Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line
Periodic and bounded solutions of the Coulomb equation of motion in the line are obtained for two and three identical negative point charges in the fields of two and three symmetrically located fixed point charges. The systems possess equilibrium configurations. The Lyapunov, Siegel, Moser, and Wein...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508111121416192 |
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| author | Skrypnik, W. I. Скрипник, В. І. |
| author_facet | Skrypnik, W. I. Скрипник, В. І. |
| author_sort | Skrypnik, W. I. |
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| description | Periodic and bounded solutions of the Coulomb equation of motion in the line are obtained for two and three identical negative point charges in the fields of two and three symmetrically located fixed point charges. The systems possess equilibrium configurations. The Lyapunov, Siegel, Moser, and Weinstein theorems are applied. |
| first_indexed | 2026-03-24T02:20:00Z |
| format | Article |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Nat. Acad. Sci. Ukraine, Kyiv)
PERIODIC AND BOUNDED SOLUTIONS
OF THE COULOMB EQUATION OF MOTION
OF TWO AND THREE POINT CHARGES WITH EQUILIBRIUM ON LINE
ПЕРIОДИЧНI ТА ОБМЕЖЕНI РОЗВ’ЯЗКИ РIВНЯНЬ РУХУ КУЛОНА
ДЛЯ ДВОХ ТА ТРЬОХ ТОЧКОВИХ ЗАРЯДIВ З РIВНОВАГОЮ НА ПРЯМIЙ
Periodic and bounded solutions of the Coulomb equation of motion on line for two and three identical negative point
charges in fields of two and three fixed point charges located symmetrically are found. The systems possess equilibrium
configurations. The Lyapunov, Siegel, Moser, and Weinstein theorems are applied.
Знайдено перiодичнi та обмеженi розв’язки рiвнянь руху Кулона на прямiй для двох та трьох однакових негативних
точкових зарядiв у полi двох та трьох точкових фiксованих зарядiв, що симетрично розташованi. Цi системи мають
рiвноважнi стани. При цьому використано теореми Ляпунова, Зiгеля, Мозера та Вайнстайна.
1. Introduction. Construction of solutions of classical electrodynamics, based on the Maxwell –
Lorentz (ML) equations for point charges, is a fundamental task of mathematics. The simplest ap-
proximations of these equations are the Coulomb and Darwin equations which do not take into
account radiation of point charges. Solutions of the former and latter were proven to exist on a finite
time interval on which there are no collisions in [1] and [17]. If one finds Coulomb systems of point
charges with equilibrium then it will be possible to apply several basic theorems, including the center
Lyapunov theorem, guaranteeing existence of solutions of the equation of motion on all the time
interval. It would be important also to establish equilibrium stability or instability.
In the system of three point charges without external fields one equilibrium configuration is well
known [2]: the three charges are located on a line, two of them have the same value −e0, the third
charge is placed between them at the same distance and has the value
e0
4
. The rigourously proved
Earnshaw theorem [2, 3] establishes instability of equilibrium in the Coulomb point charges systems
without external fields. There is no such result for the systems in an external field created by fixed
charges (charged centers). For Coulomb systems of point charges on a line in external fields, created
by fixed symmetric charges outside of the line, equilibrium may be stable. The simplest of them will
be considered by us here.
In this paper we consider Coulomb systems of two and three point charges with all masses equal
to m restricted to move along a line where there is an equilibrium. They are the following ones:
A. Two charges with the same value −e0 < 0 in the field of two symmetric point charges with
the value e′ > 0. The latter are placed at the perpendicular, crossing the origin, at the same distance
b > 0 from the line where the negative point charges move.
B. The system A with the additional negative charge −e0 which is immobile at the origin (an
additional repulsive center for two moving charges is introduced).
C. The system of three equal negative charges −e0 in the field of the two attracting positive
charges the same as in the system A.
D. The system of three point charges −e0, −e0,
e0
4
.
c© W. I. SKRYPNIK, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5 679
680 W. I. SKRYPNIK
The systems A and B are the most simple Coulomb systems with equilibrium. They are also
restricted systems meaning that their dynamics describes the dynamics of two identical charges on
a plane with the attracting centers there if their initial velocities (momenta) are directed along the
line where the two negative charges are placed initially (at the same distance from the origin for the
system B). The system B is derived from the system C since the negative charge at the origin of the
latter is immobile if its initial velocity (momentum) is zero and the other two charges start to move
at the same distance from the origin with same velocities.
We prove the existence of bounded and periodic solutions for the systems A, B, C with the help of
the semilinearization Siegel and Lyapunov, Moser, Weinstein theorems, respectively. The first three
theorems demand a resonance condition on
e0
e′
and the Weinstein theorem does not. The latter can be
applied only for a stable equilibrium since it demands a Hamiltonian to have a Hessian with positive
eigenvalues. We prove also the existence of bounded solutions for the systems with the help of the
analog of the Siegel theorem which does not demand the resonance condition. All these theorems are
formulated in the Appendix.
We prove in the fifth section that the matrix of second derivatives of the potential energy (its Hes-
sian) for the system D at its equilibrium has zero eigenvalues that are obstructions for an application
of the above theorems for the system D.
We prove the existence of bounded solutions relying upon the following general theorem which
is proved in the Appendix. It follows from the semilinearization Siegel theorem and its mentioned
analog (Theorems 6.1 and 6.2).
