Equilibrium, Regular Polygons, and Coulomb-Type Dynamics in Different Dimensions

The equation of motion in ℝ d of n generalized point charges interacting via the s -dimensional Coulomb potential, which contains for d = 2 a constant magnetic ﬁ eld, is considered. Planar exact solutions of the equation are found if either negative n − 1 > 2 charges and their masses are equal or n = 3 and the charges are di ﬀ erent. They describe a motion of negative charges along identical orbits around the positive immobile charge at the origin in such a way that their coordinates coincide with vertices of regular polygons centered at the origin. Bounded solutions converging to an equilibrium in the in ﬁ nite time for the considered equation without a magnetic ﬁ eld are also obtained. A condition permitting the existence of such solutions is proposed.


Introduction
In this paper, we find solutions of the equation of motion in ℝ d of n generalized point charges (or simply charges) e j ∈ ℝ, j = 1, ⋯n with masses m j > 0 interacting via the pair s -dimensional Coulomb potential, which is proportional to the fundamental solutions of the s-dimensional Laplacian [1]. In the case d = 2, the equation also contains a constant magnetic field h c directed along the perpendicular to the plane containing the charges. The charge coordinates x j ∈ ℝ d obey the following equation of motion: where U x n ð Þ = 〠 1≤j<k≤n e j e k φ x j − x k À Á , φ x ð Þ = 1 − δ 2,s ð Þx j j 2−s + δ 2,s ln x j j, ð2Þ _ x = dx j /dt and the symbol × determines the skewsymmetric product. For d = 2, one has ð _ x 2 is the Kronecker symbol. The systems (1) and (2) without a magnetic field and their dynamics in the general case, case s = 3, and case d = 3, s = 3 we call ðd, sÞ-Coulomb, d-dimensional Coulomb, and Coulomb, respectively.
The most important among the systems are those that describe the electrodynamics, that is, d = 1, 2, 3, s = 2, 3, and the sequence of e j , j = 1, ⋯, n contains numbers with different signs for them. We shall consider the systems with n − 1 equal negative charges −e 0 and a positive charge e n .
The profound result concerning the collision set for the dynamics determined by (3) without the magnetic field is obtained in [2] (see Remark 13).
In this paper, we find the regular polygon equilibrium, exact regular polygon solutions for d = 2, and bounded solutions converging to the equilibrium in the infinite time limit. The regular polygon equilibrium is created by the system of n − 1 equal negative charges e 0 located at all n − 1 vertices of a regular polygon centered at the origin which is occupied by the central positive charge e n = e 0 n . It is given in Theorem 7 and (32) showing that e 0 n = 2 −1 e 0 ðn − 2Þ for s = 2 and e 0 3 = 2 1−s e 0 . For n = 3, the regular polygon is represented by two opposite points on a circle.
This equilibrium allows one to find easily the exact planar regular polygon solutions. They describe an identical motion of every negative charge around the origin where the positive charge is immobile if gðe n − e 0 n Þ > 0. This condition induces a Keplerian-type motion of the negative charges and the Keplerian motion of them for the Coulomb systems (s = 3). Besides, the negative charges are tied to vertices of regular polygons centered at the origin all the time. In the simplest case, the negative charges rotate around the positive immobile charge with the same frequency. This is described by the following solution x k ðtÞ ∈ ℂ of the equation of motion for negative charges (we consider ℝ 2 as ℂ) where z k , uðtÞ, rðtÞ satisfy the first, second, and third structure equations, respectively, the first of which does not depend on time. Here, z k are coordinates of the regular polygon vertices and the equation for u means conservation of the kinetic moment of a two-dimensional mechanical system. The third structure equation is its equation of motion for the radial variable. Earlier, we obtained the same result for the Coulomb systems with the zero magnetic field [3].
For the system of the three charges, we prove the following theorem.
x is the velocity, and We were able to prove this theorem since we found all eigenvalues of the matrix ðM −1 U 0 Þ j,α;j,α , where M is the diagonal matrix in ℝ dn with elements M j,α = m j , 1 ≤ j ≤ n, and 1 ≤ α ≤ d and U 0 j,α;l,β is the matrix of the second partial derivatives U j,α;j,β of the potential energy at the regular polygon equilibrium We prove also the following theorem. (2) possess an equilibrium x 0 and Tr Then, there exists an integer p, 1 ≤ p < nd, a positive number λ, and a bounded at positive time solution xðtÞ of the equation of motion (3) with h c = 0 depending on p real parameters, which is real analytic function in them in a neighborhood of the origin, such that kx − x 0 k λ < ∞,k _ xk λ < ∞.
If s = d, then TrM −1 U 0 = 0 for an arbitrary equilibrium for our systems without a magnetic field since In our calculations of partial derivatives of the potential energy, we use the vector-valued equalities 2 Advances in Mathematical Physics Theorem 2 implies the following result. Then, the conclusion of Theorem 2 is true.
In order to estimate TrM −1 U 0 in the case s ≠ d, one has to consider a particular equilibrium. We shall choose in the next theorem the regular polygon equilibrium for n charges (see Corollary 8).

