Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges
Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two fixed and equal positive charges are found with the help of the Lyapunov center theorem.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507810937176064 |
|---|---|
| author | Skrypnik, W. I. Скрипник, В. І. |
| author_facet | Skrypnik, W. I. Скрипник, В. І. |
| author_sort | Skrypnik, W. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:31:35Z |
| description | Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two
fixed and equal positive charges are found with the help of the Lyapunov center theorem. |
| first_indexed | 2026-03-24T02:15:14Z |
| format | Article |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci., Ukraine, Kyiv)
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE
CHARGES IN THE FIELD OF FIXED TWO EQUAL POSITIVE CHARGES
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE
CHARGES IN THE FIELD OF FIXED TWO EQUAL POSITIVE CHARGES
Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two
fixed and equal positive charges are found with the help of the Lyapunov center theorem.
Знайдено перiодичнi та квазiперiодичнi розв’язки рiвнянь руху Кулона двох рiвних негативних зарядiв у полi
фiксованих двох рiвних позитивних зарядiв iз допомогою центральної теореми Ляпунова.
1. Introduction. The Coulomb systems of two and three negative equal charges e0 in the field of fixed
two equal positive charges e\prime have equilibrium configurations [1]. This fundamental fact allowed us
to construct periodic and bounded solutions close to the equilibria of the Coulomb equation of motion
for two negative charges restricted to such a line that the positive charges are located symmetrically
at a perpendicular to this line [1]. We applied the Siegel [2], Weinstein [3, 4], Moser [5] and center
Lyapunov [2, 5 – 9] theorems which demand a knowledge of the spectra of the matrix U0 of second
derivatives of the potential energy at the equilibrium(for equal masses). The last two theorems,
guaranteeing the existence of the periodic solutions in terms of convergent series, restrict the values
of
e0
e\prime
through a nonresonance condition. The Weinstein theorem establishes also the existence of
the periodic solutions but can be applied only for mechanical systems with a stable equilibrium (U0
is positive definite and the equilibrium is a minimum of the potential energy). Periodic solutions are
also found in planar Coulomb systems of n - 1, n > 2 equal negative charges and one and three
positive charges [10, 11].
In this paper we find periodic and quasiperiodic solutions of the Coulomb equation of motion for
planar and space systems of two equal negative charges in the field of two fixed positive charges.
This result is a consequence of an explicit calculation of the eigenvalues of U0 which in its turn
follows from the representation of U0 as a direct sum of two and three two-dimensional matrices in
the planar and space dynamics, respectively.
There are two different cases of the planar dynamics determined by the fact whether the positive
charges are outside of the plane with the negative charges or not. This difference is explained by
different characters of spectra of U0 and their canonical matrices, i.e., the matrices which determine
linear parts of hamiltonian vector fields at equilibria (see Appendix). For the external positive charges
there is the zero eigenvalue for all values of e0, e\prime and for some of these values the other eigenvalues
are positive. For the internal positive charges there is no such zero eigenvalue and not all the
eigenvalues are nonnegative. Besides in the first case the spectrum contains the eigenvalues of the
line dynamics. The spectrum of U0 of the space dynamics contains eigenvalues of both cases which
are proportional to u\prime =
e20
4a3
, where 2a is the equilibrium distance between two negative charges
expressed through the distance between two positive charges 2b. Such the spectrum is an obstruction
for the applications of the Lyapunov and Moser theorems.
c\bigcirc W. I. SKRYPNIK, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8 1273
1274 W. I. SKRYPNIK
We circumvent the obstruction of the zero eigenvalue with the help of the Jacobi procedure
of an elimination of node from the Celestial Mechanics (Section 18 in [2]). It is known that the
zero eigenvalue of a canonical matrix is generated by integrals of motion [2]. The main idea of
the procedure of an elimination of node is to produce a canonical transformation which turns the
integrals of motions into cyclic variables (a Hamiltonian does not depend on them). Then the linear
part of the equation of motion for them will be zero and the linear part of the equation of motion for
remaining variables will not contain the zero eigenvalue. The procedure of elimination of node can
be formulated in the following theorem (the proof of its first two statements are given in [2]).
Theorem 1.1. Let H(x, p) be a 2n-dimensional Hamiltonian, Q be its time independent integral
and w(u, p) be a generating function of a canonical transformation such that
vk =
\partial w
\partial uk
, xk =
\partial w
\partial pk
, k = 1, . . . , n, (1.1)
\partial w
\partial un
= Q(x, p), Wk,j =
\partial 2w
\partial uk\partial pj
, \mathrm{D}\mathrm{e}\mathrm{t}W \not = 0. (1.2)
(1) Then the transformed Hamiltonian H \prime (u, v) does not depend on un. Let also the canonical matrix
of H have doubly degenerate zero eigenvalue, the Q-canonical transformation and the Hamiltonian
be holomorphic at the neighborhood of the equilibrium. Then (2) the canonical matrix of the
2(n - 1)-dimensional hamiltonian equation
\.uj =
\partial H \prime
\partial vj
, \.vj = - \partial H \prime
\partial uj
, j = 1, . . . , n - 1, (1.3)
does not have the zero eigenvalue for the equilibrium value Q0 of Q and (3) eigenvalues of the
canonical matrices of H and H \prime are identical.
