{m_j}{|q_{\left(k\right)}-q_j|}}\to \sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}\mathrm{almost}\mathrm{everywhere}\mathrm{on}[0,1],k\to +\infty .\end{array}$$

Therefore, Fatou’s Lemma implies that

$$\underset{k\to +\infty }{\underline{lim}}{\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q_{\left(k\right)}-q_j|}}dt\ge {\int }_0^1\underset{k\to +\infty }{\underline{lim}}\sum _{j=1}^3{\displaystyle \frac{m_j}{|q_{\left(k\right)}-q_j|}}dt={\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}dt.$$

(9)

It is well known that the norm and its square are weakly lower semi-continuous. This fact, combined with (9), allows us to conclude that

$$\begin{array}{c}\hfill \underset{k\to +\infty }{\underline{lim}}f\left(q_{\left(k\right)}\right)\ge f\left(q\right).\end{array}$$

(ii) If $\mu \left(E\right)\ne 0$, then

$$\begin{array}{c}\hfill {\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}dt=+\infty .\end{array}$$

Combining (4) and $\mu \left(E\right)\ne 0$, there is

$$f\left(q\right)=+\infty .$$

(10)

Note that $E=\{t\in [0,1]|\exists 1\le j_0\le 3\mathrm{such}\mathrm{that}{q}_{j_0}\left(t\right)=q\left(t\right)\}$; then, using (6) along with $\mu \left(E\right)\ne 0$, it is easy to see that

$${\int }_E{\displaystyle \frac{m_{j_0}}{|q_{\left(k\right)}-q|}}dt\to +\infty \mathrm{as}k\to +\infty .$$

(11)

Since $E\subseteq [0,1]$, we have

$$\begin{array}{c}\hfill {\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q_{\left(k\right)}-q_j|}}dt\ge {\int }_0^1{\displaystyle \frac{m_{j_0}}{|q_{\left(k\right)}-q_{j_0}|}}dt\ge {\int }_E{\displaystyle \frac{m_{j_0}}{|q_{\left(k\right)}-q_{j_0}|}}dt={\int }_E{\displaystyle \frac{m_{j_0}}{|q_{\left(k\right)}-q|}}dt.\end{array}$$

(12)

Then, employing (11) and (12), we obtain

$$\begin{array}{c}\hfill {\int }_0^1\sum _{j=1}^3{\displaystyle \frac{m_j}{|q_{\left(k\right)}-q_j|}}dt\to +\infty ,k\to +\infty ,\end{array}$$

which implies that

$$\underset{k\to +\infty }{lim}f\left(q_{\left(k\right)}\right)=+\infty .$$

(13)

From $\mu \left(E\right)\ne 0$, (10) and (13), we get

$$\begin{array}{c}\hfill \underset{k\to +\infty }{\underline{lim}}f\left(q_{\left(k\right)}\right)\ge f\left(q\right).\end{array}$$

To sum up, it is easy to see that f is weakly lower semi-continuous on $\overline{{\Lambda }_{-}}$. In Lemma 1, we choose $X=W^{1,2}(\mathbb{R}/\mathbb{Z})$ and $S=\overline{{\Lambda }_{-}}$. Note that f is coercive on $\overline{{\Lambda }_{-}}$; then, by Lemma 1, it can be concluded that $f\left(q\right)$ in (4) attains its infimum on $\overline{{\Lambda }_{-}}$.

Similarly to the above proof, we can obtain the conclusion that the infimum of functional $f\left(q\right)$ in (4) can be attained on $\overline{{\Lambda }_{+}}$. □

3. The Lower-Bound Estimate

For any generalized periodic solution, q, the fourth mass may collide with one or more of the other three masses. Thus, in order to prove Theorems 1 and 2, it suffices to estimate the lower bound for the value of the Lagrangian action functional f on the boundary $\partial {\Lambda }_{\pm }$.

Lemma 8.  