Theorem 4.
For the regular polygon equilibrium for n − 1 equal negative charges −e 0 and central positive charge e 0 n , the following equality is true: where m ′ = ∑ n−1 l=1 m −1 l , An easy calculation at the end of the fourth section shows that TrM −1 U 0 < 0 for s > d if n = 3, 4, 5 (see Remark 14).
Note that the condition of the neutrality e n = e 0 ðn − 1Þ is impossible for the regular polygon equilibrium for s = 2.
The conclusion of Theorem 3 is true for the 3dimensional Coulomb systems of n + 2 and n + 1 charges n − 1 of which are equal considered in [4,5], respectively. In their equilibria, the equal negative charges are located at all the vertices of a regular polygon centered at the origin while the rest nonplanar charges are located at the perpendicular crossing the origin. In [4], a positive charge occupies the origin and the nonplanar charges are equal and may have different signs. In [5], the nonplanar charges are equal and positive. The solutions of the equation of motion are found in these papers provided the masses of the equal negative charges are equal.
Theorem 2 relies on the following basic theorem proven in [6].

Theorem 5.
Let U 0 be the symmetric matrix of second-order partial derivatives of U at an equilibrium x 0 Let also U be a real analytic function in a neighborhood of x 0 and the matrix M −1 U 0 have p negative eigenvalues σ j , j = 1, ⋯, p. Then, (1) with h c = 0 possesses a bounded at positive time solution x j ðtÞ, 1 ≤ j ≤ n depending on p real parameters which is a real analytic function in them in a neighborhood of the origin and where _ x is the velocity and For d > 1, the pair potentials and potential energies in (2) are real analytic functions in a neighborhood of a j ∈ ℝ d , 1 ≤ j ≤ n, a j ≠ a k , i.e., a set where coordinates do not coincide (a j may correspond to an equilibrium). This follows from the fact that the Taylor power expansion for x −s converges absolutely and uniformly in a neighborhood of a point from ℝ + . In the one-dimensional case, the chosen order of charges is preserved by the dynamics close to the equilibrium and one can change jx j − x k j 2−s , ln | x | by the real analytic functions So one can apply Theorem 5 to (1) and (2). Theorem 1 follows from Theorem 12 and Theorem 5. Our paper is organized as follows. In the second section, we introduce the three structure equations and explain in Proposition 6 how to construct with its help the exact solutions of (1) when there are equilibrium configurations in the form of regular polygons. Theorem 7 establishes their existence. In the third section, we consider the three charge cases and show that the exact solutions exist for which the charges are located at a common line provided the positive charge is greater than the absolute values of the rest two charges. In the fourth section, we prove Theorem 4. Eigenvalues of M −1 U 0 for the system of charges from Theorem 1 are calculated in the fifth section. This result is formulated in Theorem 12.

