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

Authors: W. I. Skrypnik
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We consider a neutral Coulomb planar system ofn 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.


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## Body

## 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 n-1, n>2 equal negative charges and one and three positive charges in [4, 5].In this paper, we consider the Coulomb planar system ofn>1 point equal negative charges in a constant magnetic field and a field of n equal fixed positive charges located at a line (jth charge is placed at rj) 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 n-center problem, which can be used for dynamics in a system of a negative charge in the field of n fixed positive charges, are obtained in [11, 12].The potential energy of our Coulomb system ofn>1 point equal negative charges e0 and n fixed positive charges e0 is given by (1)Uxn=e022-1∑1≤j≠k≤n1xj-xk-∑1≤j≤n∑1≤k≤n1xj-rk,xj=xj1,xj2,rj=rj1,0∈R2, where x is the Euclidean norm of x and x(n)=(x1,…,xn)∈R2n. The equation of motion of the charges in a magnetic field h directed perpendicular to the plane is given by (2)md2xjdt2=-∂Uxn∂xj+e0dxjdt×h,j=1,…,n, where (v×h)1=h0v2 and (v×h)2=-h0v1, h0∈R. In the explicit form, it is written as (3)d2xjdt2=e02m∑1≤k≤n,k≠jxj-xkxj-xk3-∑1≤k≤nxj-rkxj-rk3+e0mdxjdt×h.Let us introduce the new (difference) variables xj′. (4)xjt=rj+gxj′t,g3=e02m.If one omits the primes, the Coulomb equation for them is given by (5)d2xjdt2=-xjxj3+g2∑1≤k≤n,k≠jrj-rk+gxj-xkrj-rk+gxj-xk3-∑1≤k≤n,k≠jrj-rk+gxjrj-rk+gxj3+e0mdxjdt×h. If xj=xj1+ixj2, x~j=x~j1+ix~j2, xj1-ixj2=xj∗=x~j, then this equation of motion is represented in the following complex form: (6)d2xjdt2=-xj-1/2x~j-3/2+g2∑1≤k≤n,k≠jrj-rk+gxj-xk-1/2rj-rk+gx~j-x~k-3/2-rj-rk+gxj-1/2rj-rk+gx~j-3/2-ih0dxjdt,h0=h0e0m,d2x~jdt2=-x~j-1/2xj-3/2+g2∑1≤k≤n,k≠jrj-rk+gx~j-x~k-1/2rj-rk+gxj-xk-3/2-rj-rk+gx~j-1/2rj-rk+gxj-3/2+ih0dx~jdt. That is, (7)d2xjdt2=-xj-1/2x~j-3/2+fj,d2x~jdt2=-x~j-1/2xj-3/2+f~j.This equation can be considered if xj1-ixj2=xj∗≠x~j. 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, ξ and η, such that (8)dξdt=αξ,dηdt=-αη,α=i4ξη-3,(9)xj=ξ4u1+∑k,laj,klζk,l=uξ41+Aj,k-2l≥0,k>0,(10)x~j=η4u1+∑k,laj,klζk,-l=uη41+Bj,ζk,l=ζkl=ξk+2lηk-2l,where aj,kl=aj,k,l∈R, aj,k,l=0 for k<3 or k<6 for a special choice of b and (11)u=1-bζ31/3,ζk=ζk0,b∈R. The choice of u is motivated by the expression of ζk,l. It helps to derive the Siegel-type inequality (31) (see also the last sentence of the fourth section).Note thatxj∗=x~j if ξ∗=η. The last condition implies that the equations in (8) have the periodic solutions (12)ξ=eαtξ0,η=e-αtη0,α=i4ξ0η0-3 since ξη 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 aj,k,l, aj,k,-l, and prove that it has a solution that provides convergence of