For the boundaries

$$\begin{array}{c}\hfill \partial {\Lambda }_{-}=\left\{q\in W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)|\begin{array}{c}q(t+{\displaystyle \frac{1}{2}})=-q\left(t\right),\exists 1\le j_0^{-}\le 3,t_{j_0^{-}}\in [0,1]\\ s.t.q_{j_0^{-}}\left(t_{j_0^{-}}\right)=q\left(t_{j_0^{-}}\right)\end{array}\right\}\end{array}$$

and

$$\begin{array}{c}\hfill \partial {\Lambda }_{+}=\left\{q\in W^{1,2}(\mathbb{R}/\mathbb{Z},{\mathbb{R}}^2)|\begin{array}{c}q(t+{\displaystyle \frac{1}{2}})=-q\left(t\right),\exists 1\le j_0^{+}\le 3,t_{j_0^{+}}\in [0,1]s.t.\\ q_{j_0^{+}}\left(t_{j_0^{+}}\right)=q\left(t_{j_0^{+}}\right)\end{array}\right\},\end{array}$$

we have

$$\underset{q\in \partial {\Lambda }_{\pm }}{inf}f\left(q\right)\ge {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}C{M}^{-1/3}\triangleq d_1,$$

where

$$C=min\left\{\begin{array}{c}{2}^{{\displaystyle \frac{2}{3}}}{m}_1+m_2+m_3-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3),\\ 2^{{\displaystyle \frac{2}{3}}}{m}_2+m_1+m_3-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3),\\ 2^{{\displaystyle \frac{2}{3}}}{m}_3+m_1+m_2-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3)\end{array}\right\},$$

and M is in Lemma 6 (1).

Proof. 

Using (2), we have $m_1{q}_1\left(t\right)+m_2{q}_2\left(t\right)+m_3{q}_3\left(t\right)=0$; then ${\sum }_{j=1}^3{m}_j{\dot{q}}_j=0$, which implies that

$$\begin{array}{ccc}\hfill \sum _{j=1}^3{m}_j{|\dot{q}-{\dot{q}}_j|}^2& =& \sum _{j=1}^3{m}_j\left(|\dot{q}{|}^2+{|{\dot{q}}_j|}^2-2\langle \dot{q},{\dot{q}}_j\rangle \right)=M|\dot{q}{|}^2+\sum _{j=1}^3{m}_j{|{\dot{q}}_j|}^2-2〈\dot{q},\sum _{j=1}^3{m}_j{\dot{q}}_j〉\hfill \\ & =& M|\dot{q}{|}^2+\sum _{j=1}^3{m}_j{\left|{\dot{q}}_j\right|}^2,\hfill \end{array}$$

where $M=m_1+m_2+m_3$. Therefore, $|\dot{q}{|}^2=[{\sum }_{j=1}^3{m}_j\left(\right|\dot{q}-{\dot{q}}_j{|}^2-|{\dot{q}}_j{|}^2\left)\right]/M$. Hence,

$$\begin{array}{ccc}\hfill f\left(q\right)& =& {\int }_0^1\left[{\displaystyle \frac{1}{2}}{\left|\dot{q}\right|}^2+\sum _{j=1}^3{\displaystyle \frac{m_j}{|q-q_j|}}\right]dt\hfill \\ & =& {\displaystyle \frac{1}{M}}{\int }_0^1\sum _{j=1}^3{m}_j[{\displaystyle \frac{1}{2}}|\dot{q}-{\dot{q}}_j{|}^2+{\displaystyle \frac{M}{|q-q_j|}}]dt-{\displaystyle \frac{1}{2M}}{\int }_0^1\sum _{j=1}^3{m}_j{\left|{\dot{q}}_j\right|}^{2}dt.\hfill \end{array}$$

(14)

If $q\in \overline{{\Lambda }_{-}}$ is a generalized collision solution, then there exist $t_{j_0^{-}}\in [0,1]$ and $1\le j_0^{-}\le 3$ such that $q\left(t_{j_0^{-}}\right)=q_{j_0^{-}}\left(t_{j_0^{-}}\right)$. On the other hand, employing (2), for any $j\in \{1,2,3\}$ and any $t\in [0,1]$, we have $q_j(t+1/2)=-q_j\left(t\right)$ and $q(t_{j_0^{-}}+1/2)=-q\left(t_{j_0^{-}}\right)$, which implies that $q(t_{j_0^{-}}+k/2)-q_{j_0^{-}}(t_{j_0^{-}}+k/2)=0$ for any $k\in \{1,2\}$. So, by Lemma 4, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{M}}{\int }_0^1{m}_{j_0^{-}}\left[{\displaystyle \frac{1}{2}}{|\dot{q}-{\dot{q}}_{j_0^{-}}|}^2+{\displaystyle \frac{M}{|q-q_{j_0^{-}}|}}\right]dt& =& {\displaystyle \frac{2}{M}}{m}_{j_0^{-}}{\int }_0^{{\displaystyle \frac{1}{2}}}\left[{\displaystyle \frac{1}{2}}{|\dot{q}-{\dot{q}}_{j_0^{-}}|}^2+{\displaystyle \frac{M}{|q-q_{j_0^{-}}|}}\right]dt\hfill \\ & \ge & {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}{2}^{2/3}{m}_{j_0^{-}}{M}^{-1/3}.\hfill \end{array}$$

