Periodic Solutions of the 4-Body Electromagnetic Problem and Application to Li Atom

Abstract

The 4-body equations of motion are derived in our previously published paper. Here we prove the existence–uniqueness of a periodic solution by applying the fixed-point method for a suitable introduced operator. To apply the fixed-point theorem, we need to derive appropriate analytical inequalities for the right-hand sides of the equations that ensure that the operator for periodic solutions maps the set of periodic functions into itself. In this way, we prove the existence of the Bohr–Sommerfeld orbits for the 4-body problem in the relativistic case. That allows us to estimate the minimal distances between the electrons on the first and second Bohr–Sommerfeld stationary states. A natural example of such a problem is the Lithium atom, which has three electrons orbiting the nucleus.

1. Introduction

In 1845, C. F. Gauss described the conception of action at a distance propagated with a finite velocity as an alternative to the Newtonian instantaneous action. In 1921, W. Pauli [1] (cf. also G. Herglotz [2]) obtained the relativistic form of the pondermotive force using the Lienard–Wiechert retarded potentials and derived an explicit form of the interacted fields. In 1940, J. L. Synge [3] proposed a direct treatment of the 2-body problem of classical electrodynamics on the basis of the results from [1]. Synge’s equations of motion derived in the Minkowski space have the following relativistic form:

$${\displaystyle \frac{d{\lambda }_r^{(1)}}{d{s}_1}} = {\displaystyle \frac{e_1}{m_1 c^2}} F_{rs}^{(2)} {\lambda }_s^{(1)} , {\displaystyle \frac{d{\lambda }_r^{(2)}}{d{s}_2}} = {\displaystyle \frac{e_2}{m_2 c^2}}{F}_{rs}^{(1)} {\lambda }_s^{(2)}: (r=1,2,3,4)$$

where $e_1, e_2$ are the charges of the particles, $m_1, m_2$—their masses, $c$—the vacuum speed of the light, ${\lambda }_s^{(k)}(k=1,2)$—the unit tangent vectors to the world lines, and $d{s}_k (k=1,2)$—the elements of proper time. The Einstein summation convention is valid. The elements of the electromagnetics tensors $F_{rs}^{(k)}(k=1,2)$ are derived by Lenard–Wiechert retarded potentials. We note that the interaction between both particles is the basis of two groups of equations. Thus, the first group expresses the retarded influence of the second particle on the first one, while the second group expresses the retarded infcluence of the first particle on the second one.

The following problems arise: (i) to introduce delays in the equations of motion, generated by the finite propagation speed of the interaction, a fundamental assumption in Einstein’s theory of relativity; (ii) to introduce relativistic invariant radiation terms; (iii) the system (1) is overdetermined — it consists of eight equations, while the unknown functions are six in number.

In 1963, R.D. Driver [4] answered the first problem. In his mathematical formulation of J. Synge’s 2-body problem, he introduces delays in the equations of motion of neutral type, depending on the unknown trajectories, called state-dependent delays. Such a formulation allows us to make a transition between the proper Einstein time and the absolute Newton time.

To solve the second problem, we step on the original Dirac’s assumption (cf. P.A.M. Dirac [5]). He defines the radiation term as a half a difference between retarded and advanced potentials. In [6] we have obtained a new relativistic invariant form of the radiation term and, using these radiation terms in the equations of motion, we reach the following system for the 2-body problem:

$$m_1 {\displaystyle \frac{d{\lambda }_r^{(1)}}{d{s}_1}} = {\displaystyle \frac{e_1}{c^2}}\left(F_{rl}^{(12)}{\lambda }_l^{(1)} +F_{rl}^{(1)rad}{\lambda }_l^{(1)}\right), m_2 {\displaystyle \frac{d{\lambda }_r^{(2)}}{d{s}_2}} = {\displaystyle \frac{e_2}{c^2}}\left(F_{rl}^{(21)}{\lambda }_l^{(2)}+F_{rl}^{(2)rad}{\lambda }_l^{(2)}\right).$$

(1)

This system is derived in [6]. In [7] we have proved the existence–uniqueness of periodic solution (orbit). Let us compare the results obtained with the previously known ones. In any atomic physics textbook, the following can be read: according to classical electrodynamics, a 2-body system composed of a positive nucleus with electrons orbiting about it would inevitably radiate light, so the orbits of the electrons shrink and finally collapse into the nucleus. That is why N. Bohr broke with classical physics by postulating that the electrons of the atom can only exist in certain definite, so-called stationary states, which he envisaged in terms of the orbits in which the electrons circled. Still contradicting classical physics, he postulated that if an electron is in such an orbit, it will not radiate light. The results obtained in [7] confirm Bohr’s hypothesis of the existence of stationary states, i.e., the existence of electron orbits, but nevertheless, the electron radiates. Thus, a stable orbit of the hydrogen atom exists, which is a consequence of the correct formulation of the initial value problem for the Synge equations and the newly obtained relativistic form of the Dirac radiation term.

Therefore, we have shown that Bohr’s hypothesis does not violate classical electrodynamics.

The purely mathematical problem (iii) is solved in [7]. It is proved that every fourth equation is a consequence of the first three ones. So, there are only six independent equations as many as there are unknown functions.

Let us note that another possible solution is proposed by M. Planck, who applies the statistical methods of thermodynamics to electrodynamics, which generates quantum mechanics—one theory that is useful nowadays. It is known, however, that attempts to reconcile quantum mechanics with the special theory led to infinities in the quantum field theory of strong interactions.

In [8] we derive the equations of motion for N-body with radiation terms and apply the results obtained to the 4-body problem.

The present paper is the second part of paper [8]. Here we prove the existence—uniqueness of periodic solution for the 4-body problem of classical electrodynamics, namely orbits, while quantum mechanics work with orbitals by giving the probability of finding a given particle in a particular location in space.

We proceed with the system of 16 equations of motion in Minkowski space introduced in [8]

$$m_k {\displaystyle \frac{d{\lambda }_r^{(k)}}{d{s}_k}} = {\displaystyle \frac{e_k}{c^2}}\left({\displaystyle \sum _{n=1,n\ne k}^4{F}_{rl}^{(kn)}{\lambda }_l^{(k)}}+F_{rl}^{(k)rad}{\lambda }_l^{(k)}\right) (k=1,2,3,4; r=1,2,3,4).$$

(2)

There is a summation on repeating $l$, c is the vacuum speed of light, $m_k(k=1,2,3,4)$ are the masses, $e_k(k=1,2,3,4)$—the charges of the moving particles, ${\lambda }_l^{(k)}$—the unit tangent vectors to the world lines; ${〈., .〉}_4$ is the dot product in the Minkowski space, $〈.,.〉$—the usual dot product in the three-dimensional subspace. The elements of the electromagnetic tensors $F_{rl}^{(kn)}= {\displaystyle \frac{\partial {\Phi }_l^{(n)}}{\partial x_r^{(k)}}} - {\displaystyle \frac{\partial {\Phi }_r^{(n)}}{\partial x_l^{(k)}}}$ can be calculated by the retarded Lienard–Wiechert potentials ${\Phi }_r^{(n)} = - {\displaystyle \frac{e_n {\lambda }_r^{(n)}}{{〈{\lambda }^{(n)}, {\xi }^{(kn)}〉}_4}}$ (cf. [1,2,3]), while the radiation terms $F_{rl}^{(k)rad}$—as a half a difference in retarded and advanced potentials in accordance with the Dirac assumption [5] and derived in [6]

$$\begin{gathered} F_{mn}^{(k)rad}= {\displaystyle \frac{1}{2}}\left[\left({\displaystyle \frac{\partial A_n^{(k)ret}}{\partial x_m^{(k)ret}}} - {\displaystyle \frac{\partial A_m^{(n)ret}}{\partial x_n^{(k)ret}}}\right)-\left({\displaystyle \frac{\partial A_n^{(k)adv}}{\partial x_m^{(k)adv}}} - {\displaystyle \frac{\partial A_m^{(n)adv}}{\partial x_n^{(k)adv}}}\right)\right], A_n^{(k)ret}= -{\displaystyle \frac{e_k{\lambda }_n^{(k)ret}}{{〈{\lambda }^{(k)ret},{\xi }^{(k)ret}〉}_4}}, \\ A_n^{(k)adv}= -{\displaystyle \frac{e_k{\lambda }_n^{(k)adv}}{{〈{\lambda }^{(k)adv},{\xi }^{(k)adv}〉}_4}}. \end{gathered}$$

As in [7] one can prove that every fourth Equation (2) is a consequence of the first three ones. In this way we obtain 12 equations for 12 unknown velocities.

The paper consists of six sections and four appendices. Section 1 is an Introduction. Section 2 contains the system of equations of motion, derived in a previous paper, simplified for non-relativistic cases. This is possible due to the small value of Sommerfeld fine structure constant, namely 1/137.

