Mechanical systems with singular equilibria and the Coulomb dynamics of three charges
We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507377975951360 |
|---|---|
| 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 | 2019-12-05T09:19:04Z |
| description | We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria
have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium
states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation
of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three
point charges with singular equilibrium in a line. |
| first_indexed | 2026-03-24T02:08:21Z |
| format | Article |
| fulltext |
UDC 517.9
W. I. Skrypnik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA
AND THE COULOMB DYNAMICS OF THREE CHARGES
МЕХАНIЧНI СИСТЕМИ З СИНГУЛЯРНИМИ РIВНОВАГАМИ
ТА КУЛОНIВСЬКА ДИНАМIКА ТРЬОХ ЗАРЯДIВ
We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria
have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium
states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation
of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three
point charges with singular equilibrium in a line.
Розглядаються механiчнi системи, матрицi других похiдних потенцiальних енергiй яких у рiвновазi мають нульо-
вi власнi значення. Припускається, що їхнi потенцiальнi енергiї є голоморфними функцiями в цих сингулярних
рiвновагах. Для таких систем доведено iснування власних обмежених для додатного часу розв’язкiв ньютонiв-
ських рiвнянь руху, якi збiгаються до рiвноваги в границi нескiнченного часу. Цi результати застосовуються до
кулонiвських систем трьох зарядiв iз сингулярною рiвновагою на прямiй.
1. Introduction and main result. We consider n-dimensional systems with a potential energy U
which is singular at least on a set where some coordinates coincide and has a singular equilibrium
configuration meaning that the symmetric matrix U0 of partial second derivatives of the potential
energy has zero eigenvalues at the equilibrium x0. Such systems can be derived from mechanical
systems of N d-dimensional particles (charges) interacting via singular pair or manybody potentials
after a re-numeration of variables and masses with n = dN. The Newton equation of motion of the
systems looks like
\mu j
d2xj
dt2
= -
\partial U(x(n))
\partial xj
, j = 1, . . . , n, x(n) = (x1, . . . , xn) \in \BbbR n. (1.1)
The diagonal n-dimensional matrix with the elements (effective masses) \mu j , j = 1, . . . , n, will be
called by us the mass matrix and denoted by M. We assume that U is a holomorphic function in an
equilibrium neighborhood.
The motivation to consider such the systems comes from the Coulomb system of three charges
e1 = e2 = - e0 < 0, e3 =
e0
4
which has a singular equilibrium on a line with an equal distance a of
the positive charge to the negative ones. We show this in the last section of this paper.
Our aim is to find solutions of the Newton equation for the considered systems on the infinite
time interval. Not much is known about solutions on the infinite time interval for three-dimensional
Coulomb systems except the systems of two opposite charges and a charge in the field of many at-
tractive centers. Such the solutions were found for the simplest line Coulomb systems with equilibria
[1] and a planar system of n - 1 equal negative charges and a positive charge [2]. The existence of
the Coulomb dynamics without collisions of charges on a finite time interval has been proven in [3]
(see also [4]).
Instability of equilibrium in Coulomb systems is known from the Earnshaw theorem [5, 6]. This
fact and the inverse Lagrange – Dirichlet theorem imply that the Coulomb potential energy does not
attain an absolute minimum at it and U0 does not have only positive eigenvalues.
c\bigcirc W. I. SKRYPNIK, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4 519
520 W. I. SKRYPNIK
Existence of the zero eigenvalue of U0 does not allow one to apply the results concerning
the existence of periodic solutions constructed in the Lyapunov (resonance) center theorem and its
(nonresonance) generalizations proposed in [7, 8]. Singularity of U does not allow one also to apply
the results concerning the existence of (proper) bounded at positive time solutions converging to the
equilibrium in the limit of infinite time [9].
It is known from the celestial mechanics [10] that the zero eigenvalue of a linear part of a vector
field of an ordinary differential equation is generated by integral of motions. There is a procedure
of lowering of its degeneracy degree by a separation of cyclic variables with the help of a canonical
transformation (see the paragraph Application to Lagrange solutions in [10]). But it is not known
whether the degeneracy of the zero eigenvalue of U0 is generated exclusively by the integrals of
motion. Besides it is difficult to find them all.