the series in (9)-(10) for appropriate (sufficiently small) ξ, η.Let us calculate the time derivatives of the complex coordinates in (9)-(10) satisfying (7). We set for simplicity h0≥0. (13)dxjdt=4αξ4u1+∑k,ll+1aj,klζkl,d2xjdt2=4α2ξ4u1+∑k,ll+12aj,klζkl,dx~jdt=-4αη4u1+∑k,ll+1aj,klζk,-l,d2x~jdt2=4α2η4u1+∑k,ll+12aj,klζk,-l. As a result, (14)-iξ2η6u2dxjdt=ζ3u31+∑k,ll+1aj,k,lζkl,ξ2η6u2d2xjdt2=-u31+∑k,ll+12aj,klζkl,ξ2η6u2xj-1/2x~j-3/2=1+Aj-1/21+Bj-3/2=1-12Aj+⋯1-32Bj+⋯=1-12Aj-32Bj+⋯. The last three equalities and (7) yield (15)u3∑k,ll+12aj,klζkl+h0ζ3∑k,ll+1aj,klζkl+12Aj+32Bj=Dj+Fj,where (16)Dj=1+Aj-1/21+Bj-3/2-1+12Aj+32Bj,Fj=-g2ξ2η6u2∑1≤k≤n,k≠jrj-rk+gξ4uAj-Ak-1/2rj-rk+gη4uBj-Bk-3/2-rj-rk-2-rj-rk+gξ4u+gξ4uAj-1/2rj-rk+gη4u+gη4uBj-3/2-rj-rk-2-h0ζ3u3+bζ3,where ξ2η6=ζ4,-1. Note that the power expansion in the right-hand side without two last terms begins from k>5. For h0≠0, we shall put b=h0, implying that the last two terms give b2ζ6. (15) is rewritten as follows: (17)∑k,ll+12aj,klζkl+12Aj+32Bj=Dj+Fj+Cj,where (18)Cj=ζ3∑k,lbl+12-h0l+1aj,klζkl+bh0ζ6∑k,ll+1aj,klζkl. (17) is valid, since (19)ξ2η6=ζ4,-1,ξ4k1η4k2=ζ2k1+k2,k1-k2,ζk,lζk′,l′=ζk+k′,l+l′,k1,k2≥0. By Fk,l and F+, we will denote the coefficient in the power expansion of F and the power expansion corresponding to the coefficient Fk,l, respectively.The following representation is true:(20)Dj+Fj+Cj=∑k,lρj,klζkl,D~j+F~j+C~j=∑k,lρj,k,-lζkl,ρj,kl=ρj,k,l∈R.We derive from (17) and (20) the needed equation for aj,kl,aj,k,-l (we take into account ζ~k,l=ζk,-l when applying the tilde to (17)). (21)l+12+12aj,kl+32aj,k,-l=ρj,kl,aj,k0=13aj,k0,l+12+12aj,k,-l+32aj,kl=ρj,k,-l, which can be solved easily for l≠±1. Note that ρj,kl,ρj,k,-l depend on as,r,l, 1≤s≤n, r<k since the expansion for Dj begins from the second powers of Aj, Bj, since it does not contain the term with k=0 and there are coefficients ξ2η6, ζ6, ζ3 in the right-hand side of the equality in (17). The obtained complex two-dimensional linear equation is solved as (22)aj,kl=l-12+1/2ρj,kl-3/2ρj,k,-ll2l2-1,l≠0,±1,3aj,k0=ρj,k,0. From this equality that generates a solvable recursion relation for aj,kl, it follows that (23)l+1saj,kl≤2-1c0ρj,kl+ρj,k,-l,l≠±1,s=0,1,2. The equation for aj,k,±1 follows from the equations (24)l+12aj,k+3,l+12aj,k+3,l+32aj,k+3,-l-bl+12-h0l+1aj,k,l-bh0l+1aj,k-3,l=ρj,k+3,l′,-l+12aj,k+3,-l+32aj,k+3,l+12aj,k+3,-l-b-l+12-h0-l+1aj,k,-l-bh0-l+1aj,k-3,-l=ρj,k+3,-l′, where (25)ρj,k,l′=Dj+Fjk,l, which results from (17). Then (26)92aj,k+3,1+32aj,k+3,-1-4b-2h0aj,k,1-2bh0aj,k-3,1=ρj,k+3,1′,32aj,k+3,1+12aj,k+3,-1=ρj,k+3,-1′. Let us multiply the second equation by 3 and subtract the result from the first one. That is, (27)-4b-2h0aj,k,1=2bh0aj,k-3,1+ρj,k+3,1′-3ρj,k+3,-1′,32aj,k,1+12aj,k,-1=ρj,k,-1′. These two equalities give the bounds (28)aj,k,1≤4b-2h0-12bh0aj,k-3,1+ρj,k+3,1′+3ρj,k+3,-1′,aj,k,-1≤3aj,k,1+2ρj,k,-1′.