In Section 3, the operator formulation for periodic solution is given. We introduce a suitable space for periodic functions and use an operator for periodic solutions introduced by us in a previous paper. The space of periodic functions consists of infinitely smooth function, because the radiation terms contain second order derivatives. We recall some properties of the operator functions.

Section 4 begins with the Main lemma—the system of equations of motion has a unique solution if the defined operator has a unique fixed point. Then we formulate the Main theorem guaranteeing the existence–uniqueness of smooth periodic solution, which implies the existence of orbits for the 4-body problem. The proof is based on our fixed-point theorem.

In Section 5 we apply the result obtained to Li atom and calculate the minimal distance between moving electrons.

Section 6 is a Conclusion, where we confirm the results from the 2- and 3- body problem showing the existence of periodic orbits for 4-body problem.

2. Equations of Motion

Here we extend the assumption from [8] by the following one:

$$\mathsf{(}\mathsf{C}\mathsf{)}:‖{\overrightarrow{u}}^{(k)}‖=\sqrt{〈{\overrightarrow{u}}^{(k)}(t),{\overrightarrow{u}}^{(k)}(t)〉}\le c_k<c (k=1,2,3,4).$$

Following Sommerfeld [9], we denote by ${\beta }_k=c_k/c <1 (k=1,2,3,4)$ and rewrite the system of equations of motion from [8] in the form

$${\dot{u}}_{\alpha }^{(k)}(t) = {\displaystyle \sum _{n=1,n\ne k}^4{G}_{\alpha }^{(kn)}}-{\beta }_k^2{\displaystyle \sum _{\gamma =1,\gamma \ne \alpha }^3 {\displaystyle \sum _{n=1,n\ne k}^4{G}_{\gamma }^{(kn)}}}+G_{\alpha }^{(k)rad}-{\beta }_k^2{\displaystyle \sum _{\gamma =1,\gamma \ne \alpha }^3{G}_{\gamma }^{(k)rad}}\equiv U_{\alpha }^{(k)}(t) (k=1,2,3,4; \alpha =1,2,3),$$

(3)

where $u_{\alpha }^{(k)}(t) =u_{\alpha 0}^{(k)}(t) , {\dot{u}}_{\alpha }^{(k)}(t) ={\dot{u}}_{\alpha 0}^{(k)}(t) , t\le 0 (k=1,2,3,4; \alpha =1,2,3)$, and $u_{\alpha 0}^{(k)}(t)$ are prescribed initial functions.

The above 4-body system of equations of motion consists of neutral differential equations with retarded arguments with second-order derivatives generated by the radiation terms. The delays depend on the unknown trajectories. Such types of equations generate specific difficulties.

Remark 1. 

Our considerations include moving electrons in the first and second Bohr–Sommerfeld stationary states, therefore, following Sommerfeld’s notation, we introduce the notation for the dependencies ${\beta }_1=c_1/c=1/137$, ${\beta }_2={\displaystyle \frac{{\beta }_1}{2}}={\displaystyle \frac{1}{2\times 137}},{\beta }_3={\displaystyle \frac{{\beta }_1}{4}}={\displaystyle \frac{1}{4\times 137}},$ (cf. [9,10,11,12]), where $c_1$ is the velocity of the electron in the first stationary state, $c_2$-in the second stationary state and so on. Obviously, one can disregard the values.

$${{\beta }_1}^2=1/\left({137}^2\right)\approx 0;{{\beta }_2}^2=1/\left(2^2\times {137}^2\right)\approx 0;{{\beta }_3}^2=1/\left(4^2\times {137}^2\right)\approx 0.$$

In view of $U_0{e}^{{\mu }_k{T}^{(k)}}\le {\overline{c}}_k<c$, (${\overline{c}}_k/c={\beta }_k<1$) the inequalities (cf. (A1) in Appendix A)

$$\begin{gathered} \left|G_{\alpha }^{(kn)}(t)\right|\le {\displaystyle \frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}}\left({\displaystyle \frac{1}{c^2{{\tau }_{kn}}^2}}+{\displaystyle \frac{{\omega }_n{U}_0{e}^{{\mu }_n{T}^{(n)}}}{c^3{\tau }_{kn}}}\right){\displaystyle \frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k{c}^2}}\left({\displaystyle \frac{1}{{{\tau }_{kn}}^2}}+{\displaystyle \frac{{\omega }_n{\beta }_n}{{\tau }_{kn}}}\right); \\ \left|G_{\alpha }^{(k)rad}\right|\le {\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{U}_0{e}^{{\mu }_k{T}^{(k)}}}{c^3}}\le {\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{c}_k}{c^3}}={\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}} \end{gathered}$$

show that all terms $G_{\alpha }^{(kn)}$ are of the same order. Consequently, we can neglect the terms containing multipliers ${\beta }_1^2,{\beta }_2^2,{\beta }_3^2\approx 0$. In this way we consider the following simplified system of equations of motion (1):

$${\dot{u}}_{\alpha }^{(k)}(t) = {\displaystyle \sum _{n=1,n\ne k}^4{G}_{\alpha }^{(kn)}}+G_{\alpha }^{(k)rad}\equiv U_{\alpha }^{(k)}(t) (k=1,2,3,4; \alpha =1,2,3)$$

or in details

$${\dot{u}}_{\alpha }^{(k)}(t) = {\displaystyle \sum _{n=1,n\ne k}^4\frac{e_k{e}_n\left(A_{kn}{\xi }_{\alpha }^{(kn)}-B_{kn}{u}_{\alpha }^{(n)}+ C_{kn}{\dot{u}}_{\alpha }^{(n)}\right)}{m_{k}c}}-{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{u_{\alpha }^{(k)}〈{\overrightarrow{u}}^{(k)},{\ddot{\overrightarrow{u}}}^{(k)}〉+c^2{{\ddot{u}}_{\alpha }}^{(k)}}{c^5}}\equiv U_{\alpha }^{(k)}(t) (k=1,2,3,4; \alpha =1,2,3)$$

(4)

where $A_{kn},B_{kn}, C_{kn}$ are defined in Appendix A.

Recalling (cf. [8]) that ${\overrightarrow{u}}^{(k)}={\overrightarrow{u}}^{(k)}(t)$ and $u_{\alpha }^{(n)}=u_{\alpha }^{(n)}(t-{\tau }_{kn}), {\dot{u}}_{\alpha }^{(n)}={\dot{u}}_{\alpha }^{(n)}(t-{\tau }_{kn})$ we conclude that (1) is a neutral system with state-dependent delays ${\tau }_{kn}$ which are defined as solutions of the functional equations

$${\tau }_{kn}(t) = {\displaystyle \frac{1}{c}} \sqrt{〈{\overrightarrow{\xi }}^{(kn)},{\overrightarrow{\xi }}^{(kn)}〉} \equiv {\displaystyle \frac{1}{c}} \sqrt{{\displaystyle \sum _{\gamma =1}^3{\left[x_{\gamma }^{(k)}(t)-x_{\gamma }^{(n)}(t-{\tau }_{kn}(t)\right]}^2}}$$

(5)

where c is the vacuum speed of light.

3. Operator Formulation of the Periodic Problem in Suitable Function Spaces and Preliminary Results

We prove the existence–uniqueness of $T^{(k)}$-periodic solution of the system (4), jointly with the functional equations for the delays (5).

Further on, we assume that the following compatibility condition is satisfied:

$$\mathbf{(}\mathbf{CC}\mathbf{)} {\dot{u}}_{\alpha 0}^{(k)}(0) = U_{\alpha }^{(k)}(0);{\ddot{u}}_{\alpha 0}^{(k)}(0) = {\dot{U}}_{\alpha }^{(k)}(0),{\stackrel{⃛}{u}}_{\alpha 0}^{(k)}(0) = {\ddot{U}}_{\alpha }^{(k)}(0),\dots ;(\alpha =1,2,3;k=1,2,3,4).$$

By $C_{T^{(k)}}^{\infty }[0,\infty ),(k=1,2,3,4)$ we denote the set of all infinite differentiable $T^{(k)}$-periodic functions. Denoting by $T_p^{(k)}=p{T}^{(k)}(p=0,1,2,\dots )$ we introduce the following sets of functions:

$$\begin{array}{l}{M}_{\alpha }^k=\left\{u_{\alpha }^{(k)}\in C_{T^{(k)}}^{\infty }[-T^{(k)},\infty ) : \left|{\displaystyle \frac{d^m{u}_{\alpha }^{(k)}(t)}{d{t}^m}}\right|\le U_0^{(k)}{{\omega }_k}^m{e}^{{\mu }_k(t-T_p^{(k)})}, t\in [T_p^{(k)},{T_{p+1}}^{(k)}] ;\right.\\ \left.{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{u}_{\alpha }^{(k)}(t)dt=0 ,\frac{d^m{u}_{\alpha }^{(k)}(0)}{d{t}^m}=0;u_{\alpha }^{(k)}(t)=u_{\alpha 0}^{(k)}(t) ,\frac{d^m{u}_{\alpha 0}^{(k)}(0)}{d{t}^m}=\frac{d^m{u}_{\alpha 0}^{(k)}(-T^{(k)})}{d{t}^m}=0,t\in [-T^{(k)},0] }\right\},\end{array}$$

where $U_0^{(k)}, {\omega }_k, T^{(k)},{\mu }_k > {\omega }_k$ are positive constants and $u_{\alpha 0}^{(k)}(t)$ are prescribed initial functions satisfying (CC).