Theorem 1.1. Let the Hessian of a potential energy U of a mechanical system of n bodies on
a line with equal masses have nonzero eigenvalues mσj at an equilibrium x0j , j = 1, . . . , n, U be
holomorphic at it and σj < 0, j = 1, . . . , p. Let also either λj = −
√−σj , j = 1, . . . , p < n, be not
in resonance or p = n. Then the equation of motion
m
d2xj
dt2
= −
∂U(x(n))
∂xj
, j = 1, . . . , n, x(n) = (x1, . . . , xn) ∈ Rn, (1.1)
admits bounded solutions which are holomorphic functions at the origin in p real parameters and
‖x− x0‖λ <∞, ‖ẋ‖λ <∞, where
‖x‖λ = sup
t≥0
max
j∈(1,...,n)
eλtmax |xj(t)|, λ < λ0 = min
j=1,...,p
|λj |.
The existence of periodic solutions for mechanical systems with equilibria determined by nonde-
generate minima is obtained by us with the help of the following theorem which follows from the
Weinstein theorem (Theorem 6.3) in a straightforward fashion.
Theorem 1.2. Let the potential energy U of a mechanical system of n bodies on a line with equal
masses have an equilibrium at x0(n), be a holomorphic function in its neighborhood and its Hessian
determine a positive definite quadratic form in a neighborhood of the equilibrium. Then there exist n
periodic solutions of (1.1) belonging to the energy level Eh = U(x0(n)) + h with a sufficiently small
positive h whose periods are close to those of the linearized equation of motion.
These theorem can be generalized to n-body mechanical systems with different masses. The next
four sections of our paper are devoted to the above four systems A – D. In the end of each of them,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 681
except the fifth section devoted to the D system, we formulate our results in theorems concerning
stability of their equilibria and existence of periodic an bounded solutions of the Coulomb equation
of motion.
Note that since in the systems A – C the order of charges is preserved due to the infinite repulsion
we can substitute the holomorphic functions (xj − xk)−1 instead of |xj − xk|−1 and for the system
B x−1j instead of |xj |−1 in the expression for their potential energies.
2. Two charges and two attracting centers. The potential energy for the system A is given by
U(x(2)) = e20|x1 − x2|−1 − 2e0e
′
(√
x21 + b2
)−1
− 2e0e
′
(√
x22 + b2
)−1
, xj ∈ R. (2.1)
The equilibrium equations are given by
∂
∂xj
U(x(2)) = 0, j = 1, 2. Let’s insert the equalities into it
for k = 1
∂
∂x1
|x1 − x2|−k = −k
x1 − x2
|x1 − x2|k+2
,
∂
∂x1
(√
x21 + b2
)−k
= −k x1
(
√
x21 + b2)k+2
.
That is
∂
∂xj
U(x(2)) = −e20
xj − xk
|x1 − x2|3
+ 2e0e
′ xj(√
x21 + b2
)3 , j 6= k = 1, 2.
As a result we obtain the equilibrium relation putting x1 = x01 = a, x2 = x02 = −a
e0
(2a)3
=
e′
(
√
a2 + b2)3
.
The most important information is the property of the matrix of second derivatives of U. Its two
nondiagonal elements are easily calculated as
∂U(x(2))
∂x1∂x2
=
∂U(x(2))
∂x2∂x1
= −2e20|x1 − x2|−3.
Let U0
1,2 be this function at the equilibrium. Then U0
1,2 = −
e20
4a3
= −u′. Further
∂2
∂x2j
U(x(2)) =
2e20
|x1 − x2|3
+
2e0e
′(√
x2j + b2
)3 − 6e0e
′x2j(√
x2j + b2
)5 .
Let U0
j,j be this function at the equilibrium. Then
U0
1,1 = U0
2,2 =
e20
4a3
+
2e0e
′
(
√
a2 + b2)3
− 6e0e
′a2
(
√
a2 + b2)5
.
From the equilibrium relation it follows that(e0
e′
)1/3 1
2a
=
1√
a2 + b2
, a = (4− η)−1/2√ηb, η =
(e0
e′
)2/3
< 4. (2.2)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
682 W. I. SKRYPNIK
As a result
U(x0(2)) = U(−a, a) = e20
2a
− 2e0e
′
a
(e0
e′
)1/3
=
e20
2a
(
1− 4
(
e′
e0
)2/3
)
,
U0
1,1 = U0
2,2 = 2u′ − u∗, u∗ =
3e20
42a3
(e0
e′
)2/3
=
3u′
4
(e0
e′
)2/3
.
Let U0 be the matrix with the elements U0
1,1, U
0
2,2, U
0
1,2 = U0
2,1. Its eigenvalues ζ1, ζ2 are found
easily as the roots λ = ζ1, ζ2 of the equation
(2u′ − u∗ − λ)2 − u′2 = 0,
ζ1 = u′ − u∗, ζ2 = 3u′ − u∗.
(2.3)
Two eigenvalues are negative or positive if 3u′ < u∗, u
′ > u∗, respectively. The following result fol-
lows from the stability Lagrange – Dirichlet theorem [3] since the latter case means that the potential
energy attains a minimum at the equilibrium.
Theorem 2.1. The line system of two identical charges with the the potential energy (2.1) pos-
sesses the equilibrium x0 = (−a, a) with a given by (2.2) if
e0
e′
< 8 and it is stable if
e0
e′
<
8√
27
=
(
4
3
)3/2
.
If the neutrality condition e′ = e0 holds then the equilibrium is stable.
If 3u′ > u∗, u
′ < u∗ then one root is positive and the other is negative. If u′ = u∗ or 3u′ = u∗
then one root is positive or negative and the other is zero.
The linear part of the vector field at the equilibrium of the Coulomb equation of motion has the
eigenvalues ±i
√
m−1(3u′ − u∗), ±i
√
m−1(u′ − u∗) if u′ > u∗. In order to apply the Lyapunov
theorem in this case one has to exclude the resonance between ζ1, ζ2, i.e.,
ζ2
ζ1
= k2, 1 < k ∈ Z.