A separation of cyclic variables, generated by integrals of motion, in a hamiltonian equation is
also described in [12].
For the planar dynamics with external positive charges and space dynamics the one-dimensional
angular momentum Q is an integral of motion. We find w as
w =
n\sum
j=1
gk(u1, . . . , un)pk,
where n = 4 and n = 6 correspond to the plane and space dynamics, respectively, introducing
special numerations of charges coordinates and momenta (xj ; pj) = (x\alpha j ; p
\alpha
j ), j = 1, 2, \alpha \leq 3, for
these two cases. The solution of the equation for gk derived from (1.1), (1.2), which guarantees all
the conditions for the Q-canonical transformation, is taken by us from [2]. As a result a solutions of
the Coulomb equation for these two systems are given by
x\alpha j (t) =
n - 1\sum
k=1
uk(t)[\gamma k,j,\alpha + \gamma \prime k,j,\alpha \mathrm{c}\mathrm{o}\mathrm{s}(un(t)) + \gamma \prime \prime k,j,\alpha \mathrm{s}\mathrm{i}\mathrm{n}(un(t))], (1.4)
where \gamma , \gamma \prime , \gamma \prime \prime are constants and uk(t), k = 1, . . . , n, are solutions of the equation with the
Hamiltonian H \prime .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE CHARGES IN THE FIELD 1275
Let u(n - 1), v(n - 1) be a periodic solution of (1.3) with a period \tau . Then\biggl(
\partial H \prime
\partial vn
\biggr)
(u(n - 1), v(n - 1), Q0) = H \prime
n - 1(t)
is also a periodic function such that
H \prime
n - 1(t) = H0
n - 1(t) + \xi ,
t+\tau \int
t
H0
n - 1(s)ds = 0,
where \xi is a constant. This implies that
un(t) =
t\int
0
H \prime
n - 1(s)ds =
t\int
0
H0
n - 1(s)ds+ \xi t = u0(t) + \xi t,
where u0 is periodic with the period \tau . The last equality and (1.4) determine quasiperiodic (doubly
periodic) functions x\alpha j (t) if \gamma \prime \not = 0 or \gamma \prime \prime \not = 0. We will show that their Euclidean norms are periodic
functions.
Our paper is organized as follows. In the second, third and fourth sections we formulate theorems
concerning existence of periodic and quasiperiodic solutions of the Coulomb equations of motion
for planar with external positive charges, for planar with internal positive charges and the space
dynamics, respectively. In Appendix we prove the second and third statements of Theorem 1.1.
2. Two external fixed charges. Let xj = (x1j , x
2
j ) \in \BbbR 2, j = 1, 2, be coordinates of two equal
point negative charges e0 and | x| 2j = (x1j )
2 + (x2j )
2. Let also two fixed positive charges e\prime have the
coordinates (0, 0,\pm b) \in \BbbR 3. Then the potential Coulomb energy of the negative charges is given by
U(x(2)) = e20| x1 - x2| - 1 - 2e0e
\prime
\biggl[ \Bigl( \sqrt{}
| x1| 2 + b2
\Bigr) - 1
+
\Bigl( \sqrt{}
| x2| 2 + b2
\Bigr) - 1
\biggr]
, x(2) = (x1, x2). (2.1)
The equation of motion is represented by
m
d2xj
dt2
= -
\partial U(x(2))
\partial xj
, j = 1, 2. (2.2)
The first partial derivatives of U look like
\partial
\partial x\alpha 1
U(x(2)) = - e20
x\alpha 1 - x\alpha 2
| x1 - x2| 3
+ 2e0e
\prime x\alpha 1\Bigl( \sqrt{}
| x1| 2 + b2
\Bigr) 3 ,
\partial
\partial x\beta 2
U(x(2)) = - e20
x\beta 2 - x\beta 1
| x1 - x2| 3
+ 2e0e
\prime x\beta 2\Bigl( \sqrt{}
| x2| 2 + b2
\Bigr) 3 .
The equalities x11 = a, x12 = - a, x21 = 0, x22 = 0 determine the equilibrium for which these first
derivatives are zeroes if
e0
(2a)3
=
e\prime
(
\surd
a2 + b2)3
,
\Bigl( e0
e\prime
\Bigr) 1
3 1
2a
=
1\surd
a2 + b2
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1276 W. I. SKRYPNIK
That is
a = (4 - \eta \prime ) -
1
2
\sqrt{}
\eta \prime b = (3 - \eta ) -
1
2
\surd
\eta b, \eta \prime =
\Bigl( e0
e\prime
\Bigr) 2
3
< 4, \eta =
3
4
\eta \prime < 3,
U(x0(2)) = U(a, 0; - a, 0) =
e20
2a
- 2e0e
\prime
a
\Bigl( e0
e\prime
\Bigr) 1
3
=
e20
2a
\Biggl(
1 - 4
\biggl(
e\prime
e0
\biggr) 2
3
\Biggr)
=
e20
2a
(1 - 3\eta - 1). (2.3)
The second partial derivatives of the potential energy are given by
\partial U(x(2))
\partial x\alpha 1\partial x
\beta
2
=
\partial U(x(2))
\partial x\beta 2\partial x
\alpha
1
= e20
\Biggl[
\delta \alpha ,\beta
| x1 - x2| 3
- 3
(x\alpha 1 - x\alpha 2 )(x
\beta
1 - x\beta 2 )
| x1 - x2| 5
\Biggr]
, \alpha , \beta = 1, 2,
and
\partial 2U(x(2))
\partial x\beta j \partial x
\alpha
j
= -
e20\delta \alpha ,\beta
| x1 - x2| 3
+
2e0e
\prime \delta \alpha ,\beta \Bigl( \sqrt{}
| xj | 2 + b2
\Bigr) 3 -
6e0e
\prime x\alpha j x
\beta
j\Bigl( \sqrt{}
| x| 2j + b2
\Bigr) 5 + 3e20
(x\alpha 1 - x\alpha 2 )(x
\beta
1 - x\beta 2 )
| x1 - x2| 5
.