Introducing the family of pseudometrics,

$${\rho }_{\left(p,m\right)}\left(u^{(k)},{\overline{u}}^{(k)}\right)=\sup \left\{e^{-{\mu }_k(t-T_p^{(k)})}{{\omega }_k}^{-m} \left|{\displaystyle \frac{d^m{u}^{(k)}(t)}{d{t}^m}}-{\displaystyle \frac{d^m{\overline{u}}^{(k)}(t)}{d{t}^m}}\right|: t\in [T_p^{(k)},{T_{p+1}}^{(k)}]\right\}, (p=0,1,2,\dots ; m=0,1,2,\dots )$$

it is easy to see that the following inequalities

$$e^{-{\mu }_k(t-T_p^{(k)})}{{\omega }_k}^{-m} \left|{\displaystyle \frac{d^m{u}^{(k)}(t)}{d{t}^m}}-{\displaystyle \frac{d^m{\overline{u}}^{(k)}(t)}{d{t}^m}}\right|\le e^{-{\mu }_k(t-T_p^{(k)})}{{\omega }_k}^{-m} 2 {{\omega }_k}^m{e}^{{\mu }_k(t-T_p^{(k)})}{U}_0^{(k)}=2U_0^{(k)}<\infty$$

(6)

are satisfied for every $\left(p,m\right)$ and $t\in [T_p^{(k)},{T_{p+1}}^{(k)}]$. Therefore

$$\sup \left\{{\rho }_{\left(p,m\right)}\left(u,\overline{u}\right): m=0,1,2,\dots ;p=1,2,\dots \right\}<\infty .$$

Remark 2. 

Let $T^{(k)}$ be the period of the solution. We notice that all arguments of the unknown functions are $t-{\tau }_{km}$, that is, $\overrightarrow{u}={\overrightarrow{u}}^{(m)}(t-{\tau }_{km})$. Therefore, we must look for a solution on the initial set, that is, for $t-{\tau }_{km}(t)\in [{\tau }_0;0],$ where ${\tau }_0=\min \{t-{\tau }_{km}(t):t\in [0,T^{(k)}]\}$. Since $1-d{\tau }_{km}(t)/dt>0$, then $t-{\tau }_{km}(t)$ is an increasing function. We have proved that if the trajectories are $T^{(k)}$-periodic, then ${\tau }_{km}(t)$ is $T^{(k)}$-periodic, too. It follows that $-T^{(k)}-{\tau }_{km}(-T^{(k)})\le 0-{\tau }_{km}(0)\iff$ ${\tau }_{km}(0)\le T^{(k)}-{\tau }_{km}(0)$,

$-T^{(k)}-{\tau }_{km}(0)\le -{\tau }_{km}(0)$ and then

$0-{\tau }_{km}(0)\le t-{\tau }_{km}(t)\le -{\tau }_{km}(T^{(k)})=T^{(k)}-{\tau }_{km}(0)=0$. Consequently,

$-T^{(k)}\le t-{\tau }_{km}(t)\le 0$. Therefore, $t-{\tau }_{km}(t):[0,T^{(k)}]\to [-T^{(k)},0]$, that is, ${\tau }_0=-T^{(k)}$ and $t-{\tau }_{km}(t)\in [-T^{(k)},0]$.

Recall that the condition ${\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{u}_{\alpha }^{(k)}(t)dt=0 (\alpha =1,2,3);(p=0,1,2,3, \dots )}$ implies $x_{\alpha }^{(k)}(t)={\displaystyle \underset{0}{\overset{t}{\int }}{u}_{\alpha }^{(k)}(s)ds }$ is $T^{(k)}$-periodic function. We have, however, already proved in [8] that every solution of the functional equation for ${\tau }_{kn}$ has a unique continuous solution for all continuous Lipschitz trajectories and ${\tau }_{km}\ge r_{km}(t)/2c$, and if $\left(x_1^{(k)}(t),x_2^{(k)}(t),x_3^{(k)}(t)\right),(k=1,2,3,4)$ are $T^{(k)}$-periodic functions, then ${\tau }_{kn}(t)$ are $T^{(k)}$-periodic functions, too.

Define operator B for periodic solutions as a 12-tuple

$B=\left({\overrightarrow{B}}^{(1)}(u)(t), {\overrightarrow{B}}^{(2)}(u)(t),{\overrightarrow{B}}^{(3)}(u)(t),{\overrightarrow{B}}^{(4)}(u)(t)\right),$

$$B_{\alpha }^{(k)}(p,u)(t) : =\left\{\begin{array}{c}{\displaystyle \underset{0}{\overset{t}{\int }}{U}_{\alpha }^{(k)}(s)ds}-\left({\displaystyle \frac{t}{T^{(k)}}}-{\displaystyle \frac{1}{2}}\right) {\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds-\frac{1}{T^{(k)}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{\displaystyle \underset{0}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta }}}, t\in [0,T^{(k)}];\\ {\displaystyle \underset{T_p^{(k)}}{\overset{t}{\int }}{U}_{\alpha }^{(k)}(s)ds}-\left({\displaystyle \frac{t-T_p^{(k)}}{T^{(k)}}}-{\displaystyle \frac{1}{2}}\right) {\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds-\frac{1}{T^{(k)}}{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{\displaystyle \underset{T_p^{(k)}}{\overset{t}{\int }}{U}_{\alpha }^{(k)}(s)dsdt}}},t\in [T_p^{(k)},{T_{p+1}}^{(k)}], p=1,2,\dots \\ u_{\alpha 0}^{(k)}(t), t\in [-T^{(k)},0] \end{array}\right.$$

$(k=1,2,3,4; \alpha =1,2,3)$, where $u_{\alpha 0}^{(k)}(t)$ are prescribed infinite differentiable initial functions defined on $[-T^{(k)},0]$.

We use the assertions:

(1)

$u_{\alpha }^{(k)}\in M_{\alpha }^k\Rightarrow {\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{\displaystyle \underset{T_p^{(k)}}{\overset{s}{\int }}{U}_{\alpha }^{(k)}(\theta )d\theta ds}}={\displaystyle \underset{{T_{p+1}}^{(k)}}{\overset{{T_{p+2}}^{(k)}}{\int }} {\displaystyle \underset{{T_{p+1}}^{(k)}}{\overset{s}{\int }}{U}_{\alpha }^{(k)}(\theta )d\theta ds}} (p=0,1,2,\dots )$

(2)

$B_{\alpha }^{(k)}(.)\in M_{\alpha }^k$ .

4. Existence–Uniqueness of a Periodic Solution of the Equations of Motion

Lemma 1. 

(Main lemma) The $\left(T^{(1)},T^{(2)},T^{(3)},T^{(4)}\right)$-periodic problem for (4) has a unique solution from

$$M=M_1^1\times M_2^1\times M_3^1\times M_1^2\times M_2^2\times M_3^2\times M_1^3\times M_2^3\times M_3^3\times M_1^4\times M_2^4\times M_3^4$$

if the operator $\left({\overrightarrow{B}}^{(1)}(u)(t), {\overrightarrow{B}}^{(2)}(u)(t),{\overrightarrow{B}}^{(3)}(u)(t),{\overrightarrow{B}}^{(4)}(u)(t)\right)$ has a fixed point, belonging to $M$.

Prior to formulating the main result, we recall the fixed-point theorem we use to prove an existence theorem. We expose briefly some results from [13].

By $(X,\mathsf{A})$ we mean a Hausdorff sequentially complete uniform space, whose uniformity is generated by a saturated family of pseudo-metrics $\mathsf{A}$=$\left\{{\rho }_{\alpha }\left(x,y\right):\alpha \in A\right\}$, where A is an index set. By $\left(\Phi \right)=\left\{{\Phi }_{\alpha }\left(.\right):\alpha \in A\right\}$ we denote a family оf functions ${\Phi }_{\alpha }(t)=R_{+}^1\to R_{+}^1$ $\left(R_{+}^1=[0,\infty )\right)$ satisfying the properties: (Φ1) ${\Phi }_{\alpha }(t)$ is monotone increasing and continuous from the right in t for every fixed $\alpha \in A$; (Φ2) $0<{\Phi }_{\alpha }(t)<t$ for every t > 0 and $\alpha \in A$ (It follows ${\Phi }_{\alpha }(0)=0$); (Φ3) ${\Phi }_{\alpha }(t)$ are sub-additive functions, that is, ${\Phi }_{\alpha }(t_1+t_2)\le {\Phi }_{\alpha }(t_1)+{\Phi }_{\alpha }(t_2)$.