Theorem 2.2. Let
e0
e′
<
8√
27
and
e0
e′
6= 8√
27
(
k2 − 3
k2 − 1
)3/2
, 1 < k ∈ Z+ or
8√
27
<
e0
e′
< 8.
Then the Coulomb equation of motion (1.1) for n = 2 and potential energy (2.1) possesses two or
one periodic solutions such that each of them depends on a real parameter cj for some j, j = 1, 2,
or one real parameter c. These solutions and their periods τ1(c1), τ2(c2) or τ(c) are holomorphic
functions in the parameters at the origin and τ1(0) = 2π
√
m
(
u′
(
1− 3
4
(e0
e′
)2/3))−1/2
, τ2(0) =
= 2π
√
m
(
u′
(
3− 3
4
(e0
e′
)2/3))−1/2
or τ(0) = 2π
√
m
(
u′
(
3− 3
4
(e0
e′
)2/3))−1/2
.
The next theorem follows from the Theorem 1.2.
Theorem 2.3. Let
e0
e′
<
8√
27
. Then the Coulomb equation of motion (1.1) for n = 2 and poten-
tial energy (2.1) on the energy level Eh =
e20
2a
(
1− 4
(
e′
e0
)2/3
)
+h with a sufficiently small positive
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 683
h has two periodic solutions with the frequencies close to the square roots ofm−1
(
3− 3
4
(e0
e′
)2/3)
u′,
m−1
(
1− 3
4
(e0
e′
)2/3)
u′.
From the Theorem 1.1 one deduces the following theorem which corresponds to the case of
nonnegative roots in (2.3).
Theorem 2.4. Let
8√
27
<
e0
e′
< 8 . Then the Coulomb equation of motion (1.1) for n = 2 and
the potential energy (2.1) admits bounded solutions which are holomorphic functions at the origin in
one real parameter such that ‖x − x0‖λ < ∞, ‖ẋ‖λ < ∞, where λ <
√
m−1(u∗ − u′) and x0 is
the equilibrium.
The next theorem is derived from the Moser theorem (Theorem 6.5).
Theorem 2.5. Let
8√
27
<
e0
e′
< 8. Then the Coulomb equation of motion (1.1) for n = 2 and
the potential energy (2.1) admits bounded solutions which are holomorphic functions at the origin in
three real parameters cj , j = 1, 2, 3. If c3 = 0 these solutions are periodic functions with the period
τ(c1c2) which coincides at the zero with the period from the Theorem 2.2.
The neutrality condition implies the resonance condition with k = 3 in the Theorem 2.2 and this
means that if the neutrality condition e0 = e′ holds then the conclusion of the Theorem 2.3 is true.
3. Two negative charges and three centers. The system B is characterized by the potential
energy
U(x(2)) =
e1e2
|x1 − x2|
−2e′e0
2∑
k=1
(√
x2k + b2
)−1
+e20
2∑
k=1
|xk|−1, −e1 = −e2 = e0 > 0, e′ > 0.
(3.1)
As a result
∂
∂xj
U(x(2)) = −e1e2
xj − xk
|x1 − x2|3
+
2e′e0xj(√
x2j + b2
)3 − e20xj
|x|3
, j 6= k = 1, 2.
The equilibrium is given by x01 = a, x02 = −a. The equilibrium relation is determined by
5e20
4a2
− 2e′e0a
(
√
a2 + b2)3
= 0,
which follows from
∂
∂xj
U(x(2)) = 0, j = 1, 2. The matrix of second derivatives of U is easily
calculated as
∂2
∂x2j
U(x(2)) = 2e1e2|x1 − x2|−3 +
2e0e
′(√
x2j + b2
)3 − 6e0e
′x2j(√
x2j + b2
)5 +
2e20
|xj |3
,
∂2U(x(2))
∂x1∂x2
=
∂2U(x(2))
∂x2∂x1
= −2e1e2|x1 − x2|−3,
U0
2,1 = U0
1,2 = −u′ = −
e20
4a3
,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
684 W. I. SKRYPNIK
U0
1,1 = U0
2,2 = 9u′ +
2e0e
′
(
√
a2 + b2)3
− 6e0e
′a2
(
√
a2 + b2)5
.
From the equilibrium relation one derives
2e0e
′
(
√
a2 + b2)3
= 5u′, (
√
a2 + b2)−1 =
1
2a
(
5e0
e′
)1/3
, (3.2)
a = (4− η′)−1/2
√
η′b, η′ =
(
5e0
e′
)2/3
< 4,
e0
e′
<
8
5
. (3.3)
That is
U(x0(2)) = U(−a, a) = 5e20
2a
− 2e0e
′
a
√
η′,
U0
1,1 = U0
2,2 = 14u′ − u∗, u∗ =
3e20
42a3
η′ =
3u′
4
η′ < 3u′,
(3.4)
where U0
j,k are the values of partial derivatives of the potential energy at the equilibrium. The neutral-
ity condition is 2e′ = 3e0 which implies u∗ = 3
(
10
3
)2/3
u′. Let U0 be the matrix with the elements
U0
1,1, U
0
2,2, U
0
1,2 = U0
2,1. Its eigenvalues ζ1, ζ2 are found easily as the roots λ = ζ1, ζ2 of the equation
(14u′ − u∗ − λ)2 − u′2 = 0,
ζ1 = 13u′ − u∗, ζ2 = 15u′ − u∗.