Let U0 be the (4\times 4)-matrix of the second partial derivatives at the equilibrium and
u\prime =
e20
4a3
, u\prime \ast = u\ast -
3u\prime
2
, u\ast =
3u\prime
4
\Bigl( e0
e\prime
\Bigr) 2
3
= u\prime \eta
then its matrix elements look like
U0
1,\alpha ;1,\beta = U0
2,\alpha ;2,\beta = \delta \alpha ,\beta
\left( e20
(2a)3
- 6e0e
\prime a2\Bigl( \surd
a2 + b2
\Bigr) 5 \delta \alpha ,1 + 3
e20
(2a)3
\delta \alpha ,1
\right) = \delta \alpha ,\beta
\biggl(
u\prime
2
- \delta \alpha ,1u
\prime
\ast
\biggr)
,
U0
1,\alpha ;2,\beta = U0
2,\alpha ;1,\beta =
u\prime
2
\delta \alpha ,\beta (1 - 3\delta \alpha ,1).
In order to determine the spectrum of U0 we put
(1, 1) = 1, (2, 1) = 2, (1, 2) = 3, (2, 2) = 4.
As a result
U0 =
\left( u\prime
2
- u\prime \ast - u\prime
- u\prime
u\prime
2
- u\prime \ast
\right) \oplus
\left( u\prime
2
u\prime
2
u\prime
2
u\prime
2
\right)
The characteristic polynomial of U0 looks like
p(\lambda ) = \mathrm{D}\mathrm{e}\mathrm{t}( - U0 + \lambda I) =
\Biggl( \biggl(
u\prime
2
- u\prime \ast - \lambda
\biggr) 2
- u\prime 2
\Biggr) \Biggl( \biggl(
u\prime
2
- \lambda
\biggr) 2
- u\prime 2
4
\Biggr)
.
The roots \zeta j of this polynomial are given by
\zeta 1 = - u\prime
2
- u\ast = u\prime - u\ast , \zeta 2 =
3u\prime
2
- u\ast = 3u\prime - u\ast , \zeta 3 = u\prime > 0, \zeta 4 = 0
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE CHARGES IN THE FIELD 1277
or
\zeta 1 = u\prime (1 - \eta ), \zeta 2 = u\prime (3 - \eta ), \zeta 3 = u\prime , \zeta 4 = 0.
The two first eigenvalues are negative or positive if 3u\prime < u\ast , u
\prime > u\ast , respectively, or \eta > 3,
\eta < 1, respectively. The first case is excluded due to (2.3). If 3u\prime > u\ast , u
\prime < u\ast , then the second
and third roots are positive and the first root is negative. In these cases one can apply the Lyapunov
after the elimination of a node.
Now let’s describe the Q-canonical transformation announced in Theorem 1.1. Let
x11 = x1, x21 = x2, x12 = x3, x22 = x4,
p11 = p1, p21 = p2, p12 = p3, p22 = p4.
(2.4)
Then the angular moment (the integral of motion corresponding to rotation) is given (x, p are the
same as \xi , \eta in section 18 of [2])
Q = -
2\sum
j=1
(x1jp
2
j - x2jp
1
j ) =
2\sum
j=1
(x2jp2j - 1 - x2j - 1p2j).
Further the upper index will show a power. (1.2) gives
4\sum
j=1
\partial gk
\partial u4
pk =
2\sum
j=1
(g2jp2j - 1 - g2j - 1p2j)
or
\partial g2k - 1
\partial u4
= g2k,
\partial g2k
\partial u4
= - g2k - 1, (2.5)
where k = 1, 2. The simplest solution is given by
g1 = u1c, g2 = - u1s, g3 = u2c+ u3s, g4 = - u2s+ u3c, c = \mathrm{c}\mathrm{o}\mathrm{s}u4, s = \mathrm{s}\mathrm{i}\mathrm{n}u4.
This solution is generated by the nonsingular canonical transformation in the neighborhood of point
u1 = - u2 = a, u3 = u4 = 0, vj = 0, which determines the new equilibrium, since
\mathrm{D}\mathrm{e}\mathrm{t}W = \mathrm{D}\mathrm{e}\mathrm{t}G = - u1, Gj,k =
\partial gj
\partial uk
.