Let M be a subset of X and T: $M\to M$ be (in general) a nonlinear mapping. T is called $\left(\Phi ,j\right)$-contraction on M, if ${\rho }_{\alpha }(Tx,Ty)\le {\Phi }_{\alpha }({\rho }_{j(\alpha )}(x,y))$ for any fixed $\alpha \in A$ and every $x,y\in M$.

Let $j_s:A\to A, (s=1,2,\dots ,m)$ be mappings of the index set into itself with iterations defined inductively: ${j_s}^n(t)=j_s({j_s}^{n-1}(\alpha ))$,${j_s}^0(\alpha )=\alpha (n=1,2,3,\dots )$. The mapping $T:X\to X$ is called $\left(\Phi ,j_1,j_2,\dots ,j_m\right)$-contraction on X, if for every $x,y\in X$ and for any fixed $\alpha \in A$ the following inequality is satisfied

$${\rho }_{\alpha }(Tx,Ty)\le {\Phi }_{\alpha }({\rho }_{j_1(\alpha )}(x,y)+{\rho }_{j_2(\alpha )}(x,y)+\dots +{\rho }_{j_m(\alpha )}(x,y))$$

Theorem 1. 

Let.

(1) 

$T:X\to X$ be $\left(\Phi ,j_1,j_2,\dots ,j_m\right)$-contractive mapping and the mappings $j_1,j_2,\dots ,j_m$ are commutative.

(2) 

for every $\alpha \in A$ there is a function ${\overline{\Phi }}_{\alpha }(t)$ with properties (Φ1)–(Φ3) such that

$\sup \{{\Phi }_{{j_1}^{k_1}\dots {j_m}^{k_m}(\alpha )}(t):k_1+\dots +k_m=n; n=0,1,2,\dots \}\le {\overline{\Phi }}_{\alpha }(t),$ ${\overline{\Phi }}_{\alpha }(t)/t$ is non-decreasing and ${\overline{\Phi }}_{\alpha }(t)<t/m$.

(3) 

there is an element $x_0\in X$ and a constant $q(\alpha )>0$ such that

$${\rho }_{{j_1}^{k_1}\dots {j_m}^{k_m}(\alpha )}(x_0,T{x}_0)\le q(\alpha )<\infty , (k_1+\dots +k_m=n; n=0,1,2,\dots ).$$

Then T has at least one fixed point in X.

The space $(X,\mathsf{A})$ is called $\left(j_1,j_2,\dots ,j_m\right)$-bounded if for every $x,y\in (X,A), \alpha \in A$ the following inequalities hold ${\rho }_{{j_1}^{k_1}\dots {j_m}^{k_m}(\alpha ))}(x,y)\le q(\alpha )<\infty , (k_1+\dots +k_m=n; n=1,2,3,\dots )$.

Theorem 2. 

Let the conditions of Theorem 1 be valid and X be $\left(j_1,j_2,\dots ,j_m\right)$-bounded. Then the fixed point of T is unique.

Theorem 3. 

Let the following conditions be fulfilled.

(1) (IN) the initial trajectories $x_{\alpha 0}^{(k)}(t)$ and velocities $u_{\alpha 0}^{(k)}(t), t\in [-T^{(k)};0]$ are infinitely differentiable functions such that $r_{kn}(t)=\sqrt{{\displaystyle \sum _{\gamma =1}^3\left(x_{\gamma }^{(k)}(t)-x_{\gamma }^{(n)}(t)\right)}} \ge {\stackrel{⌢}{r}}_{kn}>0$,$t\in [-T^{(k)};0]$ and satisfy (CC).

(2) The following (infinite in number) inequalities are satisfied for $k=1,2,3,4$:

$$2\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{\left({r_{kn}}^{(0)}\right)}^2}+\frac{2{\omega }_n{\beta }_n}{{{\mathit{cr}}_{kn}}^{(0)}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k}}\le U_0^{(k)}$$

$$2\left(1+{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k{T}^{(k)}}}\right)\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{\left({r_{kn}}^{(0)}\right)}^2}+\frac{2{\omega }_n{\beta }_n}{{{\mathit{cr}}_{kn}}^{(0)}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]\le {\omega }_k{U}_0^{(k)}$$

$${\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|}{m_k}\left(\frac{32c({\beta }_k+{\beta }_n)}{{{\stackrel{⌢}{r}}_{kn}}^3}+\frac{8{\omega }_n{\beta }_n}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{8{{\omega }_n}^2{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{\beta }_k{{\omega }_k}^3}{c^2}}\le {{\omega }_k}^2{U}_0^{(k)}$$

and so on.

Then there is a unique $\left(T^{(1)},T^{(2)},T^{(3)},T^{(4)}\right)$-periodic solution

$$(u_1^{(1)},u_2^{(1)},u_3^{(1)},u_1^{(2)},u_2^{(2)},u_3^{(2)},u_1^{(3)},u_2^{(3)},u_3^{(3)},u_1^{(4)},u_2^{(4)},u_3^{(4)})$$

of (4) for $t\ge 0$.

Proof. 

In accordance with the Main lemma, we must prove that operator B possesses a unique fixed point, which means that the 4-body problem has a unique periodic solution. First, we show that operator B maps the set $M$ into itself. The set $M$ can be considered as a uniform space with saturated family of pseudo-metrics formed by $\left\{{\rho }_{\left(p,m\right)}\left(u_{\alpha }^{(k)},{\overline{u}}_{\alpha }^{(k)}\right): p=0,1,\dots ; m=0,1,\dots \right\}$ in the following way:

$$\left\{{\rho }_{\left(p,m\right)}\left((u_1,u_2,\dots ,u_{12}),({\overline{u}}_1,{\overline{u}}_2,\dots ,{\overline{u}}_{12})\right)={\displaystyle \sum _{q=1}^{12}{\rho }_{\left(p,m\right)}\left(u_q,{\overline{u}}_q\right)} : p=0,1,\dots ; m=0,1,\dots \right\}$$

where $(u_1,u_2,\dots ,u_{12})=(u_1^{(1)},u_2^{(1)},u_3^{(1)},u_1^{(2)},u_2^{(2)},u_3^{(2)},u_1^{(3)},u_2^{(3)},u_3^{(3)},u_1^{(4)},u_2^{(4)},u_3^{(4)}).$

The operator functions $B_{\alpha }^{(k)}(u_1^{(1)},\dots ,u_3^{(4)})(.)\in M_{\alpha }^k, (k=1,2,3 ,4 ),$ $(k=1,2,3 ,4 ),$ $\left(\alpha =1,2,3\right).$

Indeed, recall that $T^{(k)}=T_{p+1}^{(k)}-T_p^{(k)}(p=0,1,2,\dots )$. One obtains:

$$B_{\alpha }^{(k)}(u_1^{(1)},\dots ,u_3^{(4)})(0) ={\displaystyle \underset{0}{\overset{0}{\int }}{U}_{\alpha }^{(k)}(s)ds} -\left({\displaystyle \frac{0}{T^{(k)}}}-{\displaystyle \frac{1}{2}}\right) {\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds- \frac{1}{T^{(k)}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{\displaystyle \underset{0}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta =}} } {\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds- \frac{1}{T^{(k)}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{\displaystyle \underset{0}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta }} }$$

$$B_{\alpha }^k(u_1^{(1)},\dots ,u_3^{(4)})(T^{(k)}) ={\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds} -\left( {\displaystyle \frac{T^{(k)}}{T^{(k)}}}-{\displaystyle \frac{1}{2}}\right){\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds} -{\displaystyle \frac{1}{T^{(k)}}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{\displaystyle \underset{0}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta }}={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds}-{\displaystyle \frac{1}{T^{(k)}}}{\displaystyle \underset{0}{\overset{T^{(k)}}{\int }}{\displaystyle \underset{0}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta }}$$

that is $B_{\alpha }^{(k)}(u_1^{(1)},\dots ,u_3^{(4)})(0)=B_{\alpha }^k(u_1^{(1)},\dots ,u_3^{(4)})(T^{(k)}).$

