Some Properties of Motion of Atoms near a Charged Wire

Abstract

We are concerned with the classical approach to atomic motions in the vicinity of a wire charged by an oscillating voltage. This mechanism has been proposed to serve for trapping cold neutral atoms. We prove that, under some conditions, the atom must either collide with the wire or escape to infinity in the radial direction. The presented results for the atom–wire problem extend and complement some earlier ones in the literature.

1. Introduction

W. Paul [1] presented the electromagnetic trapping mechanism for the cold atoms. L. S. Brawn [2] also investigated this problem. In [3], L. V. Hau et al. studied the interaction of a neutral atom with a charged wire. The charge varies sinusoidally in time. Using numerical simulations, authors [3] predicted a range of parameters for which bounded motions occur for both the classical and quantum problems. Ch. King and A. Leśniewski [4] proved that for one range of parameters all solutions of the equation of motion either hit the wire or escape to infinity. In addition, for another range of parameters, there are solutions that remain within a finite, non-zero distance of the wire. J. Lei and M. Zhang [5] study the Lyapunov stability of the periodic motion of an atom in the vicinity of an infinite straight wire with an oscillating charge.

In this paper, we present that for some range of parameters all solutions of the equation of motion either hit the wire or escape to infinity in the radial direction. That is, there is no bounded motion in the atom–wire regime. Atom hits the wire and is probably absorbed into the surface of the wire or escapes to infinity in the radial direction. In the paper, we examine the influence of history on the motion of atoms near a charged wire. Some presented results extend and complement earlier ones in the literature. A numerical simulation is also presented.

2. Properties of Solutions of Equation of Atoms Motion

We consider a neutral atom of mass M moving in a vicinity of a rigid, straight wire carrying a uniformly distributed time-dependent charge $q\left(t\right)$. This produces the potential-energy function

$$V\left(r\right)=-\frac{2\alpha q^2}{r^2},$$

where r is the distance from the wire and $\alpha$ is the atom’s polarizability. The Hamiltonian for the atom’s radial motion is given by [3]

$$H(r,p_r)=\frac{p_r^2}{2M}-\frac{2\alpha q^2}{r^2}+\frac{L^2}{2M{r}^2},$$

with radial momentum $p_r=Mdr/dt$ and fixed angular momentum L. The radial equation of motion follows from the system

$$\begin{array}{ccc}\hfill \frac{dr}{dt}& =& \frac{\partial H(r,p_r)}{\partial p_r},\hfill \\ \hfill \frac{d{p}_r}{dt}& =& -\frac{\partial H(r,p_r)}{\partial r}.\hfill \end{array}$$

We obtain

$$\begin{array}{ccc}\hfill \frac{dr}{dt}& =& \frac{1}{M}{p}_r,\hfill \\ \hfill \frac{d{p}_r}{dt}& =& -\frac{4\alpha q^2}{r^3}+\frac{L^2}{M{r}^3}.\hfill \end{array}$$

Then we obtain

$$\begin{array}{ccc}\hfill \frac{d^{2}r}{d{t}^2}& =& \frac{1}{M}\frac{d{p}_r}{dt},\hfill \\ \hfill \frac{d^{2}r}{d{t}^2}& +& \left(\frac{4\alpha q^2}{M}-\frac{L^2}{M^2}\right)\frac{1}{r^3}=0.\hfill \end{array}$$

We put

$$p\left(t\right)=\frac{4\alpha q^2\left(t\right)}{M}-\frac{L^2}{M^2}$$

and consider the following nonlinear differential equation with the singularity

$$\frac{d^{2}r\left(t\right)}{d{t}^2}+\frac{p\left(t\right)}{r^3\left(t\right)}=0,t\ge t_0,$$

(1)

where $t_0\in R,p\in C([t_0,\infty ),R)$, i.e., the function p, is continuous on the interval $[t_0,\infty ).$

We require some lemmas for the main result.

Lemma 1. 