## 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 ξ2<q, η2<q. 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 ξ2=η2=ζ, ζ2<q, which implies that ζk,l=ζk, since ζk,l=ξk+2lηk-2l<qk if ξ2<q and η2<q. The analog of the first condition was applied by Siegel for the solution of the Hill problem in the celestial mechanics [6].Let(29)Zj=∑k,ll+1saj,klζk,s=0,1,2,Z=∑j=1nZj,where k-2l≥0; s=0, k>2 (s=2,k>5) correspond to the zero (nonzero, b=h0) magnetic field. Let also s=1 correspond to the case where b=0, h≠0. Then the following majorant bounds are true: (30)Aj≪Zj,Bj≪Zj,ξ2=η2=ζ, where f≪g means that, in the power expansion for g, the coefficients are positive and exceed absolute values of the coefficients in the power expansion for f; that is, g-f≫0.Our aim is to derive with the help of (23) and (28) the basic Siegel-type inequality [7]: (31)Z≪c′2p′z3+z4+zZ+z-3Z21-c′Z+z,z=vζ,v=u-31+bζ3,c′≥1,p′≥0.p′ turns out to be zero for b=h0≠0. We shall apply the following inequalities for x,y∈C and X,Y given by power expansions (X+,Y+ will denote the power expansions whose coefficients coincide with absolute values of the coefficients of X,Y). Let x=y, X≪Q, Y≪Q; then (32)x+X-1/2y+Y-3/2-x-1/2y-3/2≪x-32Q1-2x-1Q,(33)x+X-1/2y+Y-3/2+x-1/2y-3/2-1+12xX+32yY≪x-44Q21-2x-1Q.Equations (32)-(33) are proven in the Appendix. We shall use also the following inequalities: (34)11-g0z2Z+1≪11-Z+g0z2≪11-Z+g0z,(35)11-g0z2Z≪11-Z+g0z2≪11-Z+g0z,(36)zk≪11-z,k∈Z+,11-X+11-Y+≪11-X+-Y+,11-c′z+Z≪11-c′z+z-1Z≪11-c′z+z-3Z,c′≥1.The first inequality in (34) follows from the fact that the monomials of (g0z2(Z+1))n are determined by n monomials of (Z+g0z2)2n, (Z+g0z2)2n-1,…,(Z+g0z2)n with greater coefficients. The second one follows from the fact that the monomials of (Z+g0z2)n are determined by n monomials of (Z+g0z)2n, (Z+g0z)2n-1,…,(Z+g0z)n with greater coefficients (Cnk≤C2n-kk). Equation (35) follows from the fact that (g0z2Z)n is determined by the monomial of (Z+g0z2)2n with the coefficient C2nn. Equation (36) is easily derived. The last inequality follows from the first inequality in (34) (z-1Z and z are substituted instead of Z and z2, resp.).Using (33) with x=y=1, X=Aj, Y=Bj, we obtain (37)Dj+≪4Zj21-2Zj,∑1≤j≤nDj+≪4Z21-2Z+z≪b1z-3Z21-3Z+z,b1=4,ζ-3∑1≤j≤nDj+=z-3v3∑1≤j≤nDj+≪b2z-3Z21-3b+5Z+z,b2=41+b3,b>0. Here we used (36) and (38)v3=1+bζ331-bζ33≪1+bζ331-3bζ3≪1+bζ331-3b+1ζ≪1+b31-3b+2ζ≪1+b31-3b+2z. Let (39)r=maxj∑1≤k≤n,k≠jrj-rk-2,r0=minj,krj-rk. Then, for the term F1;j in the first square bracket in the expression for Fj, we derive, putting g0=2r0-1g and using (32) with x=rj-rk, y=x, X=gξ4u(Aj-Ak), Y=gη4u(Bj-Bk) and (35)-(36), (40)F1;j+≪rζ4g2v2g0vζ2Z1-g0vζ2Z≪n-1b3z6Z1-g0z2Z,b3=2-2n-1ng03,ζ-3F1;j+<n-1b3z3Z1-g0z2Z,∑1≤j≤nF1;j+≪b3z6Z1-Z+g0z≪b3zZ1-c∗+1Z+z,ζ-3∑1≤j≤nF1;j+≪b3z3Z1-Z+g0z≪b3zZ1-c∗+1Z+z, where c∗=1+g0, g0≤1;c∗=g0, g0>1. Here we used also b>0. (41)u=1-bζ31-bζ32/3≪1+bζ31-bζ3=v;v≪vk,k>1;1≪1-bζ3-1/3,X+,Y+≪gζ2vZj+Zk. For the term F2;j in the second square bracket in the expression for Fj, we obtain, using (32) with x=rj-rk,y=x,X=gξ4u(Aj+1),Y=gη4u(Bj+1) and (34) and (36), (42)F2;j+≪rg0g2z6Zj+11-g0z2Z+1,∑1≤j≤nF2;j+≪z6n-1b3Z+b31-Z+g0z≪b4zZ+z41-1+c∗Z+z,b4=2-2n-1b3+b3.In order to exclude the contribution of two termsζ6=ζ6,0, ζ6,-2=ξ2η10 in the bound derived from (28), we have to obtain a more accurate majorant inequality for Fj+ starting from the equality (43)F2;j=F3;j-g2ξ2η6u2∑1≤k≤n,k≠jrj-rk-2ξ4ug2rj-rkAj+1+3η4ug2rj-rkBj+1.We will need also to majorize the following function:(44)F4;j=g2ξ2η6u2∑1≤k≤n,k≠jrj-rk-2ξ4g2rj-rkuAj+u-1+3η4g2rj-rkuBj+u-1. That is, (45)F2;j=F3;j+F4;j+α1ζ6,0+α2ζ6,-2 with appropriate αj. Using (33) with x=rj-rk, y=x, X=ξ4u(Aj+1), Y=η4u(Bj+1) and (34) and (36), we obtain (46)F3;j+≪rv2ζ4g2g0vζ2Zj+121-g0vζ2Zj+1,ζ-3F3;j+≪rg2zg0z2Zj+121-g0z2Zj+1,z=vζ,ζ-3∑1≤j≤nF3;j+≪2-1b5z5Z+121-Z+g0z≪b5z-3Z2+zZ+z41-c∗+1Z+z,b5=2-1nn-1g04. Let (47)r′=maxj∑1≤k≤n,k≠jrj-rk-3. Then (48)F4;j+≪2r′v2ζ6g3vZj+v-1=2r′g3v2ζ6vZj+2bζ91-bζ3,ζ-3∑1≤j≤nF4;j+≪b6zZ+z41-1+b+c∗Z+z,b6=2-2n-1g031+2b. For the last two terms F0;j in the expression for Fj, we obtain also (49)F0;j+≪h0ζ31+bζ3+bζ3≪h0z31+bz3+bz3.For the nonzero magnetic field, we obtain (50)F0;j+≪b2ζ6≪b2z6,b=h0,∑1≤j≤nF0;j+≪b0z61-Z+g0z≪b0z41-c∗+1Z+z,b0=nh02. For the zero magnetic field, we derive (51)∑1≤j≤nF0;j+≪b′z31-Z+g0z≪b′z31-c∗+1Z+z,b′=nb. We have also (52)Cj+≪ζ3b+h0+bh0ζ6Zj,∑1≤j≤nCj+≪b7zZ1-c∗+1Z+z,b7=b+h0+bh0. The same majorant bounds hold for D~j, F~j, C~j.