In view of the (CC) we have ${\displaystyle \frac{d{B}_{\alpha }^{(k)}(u_1^{(1)},\dots ,u_3^{(3)})(0)}{dt}}=U_{\alpha }^{(k)}(0)={\dot{u}}_{\alpha }^{(k)}(0),$ ${\displaystyle \frac{d^2{B}_{\alpha }^{(k)}(u_1^{(1)},\dots ,u_3^{(3)})(0)}{d{t}^2}}={\dot{U}}_{\alpha }^{(k)}(0)={\ddot{u}}_{\alpha }^{(k)}(0)$ and so on. Since ${\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left(\frac{t-T_p^{(k)}}{T^{(k)}}-\frac{1}{2}\right)dt}=0$ we obtain

$${\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{B}_{\alpha }^{(k)}(p)(u_1^{(1)},\dots ,u_3^{(4)})(t)dt} ={\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{\displaystyle \underset{T_p}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta }} -{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left(\frac{t}{T^{(k)}}-\frac{1}{2}\right)dt} {\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds-T^{(k)} \frac{1}{T^{(k)}}{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{\displaystyle \underset{T_p^{(k)}}{\overset{\theta }{\int }}{U}_{\alpha }^{(k)}(s)dsd\theta }}= }0$$

that is, $B_{\alpha }^{(k)}(u_1^{(1)},\dots ,u_3^{(4)})(.)\in M_{\alpha }^k.$

The inequalities from Appendix A imply:

$$\begin{array}{l}\left|B_{\alpha }^{(k)}(p,u)(t) \right|\le {\displaystyle \underset{T_p^{(k)}}{\overset{t}{\int }}\left|U_{\alpha }^{(k)}(s)\right|ds} +{\displaystyle \frac{1}{2}} \left|{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds }\right|+{\displaystyle \frac{1}{2}} \left|{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds }\right|\le 2{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|U_{\alpha }^{(k)}(s)\right|ds }\le 2{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|{\displaystyle \sum _{n=1,n\ne k}^4{G}_{\alpha }^{(kn)}}+G_{\alpha }^{(k)rad}\right|ds }\\ \le 2\left({\displaystyle \sum _{n=1,n\ne k}^4{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|G_{\alpha }^{(kn)}\right|ds}}+{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|G_{\alpha }^{(k)rad}\right|ds}\right)\le 2\left[{\displaystyle \sum _{n=1,n\ne k}^4{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{1}{c^2{{\tau }_{kn}}^2}+\frac{{\omega }_n{U}_0^k{e}^{{\mu }_n{T}^{(n)}}}{c^3{\tau }_{kn}}\right)ds}}\right.\left.+{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\frac{e_k^2}{m_k}\frac{{{\omega }_k}^2{U}_0^k{e}^{{\mu }_k{T}^{(k)}}}{c^3}ds}\right]\end{array}$$

$$\begin{array}{l}\le 2\left[{\displaystyle \sum _{n=1,n\ne k}^4{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{1}{c^2{{\tau }_{kn}}^2}+\frac{{\omega }_n{\beta }_n}{c^2{\tau }_{kn}}\right)ds}}\right.\left.+{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\frac{e_k^2}{m_k}\frac{{{\omega }_k}^2{\beta }_k}{c^2}ds}\right]\end{array}$$

$$\le 2\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{\left({r_{kn}}^{(0)}\right)}^2}+\frac{2{\omega }_n{\beta }_n}{{{\mathit{cr}}_{kn}}^{(0)}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k}}\le U_0^{(k)}{e}^{{\mu }_k(t-T_p^{(k)})}$$

For the first derivatives we have

$$\begin{array}{l}\left|{\displaystyle \frac{d{B}_{\alpha }^{(k)}(t)}{dt}}\right| \le \left|U_{\alpha }^k(t)\right| +\left|{\displaystyle \frac{1}{T^{(k)}}} {\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{U}_{\alpha }^{(k)}(s)ds }\right| \le 2\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{2{\omega }_n{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]\\ +{\displaystyle \frac{2(e^{{\mu }_k{T}^{(k)}}-1)}{{\mu }_k{T}^{(k)}}}\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{2{\omega }_n{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]\le 2\left(1+{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k{T}^{(k)}}}\right)\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{2{\omega }_n{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]\le {\omega }_k{U}_0^{(k)}{e}^{{\mu }_k(t-T_p^{(k)})}.\end{array}$$

For the second derivative we have

$$\left|{\displaystyle \frac{d^2{B}_{\alpha }^{(k)}(t)}{d{t}^2}}\right| =\left|{\displaystyle \frac{d{U}_{\alpha }^{(k)}(t)}{dt}}\right| \le {\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|}{m_k}\left(\frac{24c}{{{\stackrel{⌢}{r}}_{kn}}^3}+\frac{20{\beta }_n{\omega }_n}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{4{{\omega }_n}^2{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{\beta }_k{{\omega }_k}^3}{c^2}}\le {{\omega }_k}^2{U}_0^{(k)}{e}^{{\mu }_k(t-T_p^{(k)})}$$

The last inequality is satisfied because on the right-hand side appears ${\omega }_k^2$ and the same is true for higher-order derivatives. Consequently, B maps $M=M_1^1\times M_2^1\times M_3^1\times M_1^2\times M_2^2\times M_3^2\times M_1^3\times M_2^3\times M_3^3\times M_1^4\times M_2^4\times M_3^4$ into itself.

It remains to show that B is a contraction. First, we define mappings of the index set into itself. They are generated by delay functions ${\tau }_{kn}$, that is, $j_{kn}=t-{\tau }_{kn}(t):[T_p^{(k)},{T_{p+1}}^{(k)}]\to [{T_{p-1}}^{(k)},T_p^{(k)}]$

$$(k n)=(12),(13),(14),(21),(23),(24),(31),(32),(34),(41),(42),(43).$$

It is easy to see that every interval $[T_p^{(k)},{T_{p+1}}^{(k)}]$ after finite number of iterations of $j_{kn}$ coincides with $[-T^{(k)},0]$,

$$\begin{array}{l}\left|B_{\alpha }^{(k)}(p,u)(t)-B_{\alpha }^{(k)}(p,\overline{u})(t)\right|\le {\displaystyle \underset{T_p^{(k)}}{\overset{t}{\int }}\left|U_{\alpha }^{(k)}(u)- U_{\alpha }^{(k)}(\overline{u})\right|ds}+\left|{\displaystyle \frac{t-T_p^{(k)}}{T_{(k)}}}-{\displaystyle \frac{1}{2}}\right| {\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|U_{\alpha }^{(k)}(u)- U_{\alpha }^{(k)}(\overline{u})\right|ds}\\ + {\displaystyle \frac{1}{T^{(k)}}}{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{\displaystyle \underset{T_p^{(k)}}{\overset{\theta }{\int }}\left|U_{\alpha }^{(k)}(u)- U_{\alpha }^{(k)}(\overline{u})\right|dsd\theta }} \le {\displaystyle \underset{T_p^{(k)}}{\overset{t}{\int }}\left|U_{\alpha }^{(k)}(u)- U_{\alpha }^{(k)}(\overline{u})\right|ds}+{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|U_{\alpha }^{(k)}(u)- U_{\alpha }^{(k)}(\overline{u})\right|ds}\end{array}$$

$$\le 2{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|U_{\alpha }^{(k)}(u)- U_{\alpha }^{(k)}(\overline{u})\right|ds\le }2{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}{\displaystyle \sum _{n\ne k,n=1}^4}\left|G_{\alpha }^{(kn)}(u)- G_{\alpha }^{(kn)}(\overline{u})\right|ds+2{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}\left|G_{\alpha }^{(k)rad}(u)-G_{\alpha }^{(k)rad}(\overline{u})\right|ds }}$$

$$\le 2{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}}{e}^{{\mu }_k(s-T_p^{(k)})}{\displaystyle \sum _{n=1,n\ne k}^4}{\displaystyle \frac{\left|e_k{e}_n\right|}{m_k}}\left[\left({\displaystyle \frac{8}{{{\stackrel{⌢}{r}}_{kn}}^3}}+{\displaystyle \frac{8(1+{\beta }_n){\omega }_n{U}_0^k{e}^{{\mu }_k{T}^{(k)}}}{c^2{{\stackrel{⌢}{r}}_{kn}}^2}}\right){\displaystyle \sum _{\gamma =1}^3\left(\frac{{{\omega }_k}^h}{{{\mu }_k}^{h+1}}{\rho }_{(p,h)}(u_{\alpha }^{(k)},{\overline{u}}_{\alpha }^{(k)})\right)}\right.$$

