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Advances in Mathematical Physics
Volume 2018, Article ID 2548074, 10 pages
https://doi.org/10.1155/2018/2548074
Research Article

Coulomb Planar Periodic Motion of Equal Charges in the Field of Equal Positive Charges Fixed at a Line and Constant Magnetic Field

Institute of Mathematics, The National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kyiv, Ukraine

Correspondence should be addressed to W. I. Skrypnik; ten.rku@kynpyrks_rymydolov

Received 6 November 2017; Accepted 28 March 2018; Published 3 May 2018

Academic Editor: Stephen C. Anco

Copyright © 2018 W. I. Skrypnik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a neutral Coulomb planar system of equal negative charges in a constant magnetic field and a field of equal fixed positive charges and construct a periodic solution of the equation of motion such that each negative charge is close to its own positive fixed charge.

1. Introduction

Construction of solutions of the ML (Maxwell-Lorentz) equations of motion of point charges of classical electrodynamics is a fundamental task of mathematics. The simplest approximations of these equations are the Coulomb and Darwin equations, which do not take into account radiation of the charges. Solutions of the former and latter for arbitrary number of charges were proven to exist on a finite time interval at which there are no collisions in [1, 2], respectively.

If two equal positive charges are fixed at one coordinate axis at equal distances from the other coordinate axis along which two or three negative equal charges move, then there exists an equilibrium configuration of the negative charges and periodic solutions of their Coulomb equation of motion [3]. The center Lyapunov, Moser, and Weinstein theorems are applied by us in [3] to prove this fact. The first two of them demand an exclusion of resonances, which restrict values of the charges and yield the solutions in terms of convergent series. Periodic exact solutions are also found in planar Coulomb systems of , equal negative charges and one and three positive charges in [4, 5].

In this paper, we consider the Coulomb planar system of point equal negative charges in a constant magnetic field and a field of equal fixed positive charges located at a line (th charge is placed at ) and construct periodic solution of their equations of motion such that each negative charge moves close to its own positive fixed charge. The solution is given as a convergent series. We apply a generalization of the remarkable technique of Siegel invented by him for solving the three-body problem of the celestial mechanics [6, 7]. This result is formulated in the theorem in the end of the paper.

Earlier in [8, 9] we considered the space and planar systems of two equal negative and two fixed positive charges and proved with the help of the center Lyapunov theorem the existence of their periodic dynamics close to an equilibrium. We expect that there is a bifurcation in this system as in the case of gravitation two-center problem [10]. The results for the gravitation -center problem, which can be used for dynamics in a system of a negative charge in the field of fixed positive charges, are obtained in [11, 12].

The potential energy of our Coulomb system of point equal negative charges and fixed positive charges is given by where is the Euclidean norm of and . The equation of motion of the charges in a magnetic field directed perpendicular to the plane is given by where and , . In the explicit form, it is written as Let us introduce the new (difference) variables . If one omits the primes, the Coulomb equation for them is given by If , , , then this equation of motion is represented in the following complex form: That is, This equation can be considered if . It is solved in this paper, which is organized as follows. In the second section, we introduce power expansions in two complex (Siegel) variables for the coordinates in (7) and write down and solve an equation for their coefficients. In the third section, we obtain their majorant bounds that are used for the proof of convergence of the power expansions. Our main result is formulated in Theorem 2 concerning periodic solutions of (3). In the Appendix, we prove the majorant inequalities (32)-(33) that are together with the majorant inequalities (34)–(36) basic for the majorant technique.

2. Algebraic Form of the Equation of Motion

To solve (7), we introduce two Siegel complex variables, and , such that where , for or for a special choice of and The choice of is motivated by the expression of . It helps to derive the Siegel-type inequality (31) (see also the last sentence of the fourth section).

Note that if . The last condition implies that the equations in (8) have the periodic solutions since is a conserved quantity. In other words, we seek the periodic solutions of the equation of motion (7) with the representation (9)-(10). In order to achieve that goal, we have to calculate the first and second derivatives of the coordinates in (9)-(10), substitute them together with the coordinates into (7), obtain the equation for , , and prove that it has a solution that provides convergence of the series in (9)-(10) for appropriate (sufficiently small) , .

Let us calculate the time derivatives of the complex coordinates in (9)-(10) satisfying (7). We set for simplicity . As a result, The last three equalities and (7) yield where where . Note that the power expansion in the right-hand side without two last terms begins from . For , we shall put , implying that the last two terms give . (15) is rewritten as follows: where (17) is valid, since By and , we will denote the coefficient in the power expansion of and the power expansion corresponding to the coefficient , respectively.