We have ζj > 0 since u∗ < 3 and the case of two negative roots is excluded. As a result we can
formulate only the analogs of the Theorems 2.1 and 2.2.
Theorem 3.1. The line system of two identical charges with the the potential energy (3.1) pos-
sesses the equilibrium x0 = (−a, a) with a given by (3.2) which is stable if
e0
e′
<
8
5
.
If the neutrality condition 2e′ = 3e0 holds then the equilibrium is stable.
Since k2 6=
(
15− 3
4
(
5e0
e′
)2/3
)(
13− 3
4
(
5e0
e′
)2/3
)−1
<
3
2
, k ∈ Z+ for
e0
e′
<
8
5
the
following theorem follows from the Lyapunov theorem (Theorem 6.4).
Theorem 3.2. Let
e0
e′
<
8
5
. Then the Coulomb equation of motion (1.1) for n = 2 with the
potential energy (3.1) possesses two periodic solutions such that each of them depends on a real
parameter cj for some j, j = 1, 2. These solutions and their periods τ1(c1), τ2(c2) are holomorphic
functions in the parameters at the origin and τ1(0) = 2π
√
m
(
u′
(
13− 3
4
(
5e0
e′
)2/3
))−1/2
,
τ2(0) = 2π
√
m
(
u′
(
15− 3
4
(
5e0
e′
)2/3
))−1/2
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 685
Note that if the neutrality condition 2e′ = 3e0 holds then the conclusion of the Theorem 3.2 is
true.
4. Three charges and two attracting centers. The potential energy of the system C is written
as follows:
U(x(3)) =
1
2
3∑
j 6=k=1
ejek
|xj − xk|
− 2e′e0
3∑
k=1
(√
x2k + b2
)−1
, −ej = e0 > 0, e′ > 0. (4.1)
The matrix of second derivatives of the potential energy is obtained from the equalities
∂
∂xj
U(x(3)) = −ej
3∑
k=1,k 6=j
ek
xj − xk
|xj − xk|3
+
2e′e0xj(√
x2j + b2
)3 ,
∂2
∂x2j
U(x(3)) = 2ej
3∑
k=1,k 6=j
ek|xj − xk|−3 +
2e0e
′(√
x2j + b2
)3 − 6e0e
′x2j(√
x2j + b2
)5 ,
∂2U(x(3))
∂xj∂xk
=
∂2U(x(3))
∂xk∂xj
= −2ejek|xj − xk|−3.
The equilibrium is given by x1 = x01 = −a, x2 = x02 = 0, x3 = x03 = a. The first derivative of
the potential energy for j = 2 is zero. The equilibrium relation
5e20
4a2
− 2e′e0a
(
√
a2 + b2)3
= 0
follows from
∂
∂xj
U(x(3)) = 0, j = 1, 3, and coincides with the equilibrium relation from the
previous section. At the equilibrium the matrix of second derivatives of the potential energy is
deduced from the equalities
U0
2,1 = U0
1,2 = U0
2,3 = U0
3,2 = −
2e20
a3
= −8u′, U0
3,1 = U0
1,3 = −u′,
U0
1,1 = U0
3,3 = 9u′ +
2e0e
′
(
√
a2 + b2)3
− 6e0e
′a2
(
√
a2 + b2)5
, U0
2,2 = 16u′ +
2e0e
′
(
√
a2 + b2)3
.
From (3.2) – (3.4) it follows that
U0
2,2 = 21u′, U0
1,1 = U0
3,3 = 14u′ − u∗.
Neutrality condition is 2e′ = 3e0 and it implies u∗ =
3
4
(
10
3
)2/3
u′.
Let’s calculate the diagonal k-dimensional determinants DetkU
′ = |U ′|k, k = 1, 2, of the matrix
u′−1U0 and apply the Sylvester condition for the matrix to be positive definite putting u = u∗u
′−1 <
< 3
|U ′|1 = 14− u > 0, |U ′|2 = U ′1,1U
′
2,2 − U ′1,2U ′2,1 = 21(14− u)− 64 = 230− 21u > 0,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
686 W. I. SKRYPNIK
DetU ′ = |U ′| = U ′1,1(U
′
2,2U
′
3,3 − U ′3,2U ′2,3)− U ′1,2(U ′2,1U ′3,3 − U ′2,3U ′3,1)+
+U ′1,3(U
′
2,1U
′
3,2 − U ′2,2U ′3,1) = (14− u)[21(14− u)− 64] + 8[−8(14− u)− 8]−
−(64 + 21) = 21(14− u)2 − 128(14− u)− 149.
The roots of the last equation are given by the following remarkable equality:
14− u = 64±
√
642 + 21× 149 = 64± 85.
Hence DetU ′ > 0 if 14 − u > 149 or 14 − u < −21. But u is positive and 14 − u ≤ 14 which
contradicts the first inequality. The second inequality contradicts the inequality |U ′|1 > 0. That is the
matrix U0 is not positive definite. If we find positive roots and there are no zero roots then we can
apply the center Lyapunov theorem and find periodic solutions.
The characteristic polynomial p′(λ) = −|U ′ − λI| = Det (λ− U ′) is given by
−p′(λ) = (14− u− λ)[(21− λ)(14− u− λ)− 64]+
+8[−8(14− u− λ)− 8]− (64 + 21− λ) =
= (21− λ)(14− u− λ)2 − 128(14− u− λ)− 149 + λ,
p′(λ) = λ3 + a1λ
2 + a2λ+ a3,
a1 = −2(14− u)− 21, a2 = 42(14− u) + (14− u)2 − 129,
a3 = −21(14− u)2 + 128(14− u) + 149.