In this paper we find the proper bounded solutions relying on a modification of the Siegel semi-
linearization technique (see the paragraph Lyapunov theorem in [10]). This technique is applied to
obtain partial solutions of an ordinary differential equation represented in a simple standard form
in which a linear part f0 of its vector field f is given by a diagonal matrix as in the case of the
Poincare linearization theorem [11]. The Siegel technique allows one to linearize in new variables
(at a linear invariant manifold) only a part of the many-component equation demanding a resonance
condition between eigenvalues of f0 with negative real parts to be satisfied. If the linear part of the
second order equation has the zero eigenvalue then one can not reduce it to the simple standard form.
In our version of the Siegel technique we start from another standard form of the Newton equation
which allows some variables satisfy second order equations. Then we introduce new variables with
the help of an unknown function \varphi such that the invariant manifold of the equation is given by the
zero values of some of the new variables and at it the remaining variables satisfy the new equation in
which the diagonal linear part of the vector field have negative eigenvalues. A resonance condition
is not needed since it is solved on the infinite time interval with the help of the Lyapunov theorem
[12, 13]. Finally we prove with the help the majorant method that \varphi , which satisfies a resolvent type
equation, is a vector valued holomorphic function at a neighborhood of the origin.
Our main results are formulated in Theorems 1.1 and 1.2. The first theorem was utilized by
us in [1] in a weaker version demanding eigenvalues of M - 1U0 not to be zero and its negative
eigenvalues satisfy a resonance condition.
Theorem 1.1. Let M be the mass matrix and U0 be the symmetric matrix of second derivatives
at an equilibrium x0 of a potential energy U of an n-dimensional mechanical system. Let also U be
a holomorphic function in a neighborhood of x0 and the matrix M - 1U0 have p negative eigenvalues
\sigma j , j = 1, . . . , p. Then the Newton equation of motion of this mechanical system admits a bounded
at positive time solution depending on p real parameters which is real analytic function in them in a
neighborhood of the origin and \| x - x0\| \lambda <\infty , \| \.x\| \lambda <\infty , where \.x is the velocity and
\| x\| \lambda = \mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0
\mathrm{m}\mathrm{a}\mathrm{x}
s\in (1,...,n)
e\lambda t| xs(t)| , \lambda < \lambda 0 = \mathrm{m}\mathrm{i}\mathrm{n}
j=1,...,p
\sqrt{}
- \sigma j .
We show in the last section that for the mentioned system of three charges the eigenvalues of the
matrix M - 1U0 are determined explicitly. In the planar (three-dimensional) systems this matrix has
four (six) times degenerate zero, negative and positive(doubly degenerate) eigenvalues. For the line
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 521
system it has only one negative and doubly degenerate zero eigenvalues. Such the eigenvalues and
Theorem 1.1 imply the following result.
Theorem 1.2. The Newton – Coulomb equation of motion of the three point charges e1 = - e0,
e2 = - e0, e3 =
e0
4
> 0 with masses mj , j = 1, 2, 3, admits in the line, planar and three-
dimensional systems a bounded at positive time solution which is a real analytic function in a
neighborhood of the origin in one real parameter such that \| x - x0\| \lambda < \infty , \| \.x\| \lambda < \infty and
\lambda < \lambda 0, \lambda
2
0 = e20(4a
3) - 1(m - 1
1 +m - 1
2 + 4m - 1
3 ), where x0 is an equilibrium x011 = - a, x012 = a,
x013 = 0, x0\alpha j = 0, \alpha = 2, 3.
Note that due to the equality
\surd
MM - 1U0
\bigl( \surd
M
\bigr) - 1
=
\bigl( \surd
M
\bigr) - 1
U0
\bigl( \surd
M
\bigr) - 1
the matrix M - 1U0
has the same spectrum as the matrix
\bigl( \surd
M
\bigr) - 1
U0
\bigl( \surd
M
\bigr) - 1
and is similar to the diagonal matrix with
real elements.