The following representation is true: We derive from (17) and (20) the needed equation for (we take into account when applying the tilde to (17)). which can be solved easily for . Note that depend on , , since the expansion for begins from the second powers of , , since it does not contain the term with and there are coefficients , , in the right-hand side of the equality in (17). The obtained complex two-dimensional linear equation is solved as From this equality that generates a solvable recursion relation for , it follows that The equation for follows from the equations where which results from (17). Then Let us multiply the second equation by 3 and subtract the result from the first one. That is, These two equalities give the bounds

3. Majorant Bounds

In this section, we derive majorant bounds for the terms in the right-hand side of (17), which permit proving that the series in (9)-(10) converges absolutely for , . We shall do it with the help of the advanced Cauchy majorant technique that is close to the one applied by Siegel for the solution of the generalized Hill problem in the celestial mechanics [7]. It is sufficient to prove the absolute convergence for , , which implies that , since if and . The analog of the first condition was applied by Siegel for the solution of the Hill problem in the celestial mechanics [6].

Let where ; , () correspond to the zero (nonzero, ) magnetic field. Let also correspond to the case where , . Then the following majorant bounds are true: where means that, in the power expansion for , the coefficients are positive and exceed absolute values of the coefficients in the power expansion for ; that is, .

Our aim is to derive with the help of (23) and (28) the basic Siegel-type inequality [7]: turns out to be zero for . We shall apply the following inequalities for and given by power expansions ( will denote the power expansions whose coefficients coincide with absolute values of the coefficients of ). Let , , ; then Equations (32)-(33) are proven in the Appendix. We shall use also the following inequalities:

The first inequality in (34) follows from the fact that the monomials of are determined by monomials of , with greater coefficients. The second one follows from the fact that the monomials of are determined by monomials of , with greater coefficients (). Equation (35) follows from the fact that is determined by the monomial of with the coefficient . Equation (36) is easily derived. The last inequality follows from the first inequality in (34) ( and are substituted instead of and , resp.).

Using (33) with , , , we obtain Here we used (36) and Let Then, for the term in the first square bracket in the expression for , we derive, putting and using (32) with , , , and (35)-(36), where , , . Here we used also . For the term in the second square bracket in the expression for , we obtain, using (32) with and (34) and (36),

In order to exclude the contribution of two terms , in the bound derived from (28), we have to obtain a more accurate majorant inequality for starting from the equality

We will need also to majorize the following function: That is, with appropriate . Using (33) with , , , and (34) and (36), we obtain Let Then For the last two terms in the expression for , we obtain also For the nonzero magnetic field, we obtain For the zero magnetic field, we derive We have also The same majorant bounds hold for , , .

4. Main Result

In this section, we will prove (31), which permits proving the convergence of the series in (9)-(10), and formulate our main result in Theorem 2. We will take into account the fact that , . Equation (28) gives does not contain the terms with , in the expression for . does not contribute to this inequality as well. Besides,

Taking into account (23), one obtains () where if the magnetic field is not zero (zero). As a result, Let us sum over both sides of the above inequality for and utilize the majorant inequalities obtained in the previous section for all its terms. This gives where ,, , , for the zero (nonzero, ) magnetic field.

As a result, we obtain (31) with where if the magnetic field is nonzero (zero). Here we took into account for and .

It is majorized by (see the inequality below (36)) Let , , and Then the last inequality for is majorized by Indeed, it implies that Or Applying (36), we map into . In this way, we transform the last relation into (60) with greater .

Equation (62) leads to Moreover, Hence, We also have The following bounds also hold: Let The last inequality is easily solved taking into account , , , . It follows from the stronger middle inequality (for it is obvious) that The condition holds if . Equation (71) is satisfied if From these bounds, one obtains For the difference variables of negative charges from (9), we derive the bound As a result, for the variables of negative charges , we deduce the bound since

Remark 1. If , then one has to substitute in all the bounds and use

Let us consider the case of the zero magnetic field and . Then (60) is majorized by () Let The solution of the quadratic equation is given by Or where Let This results in This condition means that is given by the convergent power expansion if and and .

Note that if , then we can put and obtain . But in our case .

If (85) and hold, then From these bounds, one obtains also That is, if the condition , which trivially follows from , holds, then This bound and (75), (85), and (87) give

Hence, we proved the following theorem.

Theorem 2. Let the magnetic field be either zero and or nonzero and . Then the difference coordinates , given by (9)-(10) are holomorphic functions in the hyperdisc , and satisfy (7) if , satisfy (8). They are also periodic functions in with the period if . The coordinates of the negative charges, expressed in terms of these difference coordinates with primes by (4) for , satisfy the Coulomb equation of motion (3). Moreover, if , , and for the zero magnetic field, then the following inequality is true: where if the magnetic field is nonzero (zero).

Note that (31) with , holds also for the case where