Suppose that function $p\in C^1([t_0,\infty ),R)$ and r is an eventually positive solution of (1). Then,

$$\begin{array}{ccc}\hfill r^{\prime }(t-\tau )-r^{\prime }\left(t\right)& =& p\left(t\right){\int }_{t-\tau }^t{r}^{-3}\left(s\right)ds+[p\left(t\right)-p(t-\tau )]{\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds\hfill \\ & -& {\int }_{t-\tau }^t{p}^{\prime }\left(s\right){\int }_{t_0}^s{r}^{-3}\left(u\right)duds,t\ge t_0+\tau ,\tau >0.\hfill \end{array}$$

(2)

Proof. 

From Equation (1), we obtain

$$r^{\prime }(t-\tau )-r^{\prime }\left(t\right)={\int }_{t-\tau }^{t}p\left(s\right)r^{-3}\left(s\right)ds,t\ge t_0+\tau .$$

(3)

Integrating that, we obtain

$$\begin{array}{ccc}& & {\int }_{t-\tau }^{t}p\left(s\right)r^{-3}\left(s\right)ds=p\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)ds-p(t-\tau ){\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds\hfill \\ & & -{\int }_{t-\tau }^t{p}^{\prime }\left(s\right){\int }_{t_0}^s{r}^{-3}\left(u\right)duds=p\left(t\right){\int }_{t-\tau }^t{r}^{-3}\left(s\right)ds\hfill \\ & & +[p\left(t\right)-p(t-\tau )]{\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds-{\int }_{t-\tau }^t{p}^{\prime }\left(s\right){\int }_{t_0}^s{r}^{-3}\left(u\right)duds,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill r^{\prime }(t-\tau )-r^{\prime }\left(t\right)& =& p\left(t\right){\int }_{t-\tau }^t{r}^{-3}\left(s\right)ds+[p\left(t\right)-p(t-\tau )]{\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds\hfill \\ & -& {\int }_{t-\tau }^t{p}^{\prime }\left(s\right){\int }_{t_0}^s{r}^{-3}\left(u\right)duds,t\ge t_0+\tau .\hfill \end{array}$$

□

We consider the function

$$G\left(t\right)=-{\int }_{t-\tau }^t{p}^{\prime }\left(s\right){\int }_{t_0}^s{r}^{-3}\left(u\right)duds,t\ge t_0+\tau .$$

Lemma 2. 

Suppose that r is an eventually bounded positive solution of (1). Assume that:

$\left(A_1\right)$ the function $p\in C^1([t_0,\infty ),R)$,

$\left(A_2\right)$ there exists $T_n\in R$ such that $p^{\prime }\left(t\right)>0$ for $t\in (T_n-\tau ,T_n)$ and $p^{\prime }\left(t\right)<0$ for $t\in (T_n,T_n+0.5\tau ],n\in N$,

$\left(A_3\right)p(T_n+0.5\tau )-p(T_n-0.5\tau )=0,n\in N$,

$\left(A_4\right)inf\left\{-{\int }_{T_n}^{T_n+0.5\tau }(t-T_n)p^{\prime }\left(t\right)dt,n\in N\right\}>0$.

Then,

$$inf\left\{G(T_n+0.5\tau ),n\in N\right\}>0.$$

Proof. 

We obtain

$$\begin{array}{ccc}& & G(T_n+0.5\tau )=-{\int }_{T_n-0.5\tau }^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & =-{\int }_{T_n-0.5\tau }^{T_n}{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt-{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & \ge -{\int }_{T_n-0.5\tau }^{T_n}{p}^{\prime }\left(t\right)dt{\int }_{t_0}^{T_n}{r}^{-3}\left(s\right)ds-{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right)dt{\int }_{t_0}^{T_n}{r}^{-3}\left(s\right)ds\hfill \\ & & -{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{T_n}^t{r}^{-3}\left(s\right)dsdt=\left(-{\int }_{T_n-0.5\tau }^{T_n}{p}^{\prime }\left(t\right)dt-{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right)dt\right)\hfill \\ & & \times {\int }_{t_0}^{T_n}{r}^{-3}\left(s\right)ds-{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{T_n}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & =[p(T_n-0.5\tau )-p(T_n+0.5\tau )]{\int }_{t_0}^{T_n}{r}^{-3}\left(s\right)ds\hfill \\ & & -{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{T_n}^t{r}^{-3}\left(s\right)dsdt=-{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{T_n}^t{r}^{-3}\left(s\right)dsdt.\hfill \end{array}$$