Our paper is organized as follows. In the second section we transform (1.1) into a standard form
(Proposition 2.1) and formulate Theorem 2.1 which substantially diminish the number of variables
in the transformed equation and permits to find its proper bounded at positive time solutions (Corol-
lary 2.1). We prove Theorem 2.1 in the third section. In the fourth section we find eigenvalues of
M - 1U0 (Theorem 4.1 describes them) for our system of three charges proving Theorem 1.2.
2. Standard form of Newton equation and its projection. If U0 has the zero eigenvalue, then
one can transform equation (1.1) into the standard form given in the following proposition (the star
in x\ast will mean the complex conjugation).
Proposition 2.1. Let \sigma j , j = 1, . . . , n, be the real eigenvalues of M - 1U0 such that \sigma j = 0,
j = n0 +1, . . . , n. Then the Newton equation of equation (1.1) can be mapped by a linear invertible
transformation S into the following standard form:
dxj
dt
= fj(x(l)) = \lambda jxj +Xj(x(l)), j = 1, . . . , l0, t \geq 0, (2.1)
d2xj
dt2
= X \prime
j(x(l)), j = l0 + 1, . . . , l, (2.2)
where l = n+ n0, l0 = 2n0,
\lambda j = -
\sqrt{}
- \sigma j , j = 1, . . . , n0, \lambda j =
\sqrt{}
- \sigma j , j = n0 + 1, . . . , 2n0,
Xj , X
\prime \ast
j = X \prime
j are holomorphic in the neighborhood of the origin such that in their power expansions
the sum of powers of xj is not less than two and Xj+n0 = - Xj = X\ast
j , x
\ast
j = xj+n0 , if \sigma j > 0, and
Xj = X\ast
j , x
\ast
j = xj , if \sigma j < 0.
Partial solutions of (2.1), (2.2) can be found with the help of the following theorem.
Theorem 2.1. Let real \lambda j , j = 1, . . . , p < l0, be negative, real parts of \lambda j , j = p + 1, . . . , l0,
be nonnegative and Xj , X
\prime
j be the same as in Proposition 2.1. Then there exist functions \varphi j(x(p)),
j = p + 1, . . . , l, which are holomorphic in a neighborhood of the origin and zero at it such that a
partial solution of (2.1), (2.2) is given for j = p+ 1, . . . , l by
xj(t) = \varphi j
\bigl(
x(p)(t)
\bigr)
,
and xj(t) for j = 1, . . . , p, satisfy the projected evolution equation
dxj
dt
= f0j (x(p)) = \lambda jxj +X0
j (x(p)), (2.3)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
522 W. I. SKRYPNIK
where
X0
j (x(p)) = Xj
\bigl(
x(p), \varphi (l\setminus p)(x(p))
\bigr)
, (l\setminus p) = p+ 1, . . . , l,
are real functions and \varphi j have the properties of Xj , X
\prime
j if (2.1), (2.2) corresponds to (1.1).
The solution of the projected evolution equation is obtained with the help of the well-known
first global Lyapunov theorem [12, 13] a well known generalization of which is formulated in [1]
(Theorem 6.2). Hence the following result is valid.
Corollary 2.1. Let the conditions of Theorem 2.1 be satisfied. Then there exists a partial solution
of (2.1), (2.2) depending on p parameters which coincide with the initial values of the variables in
(2.3). This solution is a holomorphic function in these parameters in a neighborhood of the origin
and \| x\| \lambda <\infty , where
\| x\| \lambda = \mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0
\mathrm{m}\mathrm{a}\mathrm{x}
s\in (1,...,l)
e\lambda t| xs(t)| , \lambda < \lambda 0 = \mathrm{m}\mathrm{i}\mathrm{n}
j=1,...,p
| \lambda j | ,
and determines real solutions of (1.1).