(15)

For the noncollision pair q and $q_j$ where $j\ne j_0^{-}$, we have ${\int }_0^{1}q\left(t\right)dt=0$ and ${\int }_0^1{q}_j\left(t\right)dt=0$. Therefore, ${\int }_0^1[q\left(t\right)-q_j\left(t\right)]dt=0$. Hence, by Lemma 5, we see that

$${\displaystyle \frac{1}{M}}{\int }_0^1\sum _{j\ne j_0^{-}}{m}_j\left[{\displaystyle \frac{1}{2}}{|\dot{q}-{\dot{q}}_j|}^2+{\displaystyle \frac{M}{|q-q_j|}}\right]dt\ge {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}(M-m_{j_0^{-}})M^{-1/3}.$$

(16)

Note that Lemma 6 shows us that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{r}_1={\displaystyle \frac{\sqrt{m_2^2+m_2{m}_3+m_3^2}}{M}}l={\displaystyle \frac{\sqrt{m_2^2+m_2{m}_3+m_3^2}}{M}}\sqrt[3]{{\displaystyle \frac{M}{4{\pi }^2}}}={\displaystyle \frac{\sqrt{m_2^2+m_2{m}_3+m_3^2}}{M^{\frac{2}{3}}{\left(2\pi \right)}^{\frac{2}{3}}}},\hfill \\ r_2={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_3+m_3^2}}{M}}l={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_3+m_3^2}}{M}}\sqrt[3]{{\displaystyle \frac{M}{4{\pi }^2}}}={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_3+m_3^2}}{M^{\frac{2}{3}}{\left(2\pi \right)}^{\frac{2}{3}}}},\hfill \\ r_3={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_2+m_2^2}}{M}}l={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_2+m_2^2}}{M}}\sqrt[3]{{\displaystyle \frac{M}{4{\pi }^2}}}={\displaystyle \frac{\sqrt{m_1^2+m_1{m}_2+m_2^2}}{M^{\frac{2}{3}}{\left(2\pi \right)}^{\frac{2}{3}}}},\hfill \end{array}\right.\end{array}$$

(17)

and then, for the other term of $f\left(q\right)$ in (14), using $M=m_1+m_2+m_3$, (17), and the expression of orbits $q_1,q_2,$ and $q_3$ in (2), we obtain