Since $r\left(t\right)$ is bounded, then there exists a constant $K>0$ such that $0<r^3\left(t\right)\le K,t\ge t_0+\tau$. Then,

$$\begin{array}{ccc}\hfill G(T_n+0.5\tau )& \ge & -{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{T_n}^t{r}^{-3}\left(s\right)dsdt.\hfill \\ & \ge & -\frac{1}{K}{\int }_{T_n}^{T_n+0.5\tau }(t-T_n)p^{\prime }\left(t\right)dt,n\in N.\hfill \end{array}$$

With regard to $\left(A_4\right)$, we have $inf\{G(T_n+0.5\tau ),n\in N\}>0$. □

Theorem 1. 

Suppose that $\left(A_1\right)-\left(A_4\right)$ hold, $p\left(t\right)$ is periodic with period $2\tau$ and

$$p^{\prime }(t-\tau )-p^{\prime }\left(t\right)>0,t\in (T_n,T_n+0.5\tau ),T_n=2n\tau ,n\in N.$$

(4)

Then, (1) has no solution such that

$$0<K_1\le r^3\left(t\right)\le K_2,t\ge t_0,$$

where $K_1,K_2$ are constants.

Proof. 

Assume that Equation (1) has a solution such that $0<K_1\le r^3\left(t\right)\le K_2,t\ge t_0$. For derivatives of the function $G\left(t\right)$, we obtain

$$\begin{array}{ccc}& & G^{\prime }\left(t\right)=-p^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)ds+p^{\prime }(t-\tau ){\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds\hfill \\ & & =-p^{\prime }\left(t\right){\int }_{t-\tau }^t{r}^{-3}\left(s\right)ds-p^{\prime }\left(t\right){\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds+p^{\prime }(t-\tau ){\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds\hfill \\ & & =-p^{\prime }\left(t\right){\int }_{t-\tau }^t{r}^{-3}\left(s\right)ds+[p^{\prime }(t-\tau )-p^{\prime }\left(t\right)]{\int }_{t_0}^{t-\tau }{r}^{-3}\left(s\right)ds,t\ge t_0+\tau .\hfill \end{array}$$

The condition (4) implies that $G^{\prime }\left(t\right)>0$ for $t\in (T_n,T_n+0.5\tau )$. Thus, the function $G\left(t\right)$ is increasing on $(T_n,T_n+0.5\tau ),n\in N$. Since $G\left(T_n\right)<0,n\in N$, by virtue of Lemma 2, there exist $t_n\in (T_n,T_n+0.5\tau )$ such that $G\left(t_n\right)=0,n\in N$. Set

$$H\left(t\right)=p\left(t\right)-p(t-\tau ),t\in (T_n,T_n+0.5\tau ],n\in N.$$

According to (4) and $\left(A_3\right)$, we find that

$$H^{\prime }\left(t\right)=p^{\prime }\left(t\right)-p^{\prime }(t-\tau )<0,t\in (T_n,T_n+0.5\tau )$$

and $H(T_n+0.5\tau )=0,n\in N$. Then,

$$H\left(t\right)=p\left(t\right)-p(t-\tau )>0fort\in (T_n,T_n+0.5\tau ),n\in N.$$

(5)

Now assume that $t_n\le b_n=T_n+0.5\tau -\epsilon ,n\in N$, where $0<\epsilon <0.5\tau$. With regard to (2), we obtain

$$\begin{array}{ccc}& & r^{\prime }(b_n-\tau )-r^{\prime }\left(b_n\right)=p\left(b_n\right){\int }_{b_n-\tau }^{b_n}{r}^{-3}\left(s\right)ds\hfill \\ & & +[p\left(b_n\right)-p(b_n-\tau )]{\int }_{t_0}^{b_n-\tau }{r}^{-3}\left(s\right)ds-{\int }_{b_n-\tau }^{b_n}{p}^{\prime }\left(s\right){\int }_{t_0}^s{r}^{-3}\left(u\right)duds,\hfill \\ & & b_n\ge t_0+\tau ,n\in N.\hfill \end{array}$$