To prove this we decompose the determinant of G in the elements of the first row
G =
\left(
c 0 0 - u1s
- s 0 0 - u1c
0 c s - u2s+ u3c
0 - s c - u2c - u3s
\right) ,
\mathrm{D}\mathrm{e}\mathrm{t}G = c\mathrm{D}\mathrm{e}\mathrm{t}
\left(
0 0 - u1c
c s - u2s+ u3c
- s c - u2c - u3s
\right) + u1s
\left(
- s 0 0
0 c s
0 - s c
\right) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1278 W. I. SKRYPNIK
= - u1c
2(c2 + s2) - u1s
2(c2 + s2) = - u1.
As a result
x1 = u1c, x2 = - u1s, x3 = u2c+ u3s, x4 = - u2s+ u3c, (2.6)
v1 = cp1 - sp2, v2 = cp3 - sp4, v3 = sp3 + cp4. (2.7)
Let
v0 = sp1 + cp2,
then
p1 = cv1 + sv0, p2 = cv0 - sv1, p3 = cv2 + sv3, p4 = cv3 - sv2
and
Q = v4 = u3v2 - u2v3 - u1v0, v0 = u - 1
1 (u3v2 - u2v3 - v4).
These equalities allow one to derive the invertible matrix which determines the linear part of the Q-
canonical transformation in the neighborhood of the equilibrium and calculate the new Hamiltonian.
They imply
v22 + v23 = p23 + p24, v20 + v21 = p21 + p22,
x21 + x22 = u21, x23 + x24 = u22 + u23,
(x1 - x3)
2 + (x2 - x4)
2 = ((u2 - u1)c+ u3s)
2 + ((u2 - u1)s - u3c)
2 = (u2 - u1)
2 + u23.
The new Hamiltonian in the new variables is given by (w does not depend on t)
H \prime = (2m) - 1
3\sum
j=1
v2j + (2mu21)
- 1(u3v2 - u2v3 - v4)
2 + U \prime (u(3)), (2.8)
U \prime (u(3)) = e20((u2 - u1)
2 + u23)
- 1
2 - 2e0e
\prime
\Bigl[
(u21 + b2) -
1
2 + (u22 + u23 + b2) -
1
2
\Bigr]
.
It is known that the eigenvalues of the canonical matrix of H coincide with \pm
\sqrt{}
m - 1\zeta j [1]. That is its
zero eigenvalue is doubly degenerate and we can apply Theorem 1.1. To apply the center Lyapunov
theorem for (1.3) one has to exclude resonances for U0, i.e., the equalities
\zeta j
\zeta s
= k2, k \in \BbbZ +, and
make the translation u1 \rightarrow u1 - a, u2 \rightarrow u2 + a.
Let \zeta 1 > 0. The equalities
\zeta j
\zeta 1
\not = k2, k \in \BbbZ +, imply
\eta =
3
4
\Bigl( e0
e\prime
\Bigr) 2
3 \not = k2 - 3
k2 - 1
,
k2 - 1
k2
.
Since \zeta 1 < \zeta 3 \leq \zeta 2 the equalities
\zeta 1
\zeta s
= k2, k \in \BbbZ +, s \not = 1, are not true. If \zeta 2 > \zeta 3, i.e., 2 - \eta > 0
then the resonance for s = 2 is not true and the resonance for s = 3 coincides with
\zeta 2
\zeta 3
= k2, 3 - \eta = k2
and is not true also. The condition \zeta 3 > \zeta 2, i.e., 2 - \eta < 0, contradicts \zeta 1 > 0.
The following conclusion is true:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE CHARGES IN THE FIELD 1279
I. If \eta < 1 and
\eta \not = k2 - 3
k2 - 1
,
k2 - 1
k2
, k \in \BbbZ +.
then the resonance in \zeta 1 is absent;
II. if \eta < 1 then the resonances in \zeta 2, \zeta 3 are absent.
Let \zeta 1 < 0, i.e., \eta > 1. Once again there are no resonances in \zeta 2, \zeta 3 if \zeta 2 > \zeta 3. If \zeta 3 > \zeta 2 then
the nonresonance condition in \zeta 2 is given by
\zeta 3
\zeta 2
\not = k2, \eta \not = 3k2 - 1
k2
, k \in \BbbZ +.
If this condition is not true then there is no resonance only in \zeta 3. The condition \zeta 2 < 0 is excluded
since \eta < 3.
Hence we proved the following theorem.
Theorem 2.1. Let 0 < \eta =
3
4
\Bigl( e0
e\prime
\Bigr) 2
3
< 3. Then the equation (1.3) for n = 4, v4 = 0 and H \prime
given by (2.8), which corresponds to the Coulomb equation (2.2) with xj \in \BbbR 2 and potential energy
(2.1), possesses one, two and three periodic solutions related to the following three cases
(1) \eta > 2,
(2) 0 < \eta < 2, \eta > 2, \eta \not = 3k2 - 1
k2
,
(3) \eta < 1, \eta \not = k2 - 3
k2 - 1
,
k2 - 1
k2
.
