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

Copyright © 2018 W. I. Skrypnik.This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a neutral Coulomb planar system of n equal negative charges in a constant magnetic field and a field of n 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.


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 −1,  > 2 equal negative charges and one and three positive charges in [4,5].
In this paper, we consider the Coulomb planar system of  > 1 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  > 1 point equal negative charges  0 and  fixed positive charges  0 is given by where || is the Euclidean norm of  and  () = ( 1 , . . .,   ) ∈ R 2 .The equation of motion of the charges in a magnetic field ℎ directed perpendicular to the plane is given by where 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.

Algebraic Form of the Equation of Motion
To solve (7), we introduce two Siegel complex variables,  and , such that where  , =  ,, ∈ R,  ,, = 0 for  < 3 or  < 6 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  *  = x 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) , .
As a result, The last three equalities and (7) yield where where  2  6 =  4,−1 .Note that the power expansion in the right-hand side without two last terms begins from  > 5.
For ℎ 0 ̸ = 0, we shall put  = ℎ 0 , implying that the last two terms give  2  6 .( 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 ζ, =  ,− when applying the tilde to (17)).
which can be solved easily for  ̸ = ±1.Note that  , ,  ,,− depend on  ,, , 1 ≤  ≤ ,  <  since the expansion for   begins from the second powers of   ,   , since it does not contain the term with  = 0 and there are coefficients  2  6 ,  6 ,  3 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  ,,±1 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,      . (28)

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 || 2 < , || 2 < .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 The analog of the first condition was applied by Siegel for the solution of the Hill problem in the celestial mechanics [6].Let where  − |2| ≥ 0;  = 0,  > 2 ( = 2,  > 5) correspond to the zero (nonzero,  = ℎ 0 ) magnetic field.Let also  = 1 correspond to the case where  = 0, ℎ ̸ = 0. 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,  −  ≫ 0.
For the term  2; in the second square bracket in the expression for   , we obtain, using (32) with  =   −   ,  = ,  =  4 (  + 1),  =  4 (  + 1) and ( 34) and (36), In order to exclude the contribution of two terms  6 =  6,0 ,  6,−2 =  2  10 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,  2; =   34) and (36), we obtain Let For the last two terms  0; in the expression for   , we obtain also (49) For the nonzero magnetic field, we obtain For the zero magnetic field, we derive We have also The same majorant bounds hold for D , F , C .

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  ,,1 = 0,  < 3. Equation ( 28 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 As a result, we obtain (31) with where   = 0(  = 3) if the magnetic field is nonzero (zero).

Advances in Mathematical Physics
We also have The following bounds also hold: The condition || ≤ (2 since Remark 1.If ℎ 0 < 0, then one has to substitute || = −ℎ 0 in all the bounds and use Let us consider the case of the zero magnetic field and  ̸ = 0. Then (60) is majorized by ( ≥ 0) 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 || <  −2 and   =  +  ≫ 0 and   ≪   .Note that if  2  < 1, then we can put  = 0 and obtain   ≫ 0. But in our case  2  > 1.

Conclusion
Theorem 2 determines periodic solution of the considered equation of motion such that each negative charge moves close to its own positive fixed charge and the distance between them exceeds  0 − 0 /4] 2 .This excludes collisions between the charges.
If one considers arbitrary location of the equal positive charges on a complex plane, then it is necessary to consider complex  ,, .In this case, the analog of (28) can be derived and all the used majorant bounds can be applied for the proof of the existence of periodic solution of the considered equation of motion.
We hope that the proposed result and technique will be useful for the quantum systems.Earlier classical Coulomb systems, which are relevant to quantum atomic systems, were considered in [13,14].Our systems can be useful for a description of molecules with  nuclei.

Appendix
Here we prove ( 32