$$\begin{array}{ccc}\hfill & & -{\displaystyle \frac{1}{2M}}{\int }_0^1\sum _{j=1}^3{m}_j{\left|{\dot{q}}_j\right|}^{2}dt\hfill \\ & =& -{\displaystyle \frac{1}{2M}}(m_1{r}_1^2+m_2{r}_2^2+m_3{r}_3^2){\left(2\pi \right)}^2\hfill \\ & =& -{\displaystyle \frac{1}{2M}}\{m_1{\displaystyle \frac{m_2^2+m_2{m}_3+m_3^2}{M^2}}{\left[{\displaystyle \frac{M}{{\left(2\pi \right)}^2}}\right]}^{{\displaystyle \frac{2}{3}}}+m_2{\displaystyle \frac{m_1^2+m_1{m}_3+m_3^2}{M^2}}{\left[{\displaystyle \frac{M}{{\left(2\pi \right)}^2}}\right]}^{{\displaystyle \frac{2}{3}}}\hfill \\ & & +m_3{\displaystyle \frac{m_2^1+m_1{m}_2+m_2^2}{M^2}}{\left[{\displaystyle \frac{M}{{\left(2\pi \right)}^2}}\right]}^{{\displaystyle \frac{2}{3}}}\}\hfill \\ & =& -{\displaystyle \frac{1}{2M}}{\left(2\pi \right)}^{{\displaystyle \frac{2}{3}}}{M}^{-{\displaystyle \frac{4}{3}}}[m_1(m_2^2+m_2{m}_3+m_3^2)+m_2(m_1^2+m_1{m}_3+m_3^2)\hfill \\ & & +m_3(m_1^2+m_1{m}_2+m_2^2)]\hfill \\ & =& -{\displaystyle \frac{1}{2M}}{\left(2\pi \right)}^{{\displaystyle \frac{2}{3}}}{M}^{-{\displaystyle \frac{4}{3}}}[m_1{m}_2^2+m_1{m}_2{m}_3+m_1{m}_3^2+m_2{m}_1^2+m_1{m}_2{m}_3+m_2{m}_3^2\hfill \\ & & +m_3{m}_1^2+m_1{m}_2{m}_3+m_2^2{m}_3]\hfill \\ & =& -{\displaystyle \frac{1}{2M}}{\left(2\pi \right)}^{{\displaystyle \frac{2}{3}}}{M}^{-{\displaystyle \frac{4}{3}}}[m_1(m_1{m}_2+m_2{m}_3+m_1{m}_3)+m_2(m_1{m}_2+m_2{m}_3+m_1{m}_3)\hfill \\ \hfill \\ & =& -{\displaystyle \frac{1}{2M}}{\left(2\pi \right)}^{{\displaystyle \frac{2}{3}}}{M}^{-{\displaystyle \frac{4}{3}}}[(m_1+m_2+m_3)(m_1{m}_2+m_2{m}_3+m_1{m}_3)]\hfill \\ & =& -{\displaystyle \frac{1}{2}}{\left(2\pi \right)}^{{\displaystyle \frac{2}{3}}}(m_1{m}_2+m_1{m}_3+m_2{m}_3)M^{-4/3}.\hfill \end{array}$$

(18)

Therefore, if $q\in \overline{{\Lambda }_{-}}$ is a generalized collision periodic solution, it follows from (14)–(16) and (18) that

$$\begin{array}{ccc}\hfill f\left(q\right)& =& {\displaystyle \frac{1}{M}}{\int }_0^1{m}_{j_0^{-}}\left[{\displaystyle \frac{1}{2}}{|\dot{q}-{\dot{q}}_{j_0^{-}}|}^2+{\displaystyle \frac{M}{|q-q_{j_0^{-}}|}}\right]dt\hfill \\ & & +{\displaystyle \frac{1}{M}}{\int }_0^1\sum _{j\ne j_0^{-}}{m}_j\left[{\displaystyle \frac{1}{2}}{|\dot{q}-{\dot{q}}_j|}^2+{\displaystyle \frac{M}{|q-q_j|}}\right]dt\hfill \\ & & -{\displaystyle \frac{1}{2M}}{\int }_0^1\sum _{j=1}^3{m}_j{\left|{\dot{q}}_j\right|}^{2}dt\hfill \\ & \ge & {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}{2}^{2/3}{m}_{j_0^{-}}{M}^{-1/3}+{\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}(M-m_{j_0^{-}})M^{-1/3}\hfill \\ & & -{\displaystyle \frac{1}{2}}{\left(2\pi \right)}^{{\displaystyle \frac{2}{3}}}(m_1{m}_2+m_1{m}_3+m_2{m}_3)M^{-4/3}\hfill \\ & =& {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}{M}^{-1/3}\left[2^{2/3}{m}_{j_0^{-}}+M-m_{j_0^{-}}-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3)\right].\hfill \end{array}$$

(19)

Since $1\le j_0^{-}\le 3$, let

$$\begin{array}{c}\hfill C=min\left\{\begin{array}{c}{2}^{{\displaystyle \frac{2}{3}}}{m}_1+m_2+m_3-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3),\\ 2^{{\displaystyle \frac{2}{3}}}{m}_2+m_1+m_3-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3),\\ 2^{{\displaystyle \frac{2}{3}}}{m}_3+m_1+m_2-{\displaystyle \frac{1}{3M}}(m_1{m}_2+m_1{m}_3+m_2{m}_3)\end{array}\right\},\end{array}$$

by (19), we have

$$\begin{array}{c}\hfill f\left(q\right)\ge {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}C{M}^{-1/3}\end{array}$$

on $\overline{{\Lambda }_{-}}$. Hence,

$$\begin{array}{c}\hfill \underset{q\in \partial {\Lambda }_{-}}{inf}f\left(q\right)\ge {\displaystyle \frac{3}{2}}{\left(2\pi \right)}^{2/3}C{M}^{-1/3}\triangleq d_1.\end{array}$$