These solutions and their periods \tau j(cj) are holomorphic functions at the origin in the parameters
cj , j = 1, 2, 3. The first, second and third cases are characterized by \tau 1, \tau 1, \tau 2 and \tau 1, \tau 2, \tau 3,
respectively, where
\tau - 1
1 (0) = (2\pi ) - 1
\surd
m - 1u\prime , \tau - 1
2 (0) = (2\pi ) - 1
\sqrt{}
m - 1(3 - \eta )u\prime ,
\tau - 1
3 (0) = (2\pi ) - 1
\sqrt{}
m - 1(1 - \eta )u\prime .
The associated quasiperiodic solutions of (2.2) are given by (1.4) with n = 4, where \gamma , \gamma \prime , \gamma \prime \prime
correspond to (2.4) and (2.6), and are such that | xj | 2, j = 1, 2, are periodic functions.
3. Two internal fixed charges. In this section we consider the planar system of two identical
charges in the field of two positive fixed charges located at the points b1, b2 with the potential energy
U(x(2)) = e20| x1 - x2| - 1 - e0e
\prime
\sum
j,k=1,2
| xj - bk| - 1, (3.1)
where
| x| 2 = (x1j )
2 + (x2j )
2, xj = (x1j , x
2
j ) \in \BbbR 2,
bj = (b1j , b
2
j ) \in \BbbR 2, b1j = 0, b21 = - b22 = b.
The partial derivatives of this potential energy are found from
\partial
\partial x\alpha 1
U(x(2)) = - e20
x\alpha 1 - x\alpha 2
| x1 - x2| 3
+ e0e
\prime
2\sum
k=1
x\alpha 1 - b\alpha k
| x1 - bk| 3
,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1280 W. I. SKRYPNIK
\partial
\partial x\beta 2
U(x(2)) = - e20
x\beta 2 - x\beta 1
| x1 - x2| 3
+ e0e
\prime
2\sum
k=1
x\beta 2 - b\beta k
| x2 - bk| 3
.
The equilibrium is determined by x011 = a, x012 = - a, x021 = 0, x022 = 0 equating to zero the
right-hand sides of these equalities. This gives the equilibrium relation between e0, e
\prime , a, b the same
as in the previous section.
The second derivatives of the potential energy (3.1) are given by
\partial U(x(2))
\partial x\alpha 1\partial x
\beta
2
=
\partial U(x(2))
\partial x\beta 2\partial x
\alpha
1
= e20
\Biggl[
\delta \alpha ,\beta
| x1 - x2| 3
- 3
(x\alpha 1 - x\alpha 2 )(x
\beta
1 - x\beta 2 )
| x1 - x2| 5
\Biggr]
, \alpha , \beta = 1, 2, . . . ,
and
\partial 2U(x(2))
\partial x\beta j \partial x
\alpha
j
= -
e20\delta \alpha ,\beta
| x1 - x2| 3
+
+ e0e
\prime
2\sum
k=1
\Biggl[
\delta \alpha ,\beta
| xj - bk| 3
- 3
(x\alpha j - b\alpha k )(x
\beta
j - b\beta k)
| xj - bk| 5
\Biggr]
+ 3e20
(x\alpha 1 - x\alpha 2 )(x
\beta
1 - x\beta 2 )
| x1 - x2| 5
.
Let u, u\prime , u\ast , u\prime \ast , be the same as in the previous section,
u\prime \prime \ast =
6e0e
\prime b2\Bigl( \surd
a2 + b2
\Bigr) 5 =
6e0e
\prime
(2a)5
\Bigl( e0
e\prime
\Bigr) 5
3 3 - \eta
\eta
a2 = u\prime (3 - \eta )
and
Tj(\alpha , \beta ) =
2\sum
k=1
(x\alpha j - b\alpha k )(x
\beta
j - b\beta k)
| xj - bk| 5
.
Let T 0
j (\alpha , \beta ) be the equilibrium value of T 0
j (\alpha , \beta ). Then (| x0j - bk| 2 = a2 + b2)
T 0
j (\alpha , \beta ) = 2(a2 + b2) -
5
2 \delta \alpha ,\beta (a
2\delta \alpha ,1 + b2\delta \alpha ,2).
Indeed, let
\~T 0
j (\alpha , \beta ) =
2\sum
k=1
(x0\alpha j - b\alpha k )(x
0\beta
j - b\beta k),
then
\~T 0
1 (1, 2) = - ((a - b11)b
2
1 + (a - b12)b
2
2) = - (ab - ab) = 0,
\~T 0
2 (1, 2) = - (( - a - b11)b
2
1 + ( - a - b12)b
2
2) = ab - ab = 0,
\~T 0
1 (1, 1) = (a - b11)(a - b11) + (a - b12)(a - b12) = 2a2,
\~T 0
2 (1, 1) = ( - a - b11)( - a - b11) + ( - a - b12)( - a - b12) = 2a2,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE CHARGES IN THE FIELD 1281
\~T 0
1 (2, 2) =
\~T 0
2 (2, 2) = (b21)
2 + (b22)
2 = 2b2.