## 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 aj,k,1=0, k<3. Equation (28) gives (53)∑kaj,k,1+aj,k,-1ζk≪∑k2b-h0-14bh0aj,k-3,1+2ρj,k+3,1′+3ρj,k+3,-1′+2ρj,k,-1′ζk≪42b-h0-1bh0ζ3Zj+2∑k,lρj,k,l′ζk+Gj,Gj=6ζ32b-h0-1∑k,lDk,l+F1;j,k,l+F3;j,k,l+F4;j,k,lζk.Gj does not contain the terms with ζ6=ζ6,0, ζ6,-2=ξ2η10 in the expression for F2;j. F0;j does not contribute to this inequality as well. Besides, (54)∑kaj,k,0ζk≪3-1∑kρj,k,0ζk≪3-1∑k,lρj,k,lζk.Taking into account (23), one obtains (ξ2=η2=ζ)(55)Zj≪c0+3-1∑k,lρj,k,lζk+ν2b-h0-14bh0zZj1-Z+z+∑k,lρj,k,l′ζk+Gj, where ν=4(ν=1) if the magnetic field is not zero (zero). (56)∑k,lρj,k,lζk≪Dj++Fj++Cj+,∑k,lρj,k,l′ζk≪Dj++Fj+,Fj+≪F1;j++F2;j++F0;j+,Gj=6ζ32b-h0-1Dj++F1;j++F3;j++F4;j+. As a result, (57)Zj≪2ν+c0+3-1+4νζ32b-h0-1Dj++F1;j++2ν+c0+3-1F2;j++F0;j++c0+3-1Cj++2b-h0-14νbh0zZj1-Z+z+6νζ32b-h0-1F3;j++F4;j+.Let us sum over j both sides of the above inequality for ZJ and utilize the majorant inequalities obtained in the previous section for all its terms. This gives (58)Z≪c22p′z3+z4+zZ+z-3Z21-c1z+Z, where c1=5+3b+g0,c2=b9+(2ν+c0+3-1)(b8+b1+b3+b4)+6ν2b-h0-1(b2+b3+b5+b6)+(c0+3-1)b7, b9=2b-h0-14νbh0, b8=0, 2p′=b′c2-1(b8=b0,p′=0) for the zero (nonzero, b=h0) magnetic field.As a result, we obtain (31) with (59)c′=g0+c2+b′′,2ν+3-1b1>8ν+1,c′>8ν+1+b′′+g0, where b′′=0(b′′=3b) if the magnetic field is nonzero (zero). Here we took into account b9>3b for b=h0 and 2νb1=8.It is majorized by (see the inequality below (36)) (60)Z≪c′2p∗z3+z4+zZ+z-3Z21-c′z-3Z+z,2p∗=b′c′-1,b′=nb.Let h0≠0, b=h0, and (61)Z′=z+z-3Z. Then the last inequality for p=0 is majorized by (62)Z′≪c2z+Z′24-cZ′,c=2+4c′,c>48ν+1+2+4g0.Indeed, it implies that(63)z-3Z≪czz+z-3Z-4z4-cz-3Z+z+c2z+z2+2z-2Z+z-6Z24-cz-3Z+z=2c-4z+2cz2+3cz-2Z+cz-6Z24-cz-3Z+z-1. Or (64)Z≪2c-4z4+2cz5+3czZ+cz-3Z24-cz-3Z+z-1. Applying (36), we map 2cz5 into 2cz4. In this way, we transform the last relation into (60) with greater c′.Equation (62) leads to (65)Z′≪W,W=c2z+W24-cW,cW2-2W+cz=0,W=c-11-1-c2z. Moreover, (66)1-cW2=1-c2z,2cW≪1-cW-2-1=c2z1-c2z. Hence, (67)W≪2-1cz1-c2z,z<c-2. We also have (68)Wz≤Wz<2-1cz1-c2z≤c-1,z≤2c2-1. The following bounds also hold: (69)Z′z≤Z′z≤Wz≤c-1,z≤2c2-1. Let (70)z=vζ=1+bζ31-bζ3ζ≤c-2. The last inequality is easily solved taking into account bζ<1, ζk<ζ<1, c>1, k>1. (71)ζ<c-21+b+bc-2=q.It follows from the stronger middle inequality (for ζ>0 it is obvious) that (72)1+bζ3ζ≤ζ+bζ4≤ζ+bζ<c-21-bζ<c-21-bζ3≤c-21-bζ3. The condition z≤(2c2)-1 holds if 2ζ<q. Equation (71) is satisfied if (73)ξ<q,η<q. From these bounds, one obtains (74)Zjz≤Zjz<Zz≤z4+z3Z′z≤2c-8. For the difference variables of negative charges xj′ from (9), we derive the bound (75)xj′≤ξ41+bζ31/31+Zz.As a result, for the variables of negative charges xj, we deduce the bound (76)rj-xj<gq21+bq31/31+2c-8<2gc-41+c-2<4gc-4<2r0g0134+4g0-4<134-22r016, since (77)1+bq31/3<1+c-61/3<1+c-2,c>134+4g0.Remark 1.