Similarly, if $q\in \overline{{\Lambda }_{+}}$ is a generalized periodic solution, we can get

$$\begin{array}{c}\hfill \underset{q\in \partial {\Lambda }_{+}}{inf}f\left(q\right)\ge d_1,\end{array}$$

and then Lemma 8 holds. □

4. Proof of Theorem 1

By Lemma 7, we obtain the existence of variational minimizers of $f\left(q\right)$ in (4) on $\overline{{\Lambda }_{\pm }}$. We claim that the variational minimizer in Lemma 7 is the noncollision period solution of System (1). In fact, to obtain Theorem 1, we need to seek a test loop, $\tilde{q}\in {\Lambda }_{-}$, such that $f\left(\tilde{q}\right)=d_2$, satisfying $d_2<d_1$.

Let $a>0,b>0$, $\theta \in [0,2\pi )$ and $\tilde{q}-q_1=acos(-2\pi t+\theta )+ibsin(-2\pi t+\theta )$. Note that $q_1\left(t\right),q_2\left(t\right)$ and $q_3\left(t\right)$ are in a plane around the center of masses; then, in this situation, by Definition 1, when $\tilde{q}\in {\Lambda }_{-}$, it is easy to see that $deg(\tilde{q}-q_1)=-1$, $deg(\tilde{q}-q_2)=-1$, and $deg(\tilde{q}-q_3)=-1$. Moreover, employing (2), we have

$$\begin{array}{ccc}\hfill \tilde{q}-q_2& =& \tilde{q}-q_1+q_1-q_2=(q_1-q_2)+(\tilde{q}-q_1)\hfill \\ & =& [r_{1}cos(2\pi t+{\theta }_1)-r_{2}cos(2\pi t+{\theta }_2)+acos(-2\pi t+\theta )]+i[r_{1}sin(2\pi t+{\theta }_1)\hfill \\ & & -r_{2}sin(2\pi t+{\theta }_2)+bsin(-2\pi t+\theta )]\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \tilde{q}-q_3& =& [r_{1}cos(2\pi t+{\theta }_1)-r_{3}cos(2\pi t+{\theta }_3)+acos(-2\pi t+\theta )]+i[r_{1}sin(2\pi t+{\theta }_1)\hfill \\ & & -r_{3}sin(2\pi t+{\theta }_3)+bsin(-2\pi t+\theta )].\hfill \end{array}$$

It is easy to verify that $\tilde{q}\in {\Lambda }_{-}$ and

$$\begin{array}{c}\hfill |\dot{\tilde{q}}-\dot{q_1}{|}^2={\left(2\pi \right)}^2\left[{\displaystyle \frac{a^2+b^2}{2}}-{\displaystyle \frac{a^2-b^2}{2}}cos\left(4\pi t-2\theta \right)\right],\end{array}$$

(20)

$$\begin{array}{c}\hfill |\tilde{q}-q_1|=\sqrt{{\displaystyle \frac{a^2+b^2}{2}}+{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )},\end{array}$$

(21)