As a result
U0
1,\alpha ;1,\beta = U0
2,\alpha ;2,\beta = \delta \alpha ,\beta
\biggl(
e20
(2a)3
- 6e0e
\prime
(
\surd
a2 + b2)5
(a2\delta \alpha ,1 + b2\delta \alpha ,2) + 3
e20
(2a)3
\delta \alpha ,1
\biggr)
=
= \delta \alpha ,\beta
\biggl(
u\prime
2
- \delta \alpha ,1u
\prime
\ast - \delta \alpha ,2u
\prime \prime
\ast
\biggr)
,
U0
1,\alpha ;2,\beta = U0
2,\alpha ;1,\beta =
u\prime
2
\delta \alpha ,\beta (1 - 3\delta \alpha ,1).
Let us re-numerate indexes as in the previous section. Then
U0 =
\left( u\prime
2
- u\prime \ast - u\prime
- u\prime
u\prime
2
- u\prime \ast
\right) \oplus
\left( u\prime
2
- u\prime \prime \ast
u\prime
2
u\prime
2
u\prime
2
- u\prime \prime \ast
\right) .
The characteristic polynomial of U0 looks like
p(\lambda ) = \mathrm{D}\mathrm{e}\mathrm{t}( - U0 + \lambda I) =
\Biggl( \biggl(
u\prime
2
- u\prime \ast - \lambda
\biggr) 2
- u\prime 2
\Biggr) \Biggl( \biggl(
u\prime
2
- u\prime \prime \ast - \lambda
\biggr) 2
- u\prime 2
4
\Biggr)
.
The roots \zeta j of this polynomial are given by
\zeta 1 = - u\prime
2
- u\prime \ast , \zeta 2 =
3u\prime
2
- u\prime \ast , \zeta 3 = u\prime - u\prime \prime \ast , \zeta 4 = - u\prime \prime \ast = - u\prime (3 - \eta )
or
\zeta 1 = u\prime - u\ast = u\prime (1 - \eta ), \zeta 2 = 3u\prime - u\ast = u\prime (3 - \eta ),
\zeta 3 = u\prime (\eta - 2), \zeta 4 = - u\prime (3 - \eta ) < 0.
At the interval \eta \in (1, 2) only one eigenvalue \zeta 2 is positive. At the interval (0, 1) \zeta 2 > \zeta 1 > 0
and \zeta 3 < 0. At the interval
\biggl(
2,
5
2
\biggr)
\zeta 2 > \zeta 3 > 0 but at the interval
\biggl(
5
2
, 3
\biggr)
0 < \zeta 2 < \zeta 3.
Hence the following theorem follows from the center Lyapunov theorem.
Theorem 3.1. Let \eta \not = 1, 2 belong to (0, 3)\setminus
\biggl[
5
2
, 3
\biggr]
, or
\biggl(
5
2
, 3
\biggr)
. Then the planar Coulomb
equation (2.2) with the potential energy з (3.1) possesses a periodic solution which depends on a
real parameter c. This solution and its period \tau (c) are real analytical functions at the origin in this
parameter and \tau (0) = 2\pi
\sqrt{}
m
\zeta 2
or \tau (0) = 2\pi
\sqrt{}
m
\zeta 3
.
4. Space dynamics. For two identical negative charges in \BbbR 3 in the field of two positive fixed
charges located at the points b1, b2 the Coulomb potential energy is given by
U(x(2)) = e20| x1 - x2| - 1 - e0e
\prime
\sum
j,k=1,2
| xj - bk| - 1, (4.1)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1282 W. I. SKRYPNIK
where
| x| 2 = (x1j )
2 + (x2j )
2 + (x3j )
2, xj = (x1j , x
2
j , x
3
j ) \in \BbbR 3,
bj = (b1j , b
2
j , b
3
j ) \in \BbbR 3, b1j = b3j = 0, b21 = - b22 = b.
The partial derivatives of the potential energy can be taken from the previous section with the extended
condition \alpha , \beta = 1, 2, 3. The equilibrium is determined by
x011 = a, x012 = - a, x0\alpha 1 = 0, x0\alpha 2 = 0, \alpha = 2, 3,
and the equilibrium relation has to be taken from the previous section. It is not difficult to check that
the matrix U0 of the second partial derivatives of the potential energy (4.1) at the equilibrium looks
like
U0
1,\alpha ;1,\beta = U0
2,\alpha ;2,\beta = \delta \alpha ,\beta
\biggl(
u\prime
2
- \delta \alpha ,1u
\prime
\ast - \delta \alpha ,2u
\prime \prime
\ast
\biggr)
, \alpha , \beta = 1, 2, 3,
U0
1,\alpha ;2,\beta = U0
2,\alpha ;1,\beta =
u\prime
2
\delta \alpha ,\beta (1 - 3\delta \alpha ,1).
Let’s re-numerate the indexes in the following way
(1, 1) = 1, (2, 1) = 2, (3, 1) = 3, (1, 2) = 4, (2, 2) = 5, (2, 3) = 6.
This implies that
U0 =
\left( u\prime
2
- u\prime \ast - u\prime
- u\prime
u\prime
2
- u\prime \ast
\right) \oplus
\left( u\prime
2
- u\prime \prime \ast
u\prime
2
u\prime
2
u\prime
2
- u\prime \prime \ast
\right) \oplus
\left( u\prime
2
u\prime
2
u\prime
2
u\prime
2
\right) .