$$\begin{array}{l}+{\displaystyle \frac{2{\omega }_n{U}_0^k{e}^{{\mu }_k{T}^{(k)}}}{c^3{\stackrel{⌢}{r}}_{kn}}}{\displaystyle \sum _{\gamma =1}^3\left(\frac{{{\omega }_k}^h}{{{\mu }_k}^h}{\rho }_{(p,h)}(u_{\alpha }^{(k)},{\overline{u}}_{\alpha }^{(k)})\right)}+\left({\displaystyle \frac{8}{{{\stackrel{⌢}{r}}_{kn}}^3}}+{\displaystyle \frac{8{\omega }_n{U}_0^k{e}^{{\mu }_k{T}^{(k)}}}{c^2{{\stackrel{⌢}{r}}_{kn}}^2}}\right)e^{{\mu }_k{T}^{(k)}}{\displaystyle \sum _{\gamma =1}^3\left(\frac{{{\omega }_k}^h}{{{\mu }_k}^{h+1}}{\rho }_{(p-1,h)}(u_{\alpha }^{(k)},{\overline{u}}_{\alpha }^{(k)})\right)}\\ +4{\displaystyle \frac{c^2+c{\tau }_{kn}{\omega }_n{U}_0^k{e}^{{\mu }_k{T}^{(k)}}}{c^3{{\stackrel{⌢}{r}}_{kn}}^2}}{e}^{{\mu }_k{T}^{(k)}}{\displaystyle \sum _{\gamma =1}^3}\left({\displaystyle \frac{{{\omega }_k}^h}{{{\mu }_k}^h}}{\rho }_{\left(p-1,h\right)}(u_{\gamma }^{(n)},{\overline{u}}_{\gamma }^{(n)})\right)+\left.{\displaystyle \frac{4(1+{\beta }_k+{\beta }_n)e^{{\mu }_k{T}^{(k)}}\frac{{{\omega }_k}^h}{{{\mu }_k}^{h+1}}{\displaystyle \sum _{\gamma =1}^3}{\rho }_{(p-1,h)}(u_{\gamma }^{(k)},{\overline{u}}_{\gamma }^{(k)})}{c^2{\stackrel{⌢}{r}}_{kn}}}\right]ds\end{array}$$

$$+{\displaystyle \underset{T_p^{(k)}}{\overset{{T_{p+1}}^{(k)}}{\int }}}{e}^{{\mu }_k(s-T_p^{(k)})}{\displaystyle \frac{e_k^2{{\omega }_k}^2}{m_k{c}^3}}{e}^{{\mu }_k{T}_p^{(k)}}\left({\displaystyle \frac{2{\beta }_k{U}_0^k}{c}}{\displaystyle \sum _{\gamma =1}^3}{\left({\displaystyle \frac{{\omega }_k}{{\mu }_k}}\right)}^h{\rho }_{(p,h)}({u_{\gamma }}^{(k)},{{\overline{u}}_{\gamma }}^{(k)})+{\displaystyle \sum _{\gamma =1}^3}{\displaystyle \frac{{{\omega }_k}^{h+2}}{{{\mu }_k}^h}}{\rho }_{(p,h+2)}({u_{\gamma }}^{(k)},{{\overline{u}}_{\gamma }}^{(k)})\right)ds$$

$$\le K_{(p-1,h)}{\rho }_{(p-1,h)}\left((u_1^{(1)},u_2^{(1)},\dots ,u_3^{(4)}),({\overline{u}}_1^{(1)},{\overline{u}}_2^{(1)},\dots ,{\overline{u}}_3^{(4)})\right)+K_{(p,h+2)}{\rho }_{(p,h+2)}\left((u_1^{(1)},u_2^{(1)},\dots ,u_3^{(4)}),({\overline{u}}_1^{(1)},{\overline{u}}_2^{(1)},\dots ,{\overline{u}}_3^{(4)})\right)$$

Therefore

$$\begin{array}{l}{\rho }_{(p,0)}\left(B_1^{(1)},B_2^{(1)},\dots ,B_3^{(4)}\right),\left({\overline{B}}_1^{(1)},{\overline{B}}_2^{(1)},\dots ,{\overline{B}}_3^{(4)}\right)\\ \le K_{(p-1,h)}{\rho }_{(p-1,h)}\left((u_1^{(1)},u_2^{(1)},\dots ,u_3^{(4)}),({\overline{u}}_1^{(1)},{\overline{u}}_2^{(1)},\dots ,{\overline{u}}_3^{(4)})\right)+K_{(p,h+2)}{\rho }_{(p,h+2)}\left((u_1^{(1)},u_2^{(1)},\dots ,u_3^{(4)}),({\overline{u}}_1^{(1)},{\overline{u}}_2^{(1)},\dots ,{\overline{u}}_3^{(4)})\right),\end{array}$$

where $K_{(p-1,h)}+K_{(p,h+2)}<1$ for sufficiently large $h\in N$ and ${\mu }_k>{\omega }_k$.

Define the maps of the index set into itself $j_1(p,0)\to (p-1,h),j_2(p,0)\to (p,h+2)$. It is easy to see that the maps $j_1,j_2$ commute, and in view of inequality (4) the space

$$M_1^1\times M_2^1\times M_3^1\times M_1^2\times M_2^2\times M_3^2\times M_1^3\times M_2^3\times M_3^3\times M_1^4\times M_2^4\times M_3^4$$

is $\left(j_1,j_2\right)-$ bounded in the sense introduced in [13]. Consequently, operator B is contraction and has a unique fixed point. It is a periodic solution to the 4-body problem in view of the Main lemma.

Theorem 3 is thus proved. □

Remark 3. 

From the above inequalities of the Theorem 3, we obtain

$$\begin{array}{l}2\left(1+{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k{T}^{(k)}}}\right)\left[{\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|(1+{\beta }_n)}{m_k}\left(\frac{4}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{2{\omega }_n{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{{\omega }_k}^2{\beta }_k}{c^2}}\right]\\ \le \left(1+{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k{T}^{(k)}}}\right){\displaystyle \frac{{\mu }_k}{e^{{\mu }_k{T}^{(k)}}-1}}{U}_0^{(k)}=\left(1+{\displaystyle \frac{e^{{\mu }_k{T}^{(k)}}-1}{{\mu }_k{T}^{(k)}}}\right){\displaystyle \frac{{\mu }_k{T}^{(k)}}{e^{{\mu }_k{T}^{(k)}}-1}}{\displaystyle \frac{U_0^{(k)}}{T^{(k)}}}=\left({\displaystyle \frac{{\mu }_k{T}^{(k)}}{e^{{\mu }_k{T}^{(k)}}-1}}+1\right){\displaystyle \frac{U_0^{(k)}}{T^{(k)}}}\le {\omega }_k{U}_0^{(k)}\iff {\displaystyle \frac{{\mu }_k{T}^{(k)}}{e^{{\mu }_k{T}^{(k)}}-1}}+1\le {\omega }_k{T}^{(k)}=2\pi .\end{array}$$

Indeed,

$$f(u)={\displaystyle \frac{u}{e^u-1}}+1\le 2\pi \Rightarrow f\prime (u)={\displaystyle \frac{e^u-1-u{e}^u}{{\left(e^u-1\right)}^2}}<0\iff g(u)=1+u{e}^u-e^u\Rightarrow g\prime (u)=u{e}^u>0$$

$$g(u)>g(0)\iff 1+u{e}^u-e^u>0\Rightarrow f\prime (u)<0\Rightarrow f(0)=2\Rightarrow f(u)={\displaystyle \frac{u}{e^u-1}}+1\le 2\pi ,u>0.$$

This means that the inequalities for the operator functions imply the inequalities for their derivatives.

5. Conditions of the Main Theorem Applied to the Lithium Atom

Let us consider the inequalities from Theorem 3, implying the existence–uniqueness of periodic solution for Li atom. First, we put the first particle (the nuclei) at the origin:

$$\left(x_1^{(1)}(t)=0,x_2^{(1)}(t)=0,x_3^{(1)}(t)=0\right)\Rightarrow \left(u_1^{(1)}(t)=0,u_2^{(1)}(t)=0,u_3^{(1)}(t)=0\right)\Rightarrow T^{(1)}=0,{\omega }_1=0$$

This means that the equation of motion for the first particle becomes $0=0$. Then the second, third, and fourth particles are electrons—the second and third one move along the first stationary state, while the fourth moves along the second stationary state. In view of [9,10,11,12]

${\beta }_2,{\beta }_3=\beta =1/137;{\beta }_4=\beta /2=1/274$;$c=3\times {10}^8 \mathrm{m}/\mathrm{s}$. For the charges we have

$e_1=3e_k=3\times 1.6\times {10}^{-19} C=4.8\times {10}^{-19} ;e_2=e_3=e_4=-1.6\times {10}^{-19} C$ and then

z $\left|e_1{e}_k\right|=4.8\times {10}^{-19}\times 1.6\times {10}^{-19}=7.68\times {10}^{-38}$.

The masses are $m_1=m\times 3\times 1836 =9.11\times {10}^{-31}\times 5508 \mathrm{kg}\simeq 5.02\times {10}^{-27} \mathrm{kg},$

$m=m_2=m_3=m_4=9.11\times {10}^{-31} \mathrm{kg}$.