$$\begin{array}{ccc}\hfill |\dot{\tilde{q}}-\dot{q_2}{|}^2& =& {\left(2\pi \right)}^2\{{\displaystyle \frac{a^2+b^2}{2}}-{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )+r_1^2+r_2^2-2r_1{r}_{2}cos({\theta }_2-{\theta }_1)\hfill \\ & & -(a+b)[r_{1}cos(4\pi t+{\theta }_1-\theta )-r_{2}cos(4\pi t+{\theta }_2-\theta )]\hfill \\ & & +(a-b)\left[r_{1}cos({\theta }_1+\theta )-r_{2}cos({\theta }_2+\theta )\right]\},\phantom{{\displaystyle \frac{V_{LO}^2}{2}}}\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill |\tilde{q}-q_2|& =& \{{\displaystyle \frac{a^2+b^2}{2}}+{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )+r_1^2+r_2^2-2r_1{r}_{2}cos({\theta }_2-{\theta }_1)\hfill \\ & & +(a+b)[r_{1}cos(4\pi t+{\theta }_1-\theta )-r_{2}cos(4\pi t+{\theta }_2-\theta )]\hfill \\ & & +(a-b)\left[r_{1}cos({\theta }_1+\theta )-r_{2}cos({\theta }_2+\theta )\right]{\}}^{{\displaystyle \frac{1}{2}}},\phantom{{\displaystyle \frac{V_{LO}^2}{2}}}\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill |\dot{\tilde{q}}-\dot{q_3}{|}^2& =& {\left(2\pi \right)}^2\{{\displaystyle \frac{a^2+b^2}{2}}-{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )+r_1^2+r_3^2-2r_1{r}_{3}cos({\theta }_3-{\theta }_1)\hfill \\ & & -(a+b)[r_{1}cos(4\pi t+{\theta }_1-\theta )-r_{3}cos(4\pi t+{\theta }_3-\theta )]\hfill \\ & & +(a-b)\left[r_{1}cos({\theta }_1+\theta )-r_{3}cos({\theta }_3+\theta )\right]\},\phantom{{\displaystyle \frac{V_{LO}^2}{2}}}\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill |\tilde{q}-q_3|& =& \{{\displaystyle \frac{a^2+b^2}{2}}+{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )+r_1^2+r_3^2-2r_1{r}_{3}cos({\theta }_3-{\theta }_1)\hfill \\ & & +(a+b)[r_{1}cos(4\pi t+{\theta }_1-\theta )-r_{3}cos(4\pi t+{\theta }_3-\theta )]\hfill \\ & & +(a-b)\left[r_{1}cos({\theta }_1+\theta )-r_{3}cos({\theta }_3+\theta )\right]{\}}^{{\displaystyle \frac{1}{2}}}\phantom{{\displaystyle \frac{V_{LO}^2}{2}}}\hfill \end{array}$$

(25)

and

$$|\dot{q_1}{|}^2={\left(2\pi \right)}^2{r}_1^2,|\dot{q_2}{|}^2={\left(2\pi \right)}^2{r}_2^2,{\left|\dot{q_3}\right|}^2={\left(2\pi \right)}^2{r}_3^2.$$

(26)

Therefore, by (20)–(26), we get

$$\begin{array}{ccc}\hfill f\left(\tilde{q}\right)& =& {\displaystyle \frac{1}{M}}{\int }_0^1\sum _{j=1}^3{m}_j\left[{\displaystyle \frac{1}{2}}{|\dot{\tilde{q}}-{\dot{q}}_j|}^2+{\displaystyle \frac{M}{|\tilde{q}-q_j|}}\right]dt-{\displaystyle \frac{1}{2M}}{\int }_0^1\sum _{j=1}^3{m}_j{\left|{\dot{q}}_j\right|}^{2}dt\hfill \\ & =& 2{\pi }^2\{{\displaystyle \frac{a^2+b^2}{2}}+{\displaystyle \frac{m_2+m_3-m_1}{M}}{r}_1^2-{\displaystyle \frac{2m_2{r}_{2}cos({\theta }_2-{\theta }_1)+2m_3{r}_{3}cos({\theta }_3-{\theta }_1)}{M}}{r}_1\hfill \\ & & +{\displaystyle \frac{m_2(a-b)}{M}}[r_{1}cos({\theta }_1+\theta )-r_{2}cos({\theta }_2+\theta )]+{\displaystyle \frac{m_3(a-b)}{M}}[r_{1}cos({\theta }_1+\theta )\hfill \\ & & -r_{3}cos({\theta }_3+\theta )]\}+m_1{\int }_0^1{[{\displaystyle \frac{a^2+b^2}{2}}+{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )]}^{-{\displaystyle \frac{1}{2}}}dt\hfill \\ & & +\sum _{j=2}^3{\int }_0^1{m}_j\{{\displaystyle \frac{a^2+b^2}{2}}+{\displaystyle \frac{a^2-b^2}{2}}cos(4\pi t-2\theta )+r_1^2+r_j^2-2r_1{r}_{j}cos({\theta }_j-{\theta }_1)\hfill \\ & & +(a+b)[r_{1}cos(4\pi t+{\theta }_1-\theta )-r_{j}cos(4\pi t+{\theta }_j-\theta )]\hfill \\ & & +(a-b)\left[r_{1}cos({\theta }_1+\theta )-r_{j}cos({\theta }_j+\theta )\right]{\}}^{-{\displaystyle \frac{1}{2}}}dt\phantom{{\displaystyle \frac{V_{LO}^2}{2}}}\hfill \\ & :=& d_2(a,b,\theta ,m_1,m_2,m_3),\hfill \end{array}$$

(27)

where the expressions of $r_1$, $r_2$, and $r_3$ are in (17).