(4.2)
Let us put
p = a2 −
a21
3
= −1
3
(14− u)2 + 14(14− u)− 276, q = a3 −
a1a2
3
+
2a31
27
.
It is easy to see that p < 0. The condition
q2
4
+
p3
27
≤ 0
implies that all the roots are real, as it follows from the Cardano formulas [15], and it holds in our
case since we deal with the symmetric matrix U0. To have all the roots different one has to assume
q2
4
+
p3
27
< 0. (4.3)
If it is true then it is possible to define the angle ϕ as in [15]
cosϕ = − q
√
27
2
√
|p|3
, 0 < ϕ < π.
Then the roots λ = ζ ′k, k = 1, 2, 3, of p′are written due to the Cardano formulas as
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PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 687
ζ ′k = 2
√
3−1|p| cos ϕ+ 2π(k − 1)
3
, k = 1, 2, 3.
If p(λ) is the characteristic polynomial of U0 then p(λ) = u′3p′(u′−1λ) and the eigenvalues of
U0 are given by ζk = u′ζ ′k that is
ζk = 2u′
√
3−1|p| cos ϕ+ 2π(k − 1)
3
, k = 1, 2, 3. (4.4)
To apply the theorems from the introduction we have to single out cases when the different roots
have equal signs and guarantee absence of resonances. It is not difficult to check that
1) ζ1 > 0, ζ3 > 0, ζ2 < 0 if 3−1ϕ ∈
(
0,
π
2
)
, 3−1(ϕ + 4π) ∈
(
3π
2
,
5π
2
)
, that is 3−1ϕ ∈
∈
(π
6
,
π
3
)
,
2) ζ1 > 0, ζ2 < 0, ζ3 < 0 if 3−1ϕ ∈
(
0,
π
6
)
.
Moreover in the second case
ζ3
ζ2
6= k2, k ∈ Z+, since cos
2π
3
= cos
(
π − π
3
)
= − cos
π
3
= −1
2
,
cos
(
π
6
+
2π
3
)
= cos
(
π − π
6
)
= − cos
π
6
= −
√
3
2
and cos
ϕ+ 2π
3
∈
[
−
√
3
2
,−1
2
]
. Besides
cos
(
π
6
+
4π
3
)
= cos
3π
2
= 0 and ζ3 = 0 at ϕ =
π
2
. The same argument excludes also the
resonance
ζ3
ζ1
6= k2, k ∈ Z+, in the first case since cos
ϕ
3
∈
[
−
√
3
2
,−1
2
]
.
The first result of this section is formulated in the following theorem deduced from the Lyapunov
theorem (the case 1 demands the nonresonance condition in ζ1, ζ3 for the existence of the two-
parametric family of periodic solutions). The Theorem 1.2 can not be applied since not all ζj and σj
can be positive.
Theorem 4.1. If (4.3) holds and 3−1ϕ ∈
(π
6
,
π
3
)
and
(
cos
ϕ+ 4π
3
)−1
cos
ϕ
3
6= k2, k ∈ Z+,
then the Coulomb equation of motion (1.1) for n = 3 with the potential energy (4.1) possesses
periodic solutions such that each of them depends on a real parameter cj for some j, j = 1, 2. These
solutions and their periods τ1(c1), τ2(c2) are holomorphic functions in the parameters at the origin
and τ1(0) = 2π
√
m
ζ1
, τ2(0) = 2π
√
m
ζ3
, where ζj are determined by (4.4). If 3−1ϕ ∈
(
0,
π
6
)
then the
same equation possesses a periodic solution depending on a constant c. This solution and its period
τ(c) are holomorphic functions in the parameter at the origin and τ(0) = τ1(0).
From the Theorem 1.1 we deduce the following result since (the case 2 demands the non-
resonance condition in ζ2, ζ3 for the existence of the two-parametric family of bounded solutions).
Theorem 4.2. Let (4.3) hold and either 3−1ϕ ∈
(
0,
π
6
)
and
(
cos
ϕ+ 2π
3
)−1
cos
ϕ+ 4π
3
6=
6= k2, k ∈ Z+, or 3−1ϕ ∈
[π
6
,
π
3
)
. Then the Coulomb equation of motion (1.1) for n = 3 with the
potential energy (4.1) admits bounded solutions which are holomorphic functions at the origin either
in two or one real parameters such that ‖x−x0‖λ <∞, ‖ẋ‖λ <∞, where λ < mink=2,3
√
m−1|ζk|
or λ <
√
m−1|ζ2|, the numbers ζk are given by (4.4) and x0 is the equilibrium.
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688 W. I. SKRYPNIK
In order to apply the Moser theorem (Theorem 6.5) we have to guarantee absence of quadratic
resonances between ζ1, ζ3 and ζ2, ζ3 for 3−1ϕ ∈
(π
6
,
π
2
)
and 3−1ϕ ∈
(
0,
π
6
)
, respectively. Taking
into account that
ζ3
ζ1
6= k2, k ∈ Z+, and
ζ3
ζ2
6= k2, k ∈ Z+, for such the cases we derive the following
consequence of the Moser theorem.
Theorem 4.3. If (4.3) is true and q 6= 0 then the Coulomb equation of motion (1.1) for n = 3
with the potential energy (4.1) possesses bounded solutions which are holomorphic functions at the
origin in three real parameters cj , j = 1, 2, 3. If c3 = 0 these solutions are periodic functions with
the period τ(c1c2) which coincides at the zero with the period τ1(0) from the Theorem 4.1.
The polynomial of the sixth order in 14− u in the left-hand side of (4.3) can have not more than
three real roots of even multiplicities. Lengthy calculations show that it is a polynomial of the fourth
order. That is, if the condition (4.3) is violated for real u then it is possible only for one or two values
of u.