Since $p\left(t\right)$ is periodic, there exists a constant $K_3>0$ such that $-K_3\le p\left(t\right)\le K_3,t\ge t_0+\tau$. With regard to (3), we obtain

$$r^{\prime }(t-\tau )-r^{\prime }\left(t\right)\le K_3{\int }_{t-\tau }^t{r}^{-3}\left(s\right)ds\le \frac{K_3\tau }{K_1},t\ge t_0+\tau .$$

Then, we have

$$\begin{array}{ccc}\hfill \frac{K_3\tau }{K_1}& \ge & p\left(b_n\right){\int }_{b_n-\tau }^{b_n}{r}^{-3}\left(s\right)ds\hfill \\ & +& [p\left(b_n\right)-p(b_n-\tau )]{\int }_{t_0}^{b_n-\tau }{r}^{-3}\left(s\right)ds+G\left(b_n\right),b_n\ge t_n,n\in N,\hfill \end{array}$$

$$\frac{2K_3\tau }{K_1}\ge [p\left(b_n\right)-p(b_n-\tau )]{\int }_{t_0}^{b_n-\tau }{r}^{-3}\left(s\right)ds+G\left(b_n\right),b_n\ge t_n,n\in N.$$

(6)

By virtue of (5), $T_n=2n\tau$, and periodicity of $p\left(t\right)$ we obtain

$$\begin{array}{ccc}& & p\left(b_n\right)-p(b_n-\tau )=p(T_n+0.5\tau -\epsilon )-p(T_n-0.5\tau -\epsilon )\hfill \\ & & =p(2n\tau +0.5\tau -\epsilon )-p(2n\tau -0.5\tau -\epsilon )\hfill \\ & & =p(0.5\tau -\epsilon )-p(-0.5\tau -\epsilon )>0.\hfill \end{array}$$

Then, for sufficiently large $b_n\ge t_n$, we obtain

$$\begin{array}{ccc}& & [p\left(b_n\right)-p(b_n-\tau )]{\int }_{t_0}^{b_n-\tau }{r}^{-3}\left(s\right)ds\ge [p(0.5\tau -\epsilon )-p(-0.5\tau -\epsilon )]\hfill \\ & & \times {\int }_{t_0}^{b_n-\tau }{r}^{-3}\left(s\right)ds\ge [p(0.5\tau -\epsilon )-p(-0.5\tau -\epsilon )]\frac{b_n-\tau -t_0}{K_2}>\frac{2K_3\tau }{K_1},\hfill \end{array}$$

which contradicts (6).

Now, assume that there exists a sequence $\left\{t_n\right\}$ such that $t_n\to T_n+0.5\tau ,asn\to \infty ,G\left(t_n\right)=0,t_n\in (T_n,T_n+0.5\tau ),n\in N$. Then,

$$\begin{array}{ccc}& & G(T_n+0.5\tau )=-{\int }_{T_n-0.5\tau }^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & =-{\int }_{T_n-0.5\tau }^{T_n}{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt-{\int }_{T_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & =-{\int }_{t_n-\tau }^{T_n}{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt+{\int }_{t_n-\tau }^{T_n-0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & -{\int }_{T_n}^{t_n}{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt-{\int }_{t_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & =-{\int }_{t_n-\tau }^{t_n}{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt+{\int }_{t_n-\tau }^{T_n-0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & -{\int }_{t_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt=G\left(t_n\right)+{\int }_{t_n-\tau }^{T_n-0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & -{\int }_{t_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt={\int }_{t_n-\tau }^{T_n-0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\hfill \\ & & -{\int }_{t_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt,n\in N.\hfill \end{array}$$

Since $t_n\to T_n+0.5\tau ,asn\to \infty$, then

$${\int }_{t_n-\tau }^{T_n-0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\to 0,-{\int }_{t_n}^{T_n+0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\to 0.$$

Otherwise, for example assume that for $t_n\to T_n+0.5\tau$, as $n\to \infty$,

$${\int }_{t_n-\tau }^{T_n-0.5\tau }{p}^{\prime }\left(t\right){\int }_{t_0}^t{r}^{-3}\left(s\right)dsdt\to c>0.$$

Then, $inf\{G(T_n+0.5\tau )>0,n\in N\}>0$, and since $G^{\prime }\left(t\right)>0$ for $t\in (T_n,T_n+0.5\tau ),G\left(T_n\right)<0$, there exists $t_n^{*}\in (T_n,T_n+0.5\tau )$ such that $G\left(t_n^{*}\right)=0,n\in N$, and $G\left(t\right)>0$ for $t\in (t_n^{*},T_n+0.5\tau ]$. This contradicts that $G\left(t_n\right)=0$ for some $t_n\in (t_n^{*},T_n+0.5\tau )$.