The reality of the solutions follows from the fact that they are expressed as real linear com-
binations of the variables x\prime j+n0
+ x\prime j ,
\surd - \sigma j
\bigl(
x\prime j+n0
- x\prime j
\bigr)
which are real and x\prime j coincides with
the solution of (2.1), (2.2) corresponding to (1.1). Here one have to take into account the equality
S - 1 = \~S - 1(S0) - 1 determined below. This corollary and Proposition 2.1 prove Theorem 1.1.
Proof of Proposition 2.1. We assume that the potential energy U has the equilibrium at the
point x0 =
\bigl(
x0j
\bigr)
, j = 1, . . . , n, at a neighborhood of which it is holomorphic, that is
\biggl(
\partial U
\partial xj
\biggr)
(x0) = 0.
Then in the new variables xj - x0j the dynamic equation is rewritten as
\mu j
d2xj
dt2
= -
\partial U \prime (x(n))
\partial xj
, (2.4)
where
U \prime (x(n)) = U
\bigl(
x1 + x01, . . . , xn + x0n
\bigr)
,
\biggl(
\partial U \prime
\partial xj
\biggr)
(0) = 0.
By an invertible linear transformation \~xj =
\sum n
k=1
\~Sj,kxk one diagonalizes M - 1U0, which has
eigenvalues \sigma j , that is \delta j,k\sigma j =
\bigl(
\~SM - 1U0 \~S - 1
\bigr)
j,k
and transforms (2.4) into (we omit tilde in
variables)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 523
d2xj
dt2
= - \sigma jxj + Fj(x(n)), (2.5)
where
Fj(x(n)) = -
n\sum
k=1
\~Sj,k\mu
- 1
k
\biggl(
\partial U \prime \prime
\partial xk
\biggr) \bigl(
( \~S - 1x)(n)
\bigr)
,
U \prime \prime (x(n)) = U \prime (x(n)) -
1
2
n\sum
j,l=1
U0
j,lxjxl.
That is
dxj
dt
= vj ,
dvj
dt
= - \sigma jxj + Fj(x(n)). (2.6)
Then by the linear two-dimensional transformation produced by the matrix S0
j the last equation is
mapped into (2.1), (2.2) with l0 = 2n0 and - \lambda 2j - 1 = \lambda 2j =
\surd - \sigma j , j = 1, . . . , n0. The matrix
S0
j diagonalizes the two dimensional matrix Aj , which determines the linear part of (2.6), with the
zero diagonal elements and nondiagonal elements Aj;1,2 = 1, Aj;2,1 = - \sigma j . That is S0
jAj = \^\sigma jS
0
j ,
where \^\sigma j is a diagonal matrix with the eigenvalues - \lambda 2j - 1 = \lambda 2j =
\surd - \sigma j . 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
2\kappa j
, \kappa j =
\sqrt{}
- \sigma j .
The new variables look like
x\prime 2j - 1 =
1
2
\biggl(
xj -
1
\kappa j
vj
\biggr)
, x\prime 2j =
1
2
\biggl(
xj +
1
\kappa j
vj
\biggr)
, j = 1, . . . , n0,
x\prime j = xj - n0 , j = 2n0 + 1, . . . , n+ n0.
The inverse transformation is given by
xj = x\prime 2j + x\prime 2j - 1, vj = \kappa j(x
\prime
2j - x\prime 2j - 1), j = 1, . . . , n0,
which implies that the functions Xj , X
\prime
j in (2.1), (2.2) are given by (we omit primes in variables)
X2j(x(n+n0)) = - X2j - 1(x(n+n0)) =
1
2\kappa j
Fj(x2 + x1, . . . , x2n0 + x2n0 - 1, x2n0+1, . . . , xn+n0),
where j = 1, . . . , n0 and
X \prime
j(x(n+n0)) = Fj(x2 + x1, . . . , x2n0 + x2n0 - 1, x2n0+1, . . . , xn+n0), j = 2n0 + 1, . . . , n+ n0.