Ifh0<0, then one has to substitute b=-h0 in all the bounds and use (78)u=1-bζ31-bζ32/3≪1+bζ31-bζ3=v.Let us consider the case of the zero magnetic field andb≠0. Then (60) is majorized by (a≥0) (79)Z′≪c2p+2z+Z′24-cZ′≪c2p+2z+Z′24-cZ′+4a≪c2p-4aZ′+2z+Z′2+16c-1a4-cZ′,p=2nbc. Let (80)W′=c2p-4aW′+2z+W′2+16c-1a4-cW′,cW′2-21+caW′+cz+cp+8a=0. The solution of the quadratic equation is given by (81)W′=c-11+ca-1+ca2-c2z-c2p-8ac. Or (82)W′=c-11+ca-d+W0=a′+W0,W0z=c-1d-d2-c2z=dWd-2z, where (83)d2=1+ca2-c2p-8ac>0. Let (84)1+ca2=c2p+8ac+1,a2=6c-1a+p,a-3c-12=p+9c-2. This results in (85)a=3c-1+p+9c-2,d=1,a′=a.This condition means that W′ is given by the convergent power expansion if z<c-2 and W′=a+W≫0 and Z′≪W′.Note that ifc2p<1, then we can put a=0 and obtain W′≫0. But in our case c2p>1.If (85) and z≤(2c2)-1 hold, then (86)Z′z≤Z′z≤W′z≤c-1+a. From these bounds, one obtains also (87)Zjz≤Zjz<Zz≤z4+z3Z′z≤c-6c-2+c-1+a,a<6c-1+p. That is, if the condition p<c5, which trivially follows from b<(2n)-1, holds, then (88)c-6c-2+c-1+a<c-68c-1+p<1. This bound and (75), (85), and (87) give (89)rj-xj<gq21+bq31/31+c-6c-2+c-1+a<2gc-41+c-2<4gc-4<2r0g038+4g0-4<38-22r016.Hence, we proved the following theorem.Theorem 2.
Let the magnetic field be either zero andb≠0 or nonzero and b=h0. Then the difference coordinates xj, x~j given by (9)-(10) are holomorphic functions in the hyperdisc ξ<q, η<q and satisfy (7) if ξ(t), η(t) satisfy (8). They are also periodic functions in t with the period 8π(ξ(0)η(0))3 if η=ξ∗. The coordinates xj of the negative charges, expressed in terms of these difference coordinates with primes by (4) for η=ξ∗, satisfy the Coulomb equation of motion (3). Moreover, if 2ξ≤q, 2η≤q, and p<c5 for the zero magnetic field, then the following inequality is true: (90)rj-xj<r08ν′2, where ν′=134(38) if the magnetic field is nonzero (zero).Note that (31) with v=1, ζ=z holds also for the case where b=0, h0≠0 with 2p′=b′c2-1, b′=nh0, which implies that p=2nh0c-1(this follows from (49) with b=0). That is, Theorem 2 is true for the case and the condition p<c5 which restricts values of h0,g0.


## 5. Conclusion

Theorem2 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 r0-r0/4ν′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 complexaj,k,l. 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 n nuclei.



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