5. Three charges. The potential energy of the system D is given by
U(x(3)) =
1
2
3∑
j 6=k=1
ejek
|xj − xk|
, −e1 = −e2 = e0, e3 =
e0
4
> 0. (5.1)
That is
U(x(3)) = e20|x1 − x2|−1 − e0e3[|x1 − x3|−1 + |x2 − x3|−1], xj ∈ R.
Then
∂
∂x1
U(x(3)) = −e20
x1 − x2
|x1 − x2|3
+ e0e3
x1 − x3
|x1 − x3|3
,
∂
∂x2
U(x(3)) = −e20
x2 − x1
|x1 − x2|3
+ e0e3
x2 − x3
|x2 − x3|3
,
∂
∂x3
U(x(3)) = −e0e3[
x3 − x1
|x1 − x3|3
+
x3 − x2
|x2 − x3|3
].
The equality
∂
∂x3
U(x(3)) = 0 holds for x1 = x01 = −a, x2 = x02 = a, x03 = 0. This configuration is
an equilibrium. This follows also from the equalities
∂
∂xj
U(x(3)) = 0, j = 1, 2. It is more convenient
to rewrite the derivatives of the potential energy as follows:
∂
∂xj
U(x(3)) = −ej
3∑
k=1,k 6=j
ek
xj − xk
|xj − xk|3
.
Then the second derivatives of the potential energy are calculated as follows:
∂U(x(3))
∂xj∂xk
=
∂U(x(3))
∂xk∂xj
= −2ejek|xj − xk|−3, k 6= j,
∂2
∂x2j
U(x(3)) = 2ej
3∑
k=1,k 6=j
ek|xj − xk|−3.
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PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 689
Hence the second derivatives of the potential energy at the equilibrium U0
j,k are given by
U0
1,2 = U0
2,1 = −
e20
4a3
= −u′, U0
3,1 = U0
1,3 = U0
2,3 = U0
3,2 = 2u′,
U1,1 = U2,2 = −u′, U3,3 = −4u′.
The rescaled characteristic polynomial p′, i.e., the determinant of the matrix −U ′+λI, U ′ = u′−1U0
is given by
−p′(λ) = (U ′1,1 − λ)[(U ′2,2 − λ)(U ′3,3 − λ)− U ′3,2U ′2,3]− U ′1,2[U ′2,1(U ′3,3 − λ)− U ′2,3U ′3,1]+
+U ′1,3[U
′
2,1U
′
3,2 − (U ′2,2 − λ)U ′3,1],
−p′(λ) = −(1 + λ)[(1 + λ)(4 + λ)− 4] + (4 + λ− 4) + 2[−2 + 2(1 + λ)] =
= −λ[(1 + λ)(5 + λ)− 5] = −λ2(λ+ 6).
Its roots are λ = 0, λ = −6 and the equilibrium is a degenerate maximum of the potential energy. If
p(λ) is the characteristic polynomial of U0 then p(λ) = u′3p′(u′−1λ).
Proposition 5.1. The (3 × 3)-matrix of second derivatives of the potential energy (5.1) at its
equilibrium has the zero eigenvalue which is doubly degenerate and its third eigenvalue is equal to
−6u′.
6. Appdendix. Here we formulate the basic theorems that are applied by us in the previous
sections. We begin from the semilinearization Siegel theorem (see Section 28 in [4]).
Theorem 6.1. Let λj ∈ C,
ẋj(t) = λjxj +Xj(x(l)(t)), j = 1, . . . , l, t ≥ 0, (6.1)
the function Xj(x(l)), x(l) = (x1, . . . , xl), be holomorphic at the origin such that in their power
expansions the sum of powers of xj is not less than two, real parts of λj ∈ C, j = 1, . . . , p, be
negative and the nonresonance condition hold: neither of these p numbers be a linear combination
of others with nonnegative integer coefficients whose sum exceeds unity. Then there exist functions
ϕ′j(x(p)), j = 1, . . . , p, ϕj(x(p)), j = p + 1, . . . , l, which are holomorphic at the origin and zero at
it, such that a partial solution of (1.1) is given by
xj(t) = ϕ′j(e
λ1tc1, . . . , e
λptcp), j = 1, . . . , p,
xj(t) = ϕj(x(p)(t)), j = p+ 1, . . . , l,
where cj are arbitrary constants.
This theorem for p = l is a version of the linearization Poincare theorem [5, 6] which was
generalized by Siegel in his linearization theorem [6, 7]. The statement of the Theorem 6.1 is not
formulated as a theorem in [4] and proved in Section 28. The existence of formal series ϕj , ϕ′j
demands the nonresonance condition which eliminates small denominators and allows one to solve
the equation (7) in Section 28 in [4]. The Siegel’s proof of the Theorem 6.1 is simpler than the proof
of the Poincare linearization theorem given in [5]. The proof of the holomorphic character of ϕj , ϕ′j
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690 W. I. SKRYPNIK
is based on an application of the standard inequality
|λk −
p∑
j=1
njλj | > c
p∑
j=1
nj ,
p∑
j=1
nj ≥ 2,
where nj is a nonnegative integer, c is a positive constant and a majoration technique.
A resonance version of the Theorem 6.1, which demands that real parts of all λj are nonzero, is
formulated in the following theorem.