Thus, $G(T_n+0.5\tau )\to 0,as{t}_n\to T_n+0.5\tau andn\to \infty$. This contradicts that $inf\{G(T_n+0.5\tau ),n\in N\}>0$. □

3. Applications

We now apply the above Theorem 1 on the atom–wire problem. For $q\left(t\right)=Qcos(\omega t/2)$, [3,4], with regard to

$$q^2\left(t\right)=Q^2\frac{1+cos\omega t}{2},$$

the atom–wire Hamiltonian has the form

$$H(r,p_r)=\frac{p_r^2}{2M}+\frac{1}{r^2}\left(\frac{L^2}{2M}-\alpha Q^2\right)-\frac{\alpha Q^2}{r^2}cos\omega t.$$

The Hamiltonian system is described by

$$\begin{array}{ccc}\hfill \frac{dr}{dt}& =& \frac{1}{M}{p}_r,\hfill \\ \hfill \frac{d{p}_r}{dt}& =& -\frac{2\alpha Q^2(1+cos\omega t)}{r^3}+\frac{L^2}{M{r}^3}.\hfill \end{array}$$

(7)

Then, we obtain

$$\begin{array}{ccc}& & \frac{d^{2}r}{d{t}^2}=\frac{1}{M}\frac{d{p}_r}{dt}=\frac{1}{M}\left(-\frac{2\alpha Q^2(1+cos\omega t)}{r^3}+\frac{L^2}{M{r}^3}\right),\hfill \\ & & \frac{d^{2}r}{d{t}^2}+\left(\frac{2\alpha Q^2(1+cos\omega t)}{M}-\frac{L^2}{M^2}\right)\frac{1}{r^3}=0.\hfill \end{array}$$

The equation governing the motion of the atom has the form

$$\frac{d^{2}r}{d{t}^2}+\frac{A+Bcos\omega t}{r^3}=0,t\ge t_0,$$

(8)

where

$$A=\frac{2\alpha Q^2}{M}-\frac{L^2}{M^2},B=\frac{2\alpha Q^2}{M}.$$

We apply the Theorem 1 on Equation (8).

Corollary 1. 

Suppose that $\tau =\pi /\omega >0,B>0$. Then, Equation (8) has no solution such that

$$0<K_1\le r^3\left(t\right)\le K_2,t\ge t_0,$$

where $K_1,K_2$ are constants.

Proof. 

The function $p\left(t\right)=A+Bcos\omega t$. The condition $\left(A_1\right)$ is satisfied. For the condition $\left(A_2\right)$, we take $T_n=2n\tau =2\pi n/\omega$. Then,

$$\begin{array}{ccc}& & p^{\prime }\left(t\right)=-\omega Bsin\omega t>0fort\in \left(\frac{(2n-1)\pi }{\omega },\frac{2\pi n}{\omega }\right),\hfill \\ & & p^{\prime }\left(t\right)<0fort\in \left(\frac{2\pi n}{\omega },\frac{(4n+1)\pi }{2\omega }\right],n\in N.\hfill \end{array}$$

For $\left(A_3\right)$, we have:

$$\begin{array}{ccc}& & A+Bcos\omega (T_n+0.5\tau )-A-Bcos\omega (T_n-0.5\tau )\hfill \\ & & =Bcos(2n+0.5)\pi -Bcos(2n-0.5)\pi =0,n\in N.\hfill \end{array}$$