Equilibrium and Exact Planar Dynamics
Then, the analogue of the exact Lagrange solutions of the three-body problem [7] is determined by the equalities x k = z k qðtÞ which give From the equation of motion (3) with the contribution of magnetic field −ie j h _ x j , it follows that 3

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Sometimes, the s-dimensional Coulomb potential which leads to this equation is mentioned as the Riesz potential [8] if s > 2 (see also [9]). Let and the following first structure equation hold The first structure equation reduces the equation of motion to the following single equation: Let q = e iuðtÞ rðtÞ, where u, r are real valued functions. Then, The imaginary part of (17) multiplied by e −iu has to be zero As a result, the equation of motion yields the following second and third structure equations: where η ∈ ℝ is a constant. For b = 0, they look like The third structure equation is the radial Kepler's equation for s = 3. Note that if b = 0, r = 1, then the last equalities imply u = ±tw. In the general case, if r = 1, then from (17), it follows that Equation (17) for q is the equation of motion of a twodimensional mechanical system in a constant magnetic field characterized by the kinetic moment I. Let us prove that it is defined also by and that it is an integral of motion. The skew-symmetric product is determined on C 2 as follows: That is As a result, Now, we have to prove that solutions of the first structure equation exist. A crucial role in our proofs is played by the equality for the coordinates of the regular polygon vertices which in its turn is a consequence of the equality The proposition is proved.
From (29), it follows that e n > e 0 n and e n < e 0 n for s > 2 and s = 2, respectively.
Proof. For j = n, the equilibrium condition is satisfied For j < n, it yields If one shows that 〠 n−1 k=1,k≠j a s z j − z k s = a n , where a n = 〠 n−2 then the following equality is proved: e 0 n = e 0 a n − a n ′ : To prove these equalities, one has to prove for coordinates z j of the regular polygon such that |z j | = 1 To prove (39) ((38) is proved the same way), we have to take into account that z k = z k , z = e ið2π/ðn−1ÞÞ , z −l = z n−1−l Hence, the multiplier of z j does not depend on j and it is real since 5 Advances in Mathematical Physics Hence, (29) yields since Re z k = cos ð2πk/ðn − 1ÞÞ, But 1 − cos 2x = 2 sin 2 x. This concludes the proof.
Let U correspond to the ðd, sÞ-Coulomb system and From it follows and the equilibrium is found from the equality Hence, we proved the corollary with the help of Theorem 7.

Three Nonequal Charges on a Plane
In this section, we shall consider the first structure equation for three charges e j < 0, j = 1, 2, e 3 > 0. We will assume that their inertia center is immobile: m 1 z 1 + m 2 z 2 + m 3 z 3 = 0. We treat z j as points of ℝ 2 . Note that we assume that e n > e 0 n for s > 2 and e n < e 0 n for s = 2.
Proposition 9. Solutions of the first structure equation for n = 3 do not coincide with the coordinates of vertices of a triangle.
Proof. It is possible to put the first charge at a coordinate axis. Let z 2 1 = 0. Then, the first structure equation for z 2 1 gives Since the center of inertia is immobile, this equality results in That is, z 2 2 = 0 and z 2 3 = 0. This concludes the proof.
Proof. Let the charges are placed at a first coordinate axis: z 2 j = 0, q 1 = z 1 1 < q 3 = z 1 3 < q 2 = z 1 2 and q 2 − q 1 = a, q 3 − q 1 = a ρ, q 2 − q 3 = aσ, a > 0. Then, the first structure equation leads to Let ρ = σ = 1/2, then the second equation gives q 3 = 0. The first and the third ones show that −q 1 = q 2 = a/2 and a, w satisfy the condition indicated in the formulation of this proposition. This concludes the proof.
Note that if one multiplies a by 2 in Proposition 9, then w in it and w in Proposition 6 coincide since e 0 3 = 2 1−s e 0 . If the conditions of the last proposition do not hold, that is, e 1 ≠ e 2 , then one can apply the following proposition.

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Proof. Let us put m * = m 1 + m 2 + m 3 . From the equality m 1 z 1 + m 2 z 2 + m 3 z 3 = 0, it follows that Taking this into account, we see that the first and the third equalities look like (q 1 , q 3 are excluded) Since As a result in order to solve the first and third equations from the previous proposition, one has to show that f = 0 has solutions in the interval 0 < ρ < 1 such that e 3 ρ 1−s > |e 2 | , e 3 ð1 − ρÞ 1−s > |e 1 | if e 1 , e 2 < 0, e 3 > 0 and If e 3 ≥ −e j > 0, j = 1, 2, then the equation f = 0 has such the solutions since f is a continuous function tending to − ∞, ∞ at 0, 1, respectively. q 3 is found from the second structure equation. This concludes the proof.