Theorem 6.2. Let in (6.1) Reλj < 0, j = 1, . . . , p, Reλj > 0, j = p + 1, . . . , l, Xj be a
holomorphic function in the hyperdisc
Dl(A) = {xj : |xj | ≤ A, j = 1, . . . , l}, A > 0.
|Xj | ≤M and have the power expansion the same as in the Theorem 6.1. Then there exist x̃0 ∈ Rl−p
and the solution of (6.1) for the initial data (x0(p), x̃0) with x0(p) = x(p)(0) ∈ Dp(A
′), A′ < A. This
solution is a holomorphic function in x0(p) ∈ Dp(A
′), belongs to Dl(A) and ‖x‖λ ≤ A, λ < λ0 if
M(λ0 − λ)−1 is sufficiently small, where
‖x‖λ = sup
t≥0
max
s∈(1,...,l)
eλt|xs(t)|, λ0 = min
j=1,...,l
|Reλj | .
Note that x̃0 = (xp+1(0), . . . , xl(0)) is a holomorphic vector function in x0(p) ∈ Dp(A
′) and the
solutions of (1.1) found in this theorem tend exponentially fast to the equilibrium in the infinite time
limit. If Fj is a sufficiently smooth function then the existence of a bounded solution for (1.1) on
the positive time interval is proven in [8]. Lyapunov proved this under the assumption p = l. This
theorem can be proved without difficulty if Reλj ≥ 0, j = p+1, . . . , l, and Xj(0(p), xp+1, . . . , xl) =
= 0. The existence of solutions converging to an equilibrium if there are λj = 0 and Xj ∈ C1(Rd)
is proved in [18].
Proof of Theorem 1.1. To prove the theorem one has to transform (1.1) into the simple standard
form (6.1) and apply the Theorems 6.1 and 6.2. We assume that the potential energy U has the
equilibrium at the points x0j , j = 1, . . . , n, at which it is holomorphic, that is(
∂U
∂xj
)
(x0(n)) = 0.
Then in the new variables x′j = xj − x0j the dynamic equation is rewritten as
m
d2xj
dt2
= −
∂U ′(x(n))
∂xj
, (6.2)
where
U ′(x(n)) = U(x1 + x01, . . . , xn + x0n),
(
∂U ′
∂xj
)
(0) = 0.
Let U0 be the symmetric matrix of the second derivatives of U calculated at the equilibrium. Then by
a nonsingular linear transformation x′j =
∑n
k=1
S′j,kxk one diagonalizes U0, which has real-valued
eigenvalues mσj , that is δj,kσj = (S′m−1U0S′−1)j,k and transforms (6.2) into
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PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 691
d2xj
dt2
= −σjxj(t) + Fj(x(n)), (6.3)
where
Fj(x(n)) = −m−1
n∑
k=1
S′j,k
(
∂U ′′
∂xk
)
((S′−1x)(n)), U ′′(x(n)) = U ′(x(n))−
1
2
n∑
j,k=1
U0
j,kxjxk.
That is
dxj
dt
= vj ,
dvj
dt
= −σjxj(t) + Fj(x(n)), (6.4)
where Fj has the properties of Xj . Then by the linear two dimensional transformation produced
by the matrix S0
j the last equation is mapped into (6.1) with l = 2n and −λ2j−1 = λ2j =
√−σj ,
j = 1, . . . , n. The matrix S0
j diagonalizes the two dimensional matrix Aj , which determines the linear
part of (6.4), with the zero diagonal elements and nondiagonal elements Aj;1,2 = 1, Aj;2,1 = −σj .
That is S0
jAj = σ̂jS
0
j , where σ̂j is a diagonal matrix with the eigenvalues −λ2j−1 = λ2j =
√−σj .
It is not difficult to check that
S0
j;1,1 = S0
j;2,1 =
1
2
, −S0
j;1,2 = S0
j;2,2 =
1
2κj
, κj =
√
−σj .
The new variables look like
x′2j−1 =
1
2
(
xj −
1
κj
vj
)
, x′2j =
1
2
(
xj +
1
κj
vj
)
, j = 1, . . . , n.
The inverse transform is given by
xj = x′2j + x′2j−1, vj = κj(x
′
2j − x′2j−1), j = 1, . . . , n.
The functions Xj from the Theorem 6.1 are given by (l = 2n)
X2j(x(2n)) = −X2j−1(x(2n)) =
1√
2κj
Fj(x2 + x1, . . . , x2n + x2n−1).
As a result we can apply the above two theorems and prove Theorem 1.1. For this it is more
convenient to have another numeration of variables:
(x1, x2, x3, . . . , x2n)→ (x1, x3, . . . , x2n−1, x2, x4, . . . , x2n).
In such a way (6.4) is mapped into (6.1) with
λj = −
√
−σj , j = 1, . . . , n, λj =
√
−σj , j = n+ 1, . . . , 2n. (6.4′)
We have also ‖x′‖λ <∞ iff ‖x‖λ <∞ and ‖v‖λ <∞. Hence the Theorem 1.1 is true.
Remark that if one of σj is zero then it is impossible to transform the Newton equation (1.1) into
the standard form (6.1).
(6.2) in the Hamiltonian form is rewritten as follows:
ẋj =
∂H ′
∂pj
, ṗj = −
∂H ′
∂xj
, H ′ =
n∑
j=1
p2j
2mj
+ U ′(x(n)), (6.5)
where the dot over the variables means the time derivative and mj = m. The equation (6.5) is solved
with the help of the Weinstein theorem and Lyapunov center theorem. The former is formulated as
follows [9] (in [10] one can find a proof of its generalization to a case of an ordinary differential
equation with an integral).