Consider the condition $\left(A_4\right)$. We obtain

$$\begin{array}{ccc}& & {\int }_{2\pi n/\omega }^{(2n+0.5)\pi /\omega }\left(t-\frac{2\pi n}{\omega }\right)\omega Bsin\omega tdt=-2\pi nB{\int }_{2\pi n/\omega }^{(2n+0.5)\pi /\omega }sin\omega tdt\hfill \\ & & +\omega B{\int }_{2\pi n/\omega }^{(2n+0.5)\pi /\omega }tsin\omega tdt=2\pi nB\left[\frac{1}{\omega }cos(2n+0.5)\pi -\frac{1}{\omega }cos2\pi n\right]\hfill \\ & & +\omega B[\frac{1}{{\omega }^2}sin(2n+0.5)\pi -\frac{\pi }{{\omega }^2}(2n+0.5)cos(2n+0.5)\pi -\frac{1}{{\omega }^2}sin2\pi n\hfill \\ & & +\frac{2\pi n}{{\omega }^2}cos2\pi n]=2\pi nB\left(-\frac{1}{\omega }\right)+\omega B\left(\frac{1}{{\omega }^2}+\frac{2\pi n}{{\omega }^2}\right)=\frac{B}{\omega }>0.\hfill \end{array}$$

The function $p\left(t\right)=A+Bcos\omega t$ is periodic with period $2\tau =2\pi /\omega$. We have

$$\begin{array}{ccc}& & p^{\prime }(t-\tau )-p^{\prime }\left(t\right)=\omega Bsin\omega \left(t-\frac{\pi }{\omega }\right)+\omega Bsin\omega t=-\omega Bsin(\omega t-\pi )\hfill \\ & & +\omega Bsin\omega t=2\omega Bsin\omega t>0fort\in \left(\frac{2\pi n}{\omega },\frac{(4n+1)\pi }{2\omega }\right),n\in N.\hfill \end{array}$$

All conditions of Theorem 1 are satisfied. Equation (8) has no solution such that $0<K_1\le r^3\left(t\right)\le K_2,t\ge t_0$. □

4. Numerical Simulation

Figure 1 and Figure 2 show the trajectories of atoms that bounce off the electromagnetic field of the charged wire. A plot of an atom trajectory that is escaping to infinity is depicted in Figure 1. The motion of the atom near the charged wire is governed by the system (7)

$$\begin{array}{ccc}\hfill \frac{dr}{dt}& =& \frac{1}{M}{p}_r,\hfill \\ \hfill \frac{d{p}_r}{dt}& =& \left(\frac{L^2}{M}-2\alpha Q^2-2\alpha Q^{2}cos\omega t\right)\frac{1}{r^3\left(t\right)},\hfill \end{array}$$

in the form

$$\begin{array}{ccc}\hfill \frac{dr}{dt}& =& 0.2p_r,\hfill \\ \hfill \frac{d{p}_r}{dt}& =& (0.1-0.5cos3t)\frac{1}{r^3\left(t\right)},r\left(0\right)=2,p_r\left(0\right)=0,t\in (0,8),\hfill \end{array}$$

where parameters $A<0,B>0$.

In the Figure 2, the motion of the atom is governed by the system (7) in the form

$$\begin{array}{ccc}\hfill \frac{dr}{dt}& =& 0.2p_r,\hfill \\ \hfill \frac{d{p}_r}{dt}& =& (-0.12-1.2cos1.2t)\frac{1}{r^3\left(t\right)},r\left(0\right)=1.6,p_r\left(0\right)=0.2,t\in (0,45),\hfill \end{array}$$

where parameters $A>0,B>0$. Figure 2 shows that, in the vicinity of a charged wire, the atom bounces to infinity.

Figure 3 shows the solution of Equation (8), which shows the motion of the atom. Equation (8) has the form

$$\begin{array}{c}\hfill \frac{d^{2}r}{d{t}^2}+(0.024+0.24cos1.2t)\frac{1}{r^3\left(t\right)}=0,r\left(0\right)=1.6,r^{\prime }\left(0\right)=0.04,t\in (0,43).\end{array}$$

(9)

5. Conclusions

Observe that when $\tau =\pi /\omega >0,B>0$, then Equation (8) cannot have a solution $r\left(t\right)$ such that

$$0<K_1\le r^3\left(t\right)\le K_2,t\ge t_0,$$

where $K_1,K_2$ are constants. That is, the atom either hits the wire and is probably absorbed into the surface of the wire or escapes to infinity in the radial direction. Every process has its own history. We are interested in the influence of history on the motion of atoms in the vicinity of a charged wire.