For the frequencies on the first Bohr–Sommerfeld orbit we take

$${\omega }_2={\omega }_{.}=\omega =4.1\times {10}^{16}\Rightarrow T=T^{(2)}=T^{(3)}=2\pi /\omega \approx 1.53\times {10}^{-16};{\omega }_4=\omega /8\simeq 5.03\times {10}^{15}.$$

Then $T_4=2\pi /{\omega }_4=(2\pi /\omega )\times 8=8T\approx 12.26\times {10}^{-16}$ and ${\displaystyle \frac{m}{4\left|e_2{e}_3\right|}}={\displaystyle \frac{9.11\times {10}^{-31}}{4\times 2.56\times {10}^{-38}}}\approx 8.89\times {10}^6,$ ${\displaystyle \frac{5508 m}{4\left|e_1{e}_2\right|}}\approx 4.9\times {10}^{10}.$

Let us choose ${\mu }_2={\mu }_3=\mu =4.5\times {10}^{16}$ $\Rightarrow \mu T=4.5\times {10}^{16}\times 1.53\times {10}^{-16}=6.885\Rightarrow e^{\mu T}\simeq 977.5$

and ${\displaystyle \frac{e^{\mu T}-1}{\mu T}}={\displaystyle \frac{977.5-1}{6.885}}={\displaystyle \frac{976.5}{6.885}}\approx 141.8$;${\displaystyle \frac{\mu }{e^{\mu T}-1}}={\displaystyle \frac{4.5\times {10}^{16}}{977.5-1}}\simeq 5\times {10}^{13}$. For the second stationary state (cf. [8,9]) we take ${\mu }_4$ such that ${\mu }_4{T}_4=\mu T$. This implies ${\mu }_4{T}_4=8{\mu }_{4}T=\mu T\Rightarrow {\mu }_4=\mu /8$ and then ${\mu }_4{T}_4=\mu T=688.5\Rightarrow e^{{\mu }_4{T}_4}\simeq 977.5;$ ${\displaystyle \frac{e^{{\mu }_4{T}^{(4)}}-1}{{\mu }_4}}=8{\displaystyle \frac{e^{\mu T}-1}{\mu }}=8{\displaystyle \frac{977.5-1}{{10}^{15}}}\simeq 7.812\times {10}^{-12}$.

We recall $r_{kn}=r_{nk}$. The constants $\mu ,{\mu }_4>0$ and $U_0^{(k)}>0 (k=1,2,3,4)$ we choose as control parameters with restrictions $\mu ,{\mu }_4\in (0;\infty );\mu T=const.$ and

$$U_0^{(k)}{e}^{{\mu }_k{T}^{(k)}}\le c_k\iff U_0^{(2)},U_0^{(3)}\le {\displaystyle \frac{c{e}^{-\mu T}}{137}}\le {\displaystyle \frac{3\times {10}^8}{137}}\simeq 2.19\times {10}^6;U_0^{(4)}\le {\displaystyle \frac{c{e}^{-{\mu }_4{T}^{(4)}}}{2\times 137}}\le {\displaystyle \frac{3\times {10}^8}{274}}\simeq {10}^6.$$

Assuming $1+{\beta }_n\approx 1$ we obtain the first group of inequalities for the distances between the particles of the Lithium atom:

$${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{2\omega }{137c{\stackrel{⌢}{r}}_{23}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{\omega \beta }{8c{\stackrel{⌢}{r}}_{24}}}\le {\displaystyle \frac{U_0^{(2)}}{2}}{\displaystyle \frac{m}{\left|e_2{e}_3\right|}}{\displaystyle \frac{\mu }{e^{\mu T}-1}}-{\displaystyle \frac{12}{{{\stackrel{⌢}{r}}_{12}}^2}}-{\displaystyle \frac{{\omega }^2\beta }{c^2}};$$

$${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{32}}^2}}+{\displaystyle \frac{2\omega \beta }{c{\stackrel{⌢}{r}}_{32}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}+{\displaystyle \frac{\omega \beta }{8c{\stackrel{⌢}{r}}_{34}}}\le {\displaystyle \frac{U_0^{(3)}}{2}}{\displaystyle \frac{m}{\left|e_3{e}_2\right|}}{\displaystyle \frac{\mu }{e^{\mu T}-1}}-{\displaystyle \frac{12}{{{\stackrel{⌢}{r}}_{13}}^2}}-{\displaystyle \frac{{\omega }^2\beta }{c^2}};$$

$${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{42}}^2}}+{\displaystyle \frac{2\omega \beta }{c{\stackrel{⌢}{r}}_{42}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{43}}^2}}+{\displaystyle \frac{2\omega \beta }{c{\stackrel{⌢}{r}}_{43}}}\le {\displaystyle \frac{U_0^{(4)}}{2}}{\displaystyle \frac{m}{\left|e_4{e}_2\right|}}{\displaystyle \frac{{\mu }_4}{e^{\mu T}-1}}-{\displaystyle \frac{12}{{{\stackrel{⌢}{r}}_{14}}^2}}-{\displaystyle \frac{{\omega }^2\beta }{128c^2}};{\displaystyle \frac{{e_2}^2}{m_2}}={\displaystyle \frac{{e_3}^2}{m_3}}={\displaystyle \frac{{e_4}^2}{m_4}}$$

Since we denote by

$A_k={\displaystyle \frac{m_k}{2{e_k}^2}}{\displaystyle \frac{\mu }{e^{\mu T}-1}}(k=2,3,4)\Rightarrow A_2=A_3=A_4;A_k={\displaystyle \frac{m_k}{2{e_k}^{2}T}}{\displaystyle \frac{\mu T}{e^{\mu T}-1}}\le {\displaystyle \frac{m_k(2\pi -1)}{2{e_k}^{2}T}};$

$r_{12}=r_{31}=5.29\times {10}^{-11};r_{41}=4r_{14}=4\times 5.29\times {10}^{-11}=21.16\times {10}^{-11};$ ${\displaystyle \frac{{e_k}^2}{m_k}}={\displaystyle \frac{{1.6}^2\times {10}^{-38}}{9.11\times {10}^{-31}}}=2.8\times {10}^{-8}$ and ${\displaystyle \frac{\left|e_1{e}_k\right|}{m}}={\displaystyle \frac{3\left|e_3{e}_2\right|}{m}}=3\times 2.8\times {10}^{-8}=8.4\times {10}^{-8}(k=2,3,4)$.

Then we obtain:

$${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{23}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{1.25\times {10}^5}{{\stackrel{⌢}{r}}_{24}}}\le \left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}-1.36\times {10}^{14}$$

$${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{23}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}+{\displaystyle \frac{1.25\times {10}^5}{{\stackrel{⌢}{r}}_{34}}}\le \left(8.92U_0^{(3)}-42.9\right)\times {10}^{20}-1.36\times {10}^{14}$$

$${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{24}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{34}}}\le (2.28U_0^{(4)}-268){10}^{18}-{10}^{12}.$$

Disregard the terms $1.36\times {10}^{14}$,${10}^{12}$ we obtain

(i)

${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{23}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{1.25\times {10}^5}{{\stackrel{⌢}{r}}_{24}}}\le \left(8.92U_0^{(2)}-42.9\right){10}^{20};$

(ii)

${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{23}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}+{\displaystyle \frac{1.25\times {10}^5}{{\stackrel{⌢}{r}}_{34}}}\le \left(8.92U_0^{(3)}-42.9\right){10}^{20};$

(iii)

33${\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{24}}}+{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{34}}}\le (2.28U_0^{(4)}-268){10}^{18}.$

Let us assume $U_0^{(2)}>{\displaystyle \frac{42.9}{8.92}}\approx 4.81$; $U_0^{(3)}>4.81$; $U_0^{(4)}>{\displaystyle \frac{268}{2.28}}\approx 117.55$. From inequality (i) we have:

$$\begin{array}{l}0<{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{1.25\times {10}^5}{{\stackrel{⌢}{r}}_{24}}}\le \left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}-{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}-{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{23}}}\Rightarrow \left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}{{\stackrel{⌢}{r}}_{23}}^2-2\times {10}^6{\stackrel{⌢}{r}}_{23}-4>0\Rightarrow \\ {\stackrel{⌢}{r}}_{23}\ge {\displaystyle \frac{{10}^6+\sqrt{{10}^{12}+4\left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}}}{\left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}}}\approx {\displaystyle \frac{2}{\sqrt{8.92U_0^{(2)}-42.9}}}\times {10}^{-10}.\end{array}$$