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692 W. I. SKRYPNIK
Theorem 6.3. Let a Hamiltonian H in R2n be twice differentiable and its Hessian determine
a positive definite quadratic form at a neighborhood of the origin which is an equilibrium. If H(0)
is its value at the origin then for a sufficiently small h > 0 the Hamiltonian level Eh = H(0) + h
contains at least n periodic solutions of the equation of motion
ẋj =
∂H
∂pj
, ṗj = −
∂H
∂xj
(6.6)
whose periods are close to those of the linearized equation of motion.
Theorem 6.3 can be applied to (6.5) and it yields the Theorem 1.2 since the Hessian of H ′
determines a positive definite quadratic form in a neighborhood of the origin if the Hessian of U ′ has
the same property. As a result the Theorem 1.2 is true for systems in which an equilibrium determines
a nondegenerate minimum.
The eigenvalues of the linear part of the Hamiltonian in (6.5) are given by (6.4′). If σj > 0,
j = 1, . . . , p, then λj = −√−σj , j = 1, . . . , p, are imaginary numbers and (1.1) possesses also
periodic solutions due to the Lyapunov center theorem formulated as follows.
Theorem 6.4. Let an n-dimensional Hamiltonian system have a holomorphic at the origin real-
valued Hamiltonian whose Taylor power expansion begins from quadratic terms. Let also λ1, . . . , λn,
−λ1, . . . ,−λn be different nonzero eigenvalues of the linear part of the Hamiltonian vector field such
that the simple nonresonance condition hold for all purely imaginary eigenvalues λj : λj 6= n′λs,
j 6= s = 1, . . . , k, for all integers n′. Then (6.6) possesses k periodic solutions such that each
of them depends on a real parameter cj for some j = 1, . . . , k. These solutions and their periods
τ1(c1), . . . , τk(ck) are holomorphic functions in the parameters at the origin and τj(0) =
2π
|λj |
.
Different proofs of this theorem and its generalizations can be found in [11 – 14]. There is a
possibility that (1.1) admits simultaneously both periodic and bounded solutions. It is guaranteed by
the following Moser theorem.
Theorem 6.5. Let an n-dimensional Hamiltonian system have a holomorphic at the origin real-
valued Hamiltonian, whose Taylor power expansion begins from quadratic terms and λ1, . . . , λn,
−λ1, . . . ,−λn be eigenvalues of the linear part of the Hamiltonian vector field in R2n which are all
different nonzero complex numbers. Let also λ1, λ2 be independent over the reals and the following
nonresonance condition λj 6= n1λ1 + n2λ2, j ≥ 3, hold for all integers n1, n2. Then there exists a
four parameter family of solutions of (6.6) of the form
xj(t) = ϕj(ξ1, ξ2, η1, η2), pj(t) = ψj(ξ1, ξ2, η1, η2), (6.7)
where
ξj = etajξ0j , ηj = e−tajη0j , j = 1, 2,
and ϕj , ψj , aj are holomorphic functions in a neighborhood of the origin, the latter of which depend
on the complex ξ01η
0
1, ξ
0
2η
0
2 and have the constant terms coinciding with λj . If λ1, λ2, −λ1, −λ2
contain their complex conjugates then the solution can be chosen to be real, depending on four real
parameters. Moreover the matrix
∂ϕj
∂ξs
∂ϕj
∂ηs
∂ψj
∂ξs
∂ψj
∂ηs
has rank four and if λ1 or λ2 is pure imaginary then the same is true for a1 or a2.
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PERIODIC AND BOUNDED SOLUTIONS OF THE COULOMB EQUATION OF MOTION. . . 693
If λ1 or λ2 is positive then one has to put ξ01 = 0 or ξ02 = 0 in order to have the solutions on
the infinite time interval. This theorem finds application in the three body problem in the celestial
mechanics.
All the formulated theorems except the Weinstein theorem are constructive: the solutions of
the equation of motion are expressed in terms of convergent series. The Weinstein theorem is a
generalization of the Berger theorem which demands smoothness of a potential energy [13].
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Received 16.01.13,
after revision — 20.06.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
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| id | umjimathkievua-article-2169 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:00Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0a/bbe0d554c9c9a64782dc6ed5201e030a.pdf |
| spelling | umjimathkievua-article-21692019-12-05T10:25:31Z Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line Періодичні та обмежені розв'язки рівнянь руху Кулона для двох та трьох точкових зарядів з рівновагою на прямій Skrypnik, W. I. Скрипник, В. І. Periodic and bounded solutions of the Coulomb equation of motion in the line are obtained for two and three identical negative point charges in the fields of two and three symmetrically located fixed point charges. The systems possess equilibrium configurations. The Lyapunov, Siegel, Moser, and Weinstein theorems are applied. Знайдено періодичні та обмежені розв'язки рівнянь руху Кулона на прямій для двох та трьох однакових негативних точкових зарядів у полі двох та трьох точкових фіксованих зарядів, що симетрично розташовані. Ці системи мають рівноважні стани. При цьому використано теореми Ляпунова, Зігеля, Мозера та Вайнстайна. Institute of Mathematics, NAS of Ukraine 2014-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2169 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 5 (2014); 679–693 Український математичний журнал; Том 66 № 5 (2014); 679–693 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2169/1340 https://umj.imath.kiev.ua/index.php/umj/article/view/2169/1341 Copyright (c) 2014 Skrypnik W. I. |
| spellingShingle | Skrypnik, W. I. Скрипник, В. І. Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line |
| title | Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line |
| title_alt | Періодичні та обмежені розв'язки рівнянь руху Кулона для двох та трьох точкових зарядів з рівновагою на прямій |
| title_full | Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line |
| title_fullStr | Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line |
| title_full_unstemmed | Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line |
| title_short | Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line |
| title_sort | periodic and bounded solutions of the coulomb equation of motion of two and three point charges with equilibrium in the line |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2169 |
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