The characteristic polynomial of U0 is given by
p(\lambda ) = \mathrm{D}\mathrm{e}\mathrm{t}( - U0 + \lambda I) =
=
\Biggl( \biggl(
u\prime
2
- u\prime \ast - \lambda
\biggr) 2
- u\prime 2
\Biggr) \Biggl( \biggl(
u\prime
2
- u\prime \prime \ast - \lambda
\biggr) 2
- u\prime 2
4
\Biggr) \Biggl( \biggl(
u\prime
2
- \lambda
\biggr) 2
-
\biggl(
u\prime
2
\biggr) 2\Biggr)
.
Its roots look like
\zeta 1 = u\prime - u\ast , \zeta 2 = 3u\prime - u\ast , \zeta 3 = u\prime - u\prime \prime \ast , \zeta 4 = - u\prime \prime \ast , \zeta 5 = u\prime , \zeta 6 = 0,
that is
\zeta 1 = u\prime (1 - \eta ), \zeta 2 = u\prime (3 - \eta ), \zeta 3 = u\prime (\eta - 2),
\zeta 4 = - u\prime (3 - \eta ), \zeta 5 = u\prime , \zeta 6 = 0.
Now let’s describe the Q-canonical transformation. For that purpose it is more convenient to use the
following numeration of variables
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE CHARGES IN THE FIELD 1283
x21 = x1, x22 = x2 x11 = x3, x31 = x4, x12 = x5, x32 = x6, (4.2)
and the same for momenta. Then the angular moment (the integral of motion corresponding to
rotation in the (1.3)-plane) is given by
Q = -
2\sum
j=1
(x1jp
3
j - x3jp
1
j ) =
3\sum
j=2
(x2jp2j - 1 - x2j - 1p2j),
The generating function w(u(6), p(6)) is given by
w =
6\sum
j=3
gk(u3, . . . u6)pk + u1p1 + u2p2.
(1.2) gives
6\sum
j=3
\partial gk
\partial u6
pk =
3\sum
j=2
(g2jp2j - 1 - g2j - 1p2j)
which results in (2.5) with k = 2, 3.
Repeating arguments from the second section we derive the expressions for gk, xk, pk, uk, vk,
k > 2, translating lower indexes of the variables by 2. In particular
x23 + x24 = u23, x25 + x26 = u24 + u25,
x1 = u1, x2 = u2, x3 = u3c, x4 = - u3s, x5 = u4c+ u5s, x6 = - u4s+ u5c,
(4.3)
where c = \mathrm{c}\mathrm{o}\mathrm{s}u6, s = \mathrm{s}\mathrm{i}\mathrm{n}u6.
The new equilibrium is determined by the equalities u3 = - u4 = a, uj = 0, j \not = 3,4, vj = 0. It
is not difficult to check that the (6 \times 6) matrix W is the direct sum of the (2 \times 2) unit matrix and
the 4\times 4 matrix W from the second section. This shows that \mathrm{D}\mathrm{e}\mathrm{t}W = - u3. The new Hamiltonian
i given by
H \prime = (2m) - 1
5\sum
j=1
v2j + (2mu23)
- 1(u5v4 - u4v5 - v6)
2 + U \prime (u(5)), v6 = Q,
(4.4)
U \prime (u(5)) = e20((u2 - u1)
2 + (u4 - u3)
2 + u25)
- 1
2 - e0e
\prime [(u23 + (u1 - b)2) -
1
2+
+(u23 + (u1 + b)2) -
1
2 + (u24 + u25 + (u2 - b)2) -
1
2 +
\Bigl[
(u24 + u25 + (u2 + b)2) -
1
2
\Bigr]
.
Now let us find periodic solutions of the equation of motion (1.3) and the associated solutions of
(2.2) taking into account that the zero eigenvalue of the canonical matrix of H is doubly degenerate.
For that we have to exclude resonances between the eigenvalues \zeta j of U0.
At the intervals \eta \in (0, 2), \eta \in (2, 3), \zeta 2 > \zeta 5, \zeta 5 > \zeta 2, respectively. At the interval
\biggl(
2,
5
2
\biggr)
\zeta 5 > \zeta 2 > \zeta 3 > 0 and at the interval
\biggl(
5
2
, 3
\biggr)
0 < \zeta 2 < \zeta 3 < \zeta 5. That is there are no resonances in
\zeta 2, \zeta 5 (see Section 2) and \zeta 5 at \eta \in (0, 2) and \eta \in (2, 3), respectively, since \zeta 3, \zeta 4 are negative at
\eta \in (0, 2). This condition also shows that at the interval \eta \in (0, 1) the nonresonance condition may
be taken from the Theorem 2.1. Now we can prove the following theorem which follows from the
Lyapunov center theorem for (1.3).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1284 W. I. SKRYPNIK
Theorem 4.1. Let 0 < \eta =
3
4
\Bigl( e0
e\prime
\Bigr) 2
3
< 3. Then the Coulomb equation of motion (1.3) for
n = 6, v6 = 0 and H \prime given by (4.4), which corresponds to (2.2) with xj \in \BbbR 3 and potential energy
(4.1), possesses one, two and three periodic solutions related to the following three cases
(1) \eta > 2,
(2) 0 < \eta < 2,
(3) \eta < 1, \eta \not = k2 - 3
k2 - 1
,
k2 - 1
k2
.