Let $a=0.49,b=0.15,\theta =\pi ,m_1=0.38,m_2=0.29$, and $m_3=0.33$; then, in (27), with an error tolerance of less than ${10}^{-6}$, using the numerical quadrature in Mathematica (v7.1), we have $d_1=5.413540,d_2=5.412712$, which implies that $d_2<d_1$. Therefore, for the three unequal masses, $m_1,m_2$, and $m_3$, by Lemma 8 and Remark 1, we conclude that the minimizer of f on $\overline{{\Lambda }_{-}}$ is a noncollision periodic solution of System (1). Moreover, let $a=0.61,b=0.23,\theta =\pi ,m_1=m_2=m_3=1$, then using the numerical quadrature in Mathematica (v7.1) again, one computes that $d_1=11.523843$ and $d_2=11.516685$, and we also have $d_2<d_1$. Thus, for the three equal masses $m_1,m_2$, and $m_3$, with the aid of Lemma 8 and Remark 1, the minimizer of f on $\overline{{\Lambda }_{-}}$ must be a noncollision periodic solution of System (1), too.

5. Proof of Theorem 2

To obtain Theorem 2, we need to find a test loop, $\overline{q}\in {\Lambda }_{+}$, such that $f\left(\overline{q}\right)=d_3$, satisfying $d_3<d_1$.

Let $a>0$, $\theta \in [0,2\pi )$ and $\overline{q}-q_1=a{e}^{\sqrt{-1}(2\pi t+\theta )}$. Therefore, using (2), we have

$$\begin{array}{c}\hfill \overline{q}-q_2=q_1+a{e}^{\sqrt{-1}(2\pi t+\theta )}-q_2=r_1{e}^{\sqrt{-1}(2\pi t+{\theta }_1)}-r_2{e}^{\sqrt{-1}(2\pi t+{\theta }_2)}+a{e}^{\sqrt{-1}(2\pi t+\theta )}\end{array}$$

and

$$\begin{array}{c}\hfill \overline{q}-q_3=q_1+a{e}^{\sqrt{-1}(2\pi t+\theta )}-q_3=r_1{e}^{\sqrt{-1}(2\pi t+{\theta }_1)}-r_3{e}^{\sqrt{-1}(2\pi t+{\theta }_3)}+a{e}^{\sqrt{-1}(2\pi t+\theta )}.\end{array}$$

Incorporating (2), it is not difficult to verify that when $\overline{q}\in {\Lambda }_{+}$, we get $deg(\overline{q}-q_1)=1$, $deg(\overline{q}-q_2)=1$, $deg(\overline{q}-q_3)=1$,

$$\begin{array}{c}\hfill |\dot{\overline{q}}-\dot{q_1}{|}^2={\left(2\pi \right)}^2{a}^2,|\overline{q}-q_1|=a,\end{array}$$

(28)

$$\begin{array}{ccc}\hfill |\dot{\overline{q}}-\dot{q_2}{|}^2& =& {\left(2\pi \right)}^2[a^2+r_1^2+r_2^2-2r_1{r}_{2}cos({\theta }_2-{\theta }_1)+2a{r}_{1}cos({\theta }_1-\theta )\hfill \\ & & -2a{r}_{2}cos({\theta }_2-\theta )],\hfill \end{array}$$

(29)

$$\begin{array}{c}\hfill |\overline{q}-q_2|={\left[a^2+r_1^2+r_2^2-2r_1{r}_{2}cos({\theta }_2-{\theta }_1)+2a{r}_{1}cos({\theta }_1-\theta )-2a{r}_{2}cos({\theta }_2-\theta )\right]}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(30)

$$\begin{array}{ccc}\hfill |\dot{\overline{q}}-\dot{q_3}{|}^2& =& {\left(2\pi \right)}^2[a^2+r_1^2+r_3^2-2r_1{r}_{3}cos({\theta }_3-{\theta }_1)+2a{r}_{1}cos({\theta }_1-\theta )\hfill \\ & & -2a{r}_{3}cos({\theta }_3-\theta )],\hfill \end{array}$$