Equilibrium Character
In this section, we calculate the trace where U 0 j,α;l,β is the matrix of the second derivatives of the potential energy at the regular polygon equilibrium created by the system of n − 1 equal negative charges located at all n − 1 vertices of a regular polygon centered at the origin where the central positive charge e 0 n is located. We have Further As a result, For the equilibrium, we have Then, one derives Besides, we have Now, let us apply the equalities For n = 3, s = 3, d = 2, we have The last two equalities are derived in [6] with the help of the found eigenvalues of M −1 U 0 (see also Theorem 12).

Eigenvalues of M −1 U 0 for Three Charges
The simplest example of a Coulomb system with equilibrium is the d-dimensional system of the three-point charges e 1 = −e 0 , e 2 = −e 0 , e 3 = 2 1−s e 0 > 0 with masses m 1 , m 2 , m 3 and the potential energy in (2) for n = 3. Its equilibrium x 0 is determined by x 01 1 = −a, x 01 2 = a, x 01 3 = 0, x 0α j = 0, j = 1, 2, 3, 1 < α ≤ d. The main result of this section is the proof of the following theorem.
Theorem 12. In the one-dimensional system, M −1 U 0 has the doubly degenerate zero eigenvalue and the eigenvalue In the d -dimensional systems, d > 1, M −1 U 0 has the zero eigenvalue, which is 2d times degenerate and the eigenvalues − Proof. We find eigenvalues of U 0 at first for the onedimensional case. We have that is The equality ð∂/∂x 3 ÞUðx ð3Þ Þ = 0 holds for x 1 = x 0 1 = −a, x 2 = x 0 2 = a, x 0 3 = 0. This configuration is an equilibrium. This follows also from the equalities ð∂/∂x j ÞUðx ð3Þ Þ = 0, j = 1, 2.
The second derivatives of the potential energy are calculated as follows: Hence, the second derivatives of the potential energy at the equilibrium U 0 j,k are given by That is Let us put Then, taking into consideration For the matrix of the second derivatives at the equilibrium, we derive U 0 1,α;1,β = U 0 2,α;2,β = ge 2 0 δ α,β − where the first and second numbers in the round brackets correspond to the lower and upper indices of variables. Then, This means where U ′ is given by (75). Let M′′ = M′ ⊕ M′ and M′ be the 3 × 3 diagonal matrix with the elements m 1 , m 2 , m 3 . Then, From this equality and (79), we derive This concludes the proof for the two-dimensional case.
All the formulas concerning partial derivatives of the potential energy of this sections will be true if one extends the previous condition α, β = 1, 2 to the condition α, β = 1, 2 , 3, ⋯, d.
Remark 13. Existence of global bounded solutions of nonsingular ordinary differential equation is well-known [10].
Remark 14. For the systems (1) and (2) with d > 1, n = 3, s > 1 , the speed of the converges to the regular polygon equilibrium does not depend on d since M −1 U 0 has only three eigenvalues for all d > 1. The explicit dependence of M −1 U 0 on d, sis given by (70) for s > 1, n = 3. One can also calculate it without difficulty for s > 1, n = 4, 5.

Conclusion
We showed that for the regular polygon equilibrium for (2) determined by Theorem 7 and s = d, the conclusion of Theorem 2 is true.
Our results that follow from Theorem 2 and Theorem 4 for Coulomb systems and s=d are formulated in the following way.
Let s = 3, d = 2; s = 2, d = 3, and n = 3, 4, 5. Let also s = 3, d = 1, n = 3. Then, there exists an integer p, 1 ≤ p < nd, a positive number λ, and a bounded at positive time solution xðtÞ of the equation of motion (3) with h c = 0 depending on p real parameters, which is a real analytic function in them in a neighborhood of the origin, such that kx − x 0 k λ < ∞, k _ xk λ < ∞, where x 0 is a regular polygon equilibrium. This result for s = 3, d = 1, n = 3 follows also from Theorem (1).
Theorem 12 shows that (1) for the systems (1) and (2) with d = 1, n = 3, s = 2, the conclusion of Theorem 2 is not true since M −1 U 0 does not have negative eigenvalues in ℝ 3 (2) for the systems (1) and (2) with d > 1, n = 3, s > 1, the speed of the converges to the regular polygon equilibrium does not depend on d since M −1 U 0 has only three eigenvalues for all d > 1

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