These solutions and their periods \tau j(cj) are holomorphic functions at the origin in the parameters
cj , j = 1, 2, 3. The first, second and third cases are characterized by \tau 1, \tau 1, \tau 2 and \tau 1, \tau 2,
\tau 3, respectively, where \tau - 1
1 (0) = (2\pi ) - 1
\surd
m - 1u\prime , \tau - 1
2 (0) = (2\pi ) - 1
\sqrt{}
m - 1(3 - \eta )u\prime , \tau - 1
3 (0) =
= (2\pi ) - 1
\sqrt{}
m - 1(1 - \eta )u\prime . The associated quasiperiodic solutions of (2.2) are given by (1.4) with
n = 6, where \gamma , \gamma \prime , \gamma \prime \prime correspond to (4.2), (4.3), and are such that | xj | 2, j = 1, 2, are periodic
functions.
5. Appendix. Here we prove the third and second statements of Theorem 1.1. We deal with the
Hamiltonian H in \BbbR 2n which is a holomorphic function at its equilibrium q0
H(q) =
1
2
(h0(q - q0), (q - q0)) + . . . , q = (x; p), x \in \BbbR n, p \in \BbbR n,
where (.,.) is the Euclidean scalar product in \BbbR 2n, h0 is a symmetric matrix and the three dots imply
higher power terms in qj in the Taylor expansion. The canonical matrix Jh0 is found from the linear
part of the equation of motion \.q = J\partial H, where
J =
\Biggl(
0 - I
I 0
\Biggr)
.
I is the n \times n unit matrix and \partial is the vector of first partial derivatives. The direct and inverse
canonical transformations of q(n) = (x, p)(n) into q\prime (n) = (x\prime , p\prime )(n) are given by
q\prime j - q\prime 0j =
2n\sum
k=1
M - 1
j,k (qk - q0k) + . . . , qj - q0j =
2n\sum
k=1
Mj,k(q
\prime
k - q\prime 0k ) + . . . ,
where Mj,k is an invertible matrix of the linear symplectic transformation in \BbbR 2n and q\prime 0 is the new
equilibrium. If M\ast j,k = Mk,j then (see Sections 2 and 15 in [2])
H \prime (q\prime ) =
1
2
(h\prime 0(q\prime - q\prime 0), q\prime - q\prime 0) + . . . ,
where
h\prime 0 = M\ast h
0M, M\ast JM = J, J = - J - 1
and J determines the symplectic structure in \BbbR 2n. This yields
- \mathrm{D}\mathrm{e}\mathrm{t}(\lambda I - Jh0) = \mathrm{D}\mathrm{e}\mathrm{t}(\lambda J + h0) = (\mathrm{D}\mathrm{e}\mathrm{t}M) - 2\mathrm{D}\mathrm{e}\mathrm{t}(\lambda J + h\prime 0).
That is the characteristic polynomial of the transformed Hamiltonian has the same roots as the initial
Hamiltonian.
Statement 2 follows from \.vn = 0, the fact that the characteristic polynomial of a canonical
matrix is an even function in the spectral parameter \lambda (see section 15 in [2]) and the fact that the
characteristic polynomial of Jh0 is proportional to \lambda 2.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
COULOMB DYNAMICS NEAR EQUILIBRIUM OF TWO EQUAL NEGATIVE CHARGES IN THE FIELD 1285
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Received 20.06.14,
after revision — 24.05.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
|
| id | umjimathkievua-article-1920 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:14Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/12/7ee9afb26e07d10fecd71d2e8f295712.pdf |
| spelling | umjimathkievua-article-19202019-12-05T09:31:35Z Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges Skrypnik, W. I. Скрипник, В. І. Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two fixed and equal positive charges are found with the help of the Lyapunov center theorem. Знайдено перiодичнi та квазiперiодичнi розв’язки рiвнянь руху Кулона двох рiвних негативних зарядiв у полi фiксованих двох рiвних позитивних зарядiв iз допомогою центральної теореми Ляпунова. Institute of Mathematics, NAS of Ukraine 2016-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1920 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 9 (2016); 1273-1285 Український математичний журнал; Том 68 № 9 (2016); 1273-1285 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1920/902 Copyright (c) 2016 Skrypnik W. I. |
| spellingShingle | Skrypnik, W. I. Скрипник, В. І. Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges |
| title | Coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| title_alt | Coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| title_full | Coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| title_fullStr | Coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| title_full_unstemmed | Coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| title_short | Coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| title_sort | coulomb dynamics near equilibrium of two equal negative charges in the field
of fixed two equal positive charges |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1920 |
| work_keys_str_mv | AT skrypnikwi coulombdynamicsnearequilibriumoftwoequalnegativechargesinthefieldoffixedtwoequalpositivecharges AT skripnikví coulombdynamicsnearequilibriumoftwoequalnegativechargesinthefieldoffixedtwoequalpositivecharges |