(31)

$$\begin{array}{c}\hfill |\overline{q}-q_3|={\left[a^2+r_1^2+r_3^2-2r_1{r}_{3}cos({\theta }_3-{\theta }_1)+2a{r}_{1}cos({\theta }_1-\theta )-2a{r}_{3}cos({\theta }_3-\theta )\right]}^{{\displaystyle \frac{1}{2}}}\end{array}$$

(32)

and

$$|\dot{q_1}{|}^2={\left(2\pi \right)}^2{r}_1^2,|\dot{q_2}{|}^2={\left(2\pi \right)}^2{r}_2^2,{\left|\dot{q_3}\right|}^2={\left(2\pi \right)}^2{r}_3^2.$$

(33)

Therefore, by (28)–(33), one has

$$\begin{array}{ccc}\hfill f\left(\overline{q}\right)& =& {\displaystyle \frac{1}{M}}{\int }_0^1\sum _{j=1}^3{m}_j\left[{\displaystyle \frac{1}{2}}{|\dot{\overline{q}}-{\dot{q}}_j|}^2+{\displaystyle \frac{M}{|\overline{q}-q_j|}}\right]dt-{\displaystyle \frac{1}{2M}}{\int }_0^1\sum _{j=1}^3{m}_j{\left|{\dot{q}}_j\right|}^{2}dt\hfill \\ & =& 2{\pi }^2[a^2+{\displaystyle \frac{m_2+m_3-m_1}{M}}{r}_1^2-{\displaystyle \frac{2m_2{r}_{2}cos({\theta }_2-{\theta }_1)+2m_3{r}_{3}cos({\theta }_3-{\theta }_1)}{M}}{r}_1\hfill \\ & & +{\displaystyle \frac{2(m_2+m_3)}{M}}a{r}_{1}cos({\theta }_1-\theta )-{\displaystyle \frac{2m_2{r}_{2}cos({\theta }_2-\theta )+2m_3{r}_{3}cos({\theta }_3-\theta )}{M}}a]\hfill \\ & & +{\displaystyle \frac{m_1}{a}}+m_2{\int }_0^1[a^2+r_1^2+r_2^2-2r_1{r}_{2}cos({\theta }_2-{\theta }_1)+2a{r}_{1}cos({\theta }_1-\theta )\hfill \\ & & -2a{r}_{2}cos({\theta }_2-\theta ){]}^{-{\displaystyle \frac{1}{2}}}dt+m_3{\int }_0^1[a^2+r_1^2+r_3^2-2r_1{r}_{3}cos({\theta }_3-{\theta }_1)\hfill \\ & & +2a{r}_{1}cos({\theta }_1-\theta )-2a{r}_{3}cos({\theta }_3-\theta ){]}^{-{\displaystyle \frac{1}{2}}}dt\hfill \\ & :=& d_3(a,\theta ,m_1,m_2,m_3),\hfill \end{array}$$

(34)

where the expressions of $r_1$, $r_2$, and $r_3$ are in (17).

In order to estimate $d_3$, we also computed the numerical values of $d_3=f\left(\overline{q}\right)$ by using some selected test loops. The computation of the integral that appears in (34) was conducted by applying the numerical quadrature in Mathematica (v7.1), and the error tolerance was less than ${10}^{-6}$). Let $a=0.17,\theta ={\displaystyle \frac{\pi }{2}},m_1=0.15,m_2=0.60$, and $m_3=0.25$; we have $d_1=5.085112$ and $d_3=5.047242$, which implies that $d_3<d_1$. When $a=0.35,\theta ={\displaystyle \frac{\pi }{3}}$ and $m_1=m_2=m_3=1$; then $d_1=11.523843$ and $d_3=11.302997$, which implies that $d_3<d_1$. Thus, using Lemma 8 and Remark 1, we complete the proof of Theorem 2.

6. Conclusions

We consider planar circular restricted four-body Newtonian problems, where three masses are positioned at the vertices of a Lagrange equilateral triangle configuration. These masses move in circular orbits, denoted as $q_1\left(t\right),q_2\left(t\right)$, and $q_3\left(t\right)$, within a plane centered around their common center of mass, and the fourth body has sufficiently small mass so that it does not influence the motion of the three primary bodies, but its own motion is influenced by the three primary bodies. Using variational minimization methods, we obtain the existence of noncollision periodic solutions with some fixed winding numbers, and we demonstrate that these noncollision solutions exist for both equal and unequal mass values for the three primary bodies.