In a similar way we have

$$\begin{array}{l}0<{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{23}}}\le \left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}-{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}-{\displaystyle \frac{1.25\times {10}^5}{{\stackrel{⌢}{r}}_{24}}}\Rightarrow \left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}{{\stackrel{⌢}{r}}_{24}}^2-1.25\times {10}^5{\stackrel{⌢}{r}}_{24}-4>0\Rightarrow \\ {\stackrel{⌢}{r}}_{24}\ge {\displaystyle \frac{1.25\times {10}^5+\sqrt{{1.25}^2\times {10}^{10}+16\left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}}}{2\times \left(8.92U_0^{(2)}-42.9\right)\times {10}^{20}}}\approx {\displaystyle \frac{2}{\sqrt{8.92U_0^{(2)}-42.9}}}\times {10}^{-10}.\end{array}$$

From inequality (ii) we obtain

$${\stackrel{⌢}{r}}_{23}\ge {\displaystyle \frac{2}{\sqrt{8.92U_0^{(3)}-42.9}}}\times {10}^{-10},{\stackrel{⌢}{r}}_{34}\ge {\displaystyle \frac{2}{\sqrt{8.92U_0^{(3)}-42.9}}}\times {10}^{-10}.$$

From inequality (iii) we obtain:

$$\begin{array}{l}0<{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{34}}}\le (2.28U_0^{(4)}-268){10}^{18}-{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}-{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{24}}}\Rightarrow (2.28U_0^{(4)}-268){10}^{18}{{\stackrel{⌢}{r}}_{24}}^2-2\times {10}^6{\stackrel{⌢}{r}}_{24}-4>0\Rightarrow \\ {\stackrel{⌢}{r}}_{24}\ge {\displaystyle \frac{{10}^6+\sqrt{{10}^{12}+4(2.28U_0^{(4)}-268){10}^{18}}}{(2.28U_0^{(4)}-268){10}^{18}}}\approx {\displaystyle \frac{2}{\sqrt{2.28U_0^{(4)}-268}}}\times {10}^{-9}\end{array}$$

and

$$0<{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{24}}^2}}+{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{24}}}\le (2.28U_0^{(4)}-268){10}^{18}-{\displaystyle \frac{4}{{{\stackrel{⌢}{r}}_{34}}^2}}-{\displaystyle \frac{2\times {10}^6}{{\stackrel{⌢}{r}}_{34}}}\Rightarrow {\stackrel{⌢}{r}}_{34}\ge {\displaystyle \frac{2}{\sqrt{2.28U_0^{(4)}-268}}}\times {10}^{-9}.$$

Consequently,

$${\stackrel{⌢}{r}}_{23}\ge \max \left\{{\displaystyle \frac{2\times {10}^{-10}}{\sqrt{8.92U_0^{(2)}-42.9}}};{\displaystyle \frac{2\times {10}^{-10}}{\sqrt{8.92U_0^{(3)}-42.9}}}\right\}$$

$${\stackrel{⌢}{r}}_{24}\ge \max \left\{{\displaystyle \frac{2\times {10}^{-10}}{\sqrt{8.92U_0^{(2)}-42.9}}},{\displaystyle \frac{2\times {10}^{-9}}{\sqrt{2.28U_0^{(4)}-268}}}\right\}$$

(7)

$${\stackrel{⌢}{r}}_{34}\ge \max \left\{{\displaystyle \frac{2\times {10}^{-10}}{\sqrt{8.92U_0^{(3)}-42.9}}},{\displaystyle \frac{2\times {10}^{-9}}{\sqrt{2.28U_0^{(4)}-268}}}\right\}$$

(8)

Finally, we check the inequalities from the Main theorem for the second derivatives:

$${\displaystyle \sum _{n=1,n\ne k}^4\frac{\left|e_k{e}_n\right|}{m_k}\left(\frac{32c({\beta }_k+{\beta }_n)}{{{\stackrel{⌢}{r}}_{kn}}^3}+\frac{8{\omega }_n{\beta }_n}{{{\stackrel{⌢}{r}}_{kn}}^2}+\frac{8{{\omega }_n}^2{\beta }_n}{c{\stackrel{⌢}{r}}_{kn}}\right)}+{\displaystyle \frac{e_k^2}{m_k}}{\displaystyle \frac{{\beta }_k{{\omega }_k}^3}{c^2}}\le {{\omega }_k}^2{U}_0^{(k)}(k=2,3,4).$$

Indeed, the first one is satisfied

$${\displaystyle \frac{\left|e_2{e}_3\right|}{137m_2}}\left({\displaystyle \frac{96c}{{\omega }^2{{\stackrel{⌢}{r}}_{21}}^3}}+{\displaystyle \frac{64c}{{\omega }^2{{\stackrel{⌢}{r}}_{23}}^3}}+{\displaystyle \frac{8}{\omega {{\stackrel{⌢}{r}}_{23}}^2}}+{\displaystyle \frac{8}{c{\stackrel{⌢}{r}}_{23}}}+{\displaystyle \frac{48c}{{\omega }^2{{\stackrel{⌢}{r}}_{24}}^3}}+{\displaystyle \frac{1}{2{{\stackrel{⌢}{r}}_{24}}^2\omega }}+{\displaystyle \frac{1}{16c{\stackrel{⌢}{r}}_{24}}}+{\displaystyle \frac{\omega }{c^2}}\right)\le U_0^{(2)}$$

Because

$${10}^{-10}\left(1.16\times {10}^8+3.19\times {10}^6+0.85\times {10}^4+1.74\times {10}^2+1.4\times {10}^4+1.7\times 10+0.24+0.456\right)\le 5.$$

The second one can be checked in a similar way. The last one becomes

$${10}^{-4}\left(1.12+0.0167+0.00034+0.00004+0.189+0.0017+0.000089+0.00000000056\right)\le 1.246.$$

To assure the inequalities of higher order derivatives we notice that for the third derivative we have

$${\displaystyle \frac{1}{{\omega }^3{{\stackrel{⌢}{r}}_{21}}^4}}\approx {\displaystyle \frac{1}{{4.1}^3\times {10}^{48}}}{\displaystyle \frac{1}{{5.29}^4\times {10}^{-44}}}$$

and for the n-th derivative we obtain the inequality

$${\displaystyle \frac{1}{{\omega }^n{{\stackrel{⌢}{r}}_{21}}^{n+1}}}\approx {\displaystyle \frac{1}{{4.1}^n\times {10}^{16n}}}{\displaystyle \frac{1}{{5.29}^{n+1}\times {10}^{-11(n+1)}}}={\displaystyle \frac{1}{5.29\times {(21.68)}^n\times {10}^{5n-11}}}\le {\displaystyle \frac{1}{5.29\times {(2)}^n\times {10}^{5n-10}}}(n=2,3,\dots )$$

Remark 4. 

To illustrate the role of the parameters we check the distance relation ${\stackrel{⌢}{r}}_{14}={\stackrel{⌢}{r}}_{12}+{\stackrel{⌢}{r}}_{24}=4{\stackrel{⌢}{r}}_{12}\Rightarrow {\stackrel{⌢}{r}}_{24}=3{\stackrel{⌢}{r}}_{12}$ known from quantum mechanics (cf. [9,10,11,12]).

Inequality (7) implies

$${\stackrel{⌢}{r}}_{24}={\displaystyle \frac{2}{\sqrt{2.28U_0^{(4)}-268}}}\times {10}^{-9}=3\times 5.29\times {10}^{-11}=3{\stackrel{⌢}{r}}_{12}\iff U_0^{(4)}={\displaystyle \frac{{12.6}^2+268}{2.28}}\approx 187.18>117.55$$

or ${\displaystyle \frac{2\times {10}^{-10}}{\sqrt{8.92U_0^{(2)}-42.9}}}=3\times 5.29\times {10}^{-11}\iff U_0^{(2)}=4.987>4.81$.

Inequality (6) yields ${\stackrel{⌢}{r}}_{34}={\displaystyle \frac{2\times {10}^{-9}}{\sqrt{2.28U_0^{(4)}-268}}}=3\times 5.29\times {10}^{-11}=3{\stackrel{⌢}{r}}_{13}\Rightarrow U_0^{(4)}=187.2$ or

$${\stackrel{⌢}{r}}_{34}={\displaystyle \frac{2\times {10}^{-10}}{\sqrt{8.92U_0^{(3)}-42.9}}}=3\times 5.29\times {10}^{-11}=3{\stackrel{⌢}{r}}_{13}\iff U_0^{(3)}=4.987>4.809.$$

6. Conclusions

The interest in the topic is evident from the articles [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. We have obtained the existence–uniqueness of general $\left(T^{(1)},T^{(2)},T^{(3)},T^{(4)}\right)-$ periodic solution of the 4-body problem extending the approach from the case of the 2- and 3-body problems. In fact, we apply the same form of radiation terms following Dirac’s physical assumption justified by nonstandard analysis (cf. [8]). Our advantage is that we obtain the estimates of distances between the moving charged particles which include as a particular case the values from quantum mechanics. Consequently, classical electrodynamics as a theory of the structure of an elementary charged particle and its interaction with other elementary particles is more general than quantum mechanics.