That is Xj = - X\ast
j , x
\ast
2j = x2j - 1, if \sigma j > 0, and Xj = X\ast
j , x
\ast
j = xj , if \sigma j < 0. Let us use another
numeration of variables: (x1, x2, x3, . . . , x2n0) \rightarrow (x1, x3, . . . , x2n0 - 1, x2, x4, . . . , x2n0). In such a
way (2.6) is mapped into (2.1), (2.2) with
\lambda j = -
\sqrt{}
- \sigma j , j = 1, . . . , n0, \lambda j =
\sqrt{}
- \sigma j , j = n0 + 1, . . . , 2n0.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
524 W. I. SKRYPNIK
As a result Xj+n0 = - Xj = X\ast
j , x
\ast
j = xj+n0 , if \sigma j > 0, and Xj = X\ast
j , x
\ast
j = xj , if \sigma j < 0 and
S = S0 \~S, where
(S0x)j = xj , j = 2n0 + 1, . . . , n+ n0,\bigl(
(S0) - 1x
\bigr)
j
= xj + xj+n0 , j = 1, . . . , n0,\bigl(
(S0) - 1x
\bigr)
j+n0
=
\sqrt{}
- \sigma j(xj+n0 - xj), j = 1, . . . , n0.
Proposition 2.1 is proved.
3. Proof of Theorem 2.1. To prove Theorem 2.1 we introduce at first the new variables uk
inspired by [10]
uj = xj - \varphi j(x(p)), j = 1, . . . , l,
and
uj = xj , \varphi j(x(p)) = 0, j \leq p.
Here the functions \varphi j are given by a power expansion in xn1
1 . . . x
np
1 with n1 + . . . + np > 1 and
coefficients Pj;n1,...,np , j = p + 1, . . . , l0, P
\prime
j;n1,...,np
, j = l0 + 1, . . . , l. They will be real if j
correspond to real \lambda j . The former variables are expressed in terms of the new ones as follows:
xj = uj + \varphi j(u(p)).
The new variables obey the following equations:
duj
dt
= \lambda juj +Gj(u(l)), j = 1, . . . , l0, (3.1)
d2uj
dt2
= G\prime
j(u(l)), j = l0 + 1, . . . , l. (3.2)
If one shows that the equalities
Gj(u(p), 0, . . . , 0) = 0, j = p+ 1, . . . , l0, G\prime
j(u(p), 0, . . . , 0) = 0, j = l0 + 1, . . . , l, (3.3)
are true then a partial solution of (2.1), (2.2) is given by (2.3) since
uj = 0, j = p+ 1, . . . , l,
is a partial solution of (3.1), (3.2). This will prove the theorem if \varphi j is a holomorphic function at the
origin. Now we shall prove this character of \varphi j . Let
\varphi jxk
=
\partial \varphi j
\partial xk
, \varphi jxkxr =
\partial 2\varphi j
\partial xk\partial xr
.
Then
Gj = Xj + \lambda j\varphi j -
p\sum
k=1
fk\varphi jxk
,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 525
G\prime
j = X \prime
j -
p\sum
r,k=1
(fk\varphi jxkxr + \varphi jxk
fkxr)fr.
The first and second equations in (3.3) will be called the first and second structure equations. The
second structure equation is rewritten as follows:
p\sum
r,k=1
xrxk\lambda r\lambda k\varphi jxkxr +
p\sum
k=0
\varphi jxk
\lambda 2kxk = X \prime
j -
p\sum
r,k=1
\bigl[
Xr(Xk + 2\lambda kxk)\varphi jxkxr+
+\varphi jxk
(XkxrXr +Xkxrxr\lambda r + \lambda k\delta k,rXr)
\bigr]
,
where
xk = \varphi k(x(p)), k = p+ 1, . . . , l, (3.4)
and \delta k,r has the (unit) non-zero value only for k = r. This equation is reduced to the following
recursion relation for the coefficients in the expansion of powers of variables (the sum of their
powers exceeds unity): \left[ \Biggl( p\sum
k=1
nk\lambda k
\Biggr) 2
+
p\sum
k=1
nk\lambda
2
k
\right] P \prime
j;n1,...,np
= \Gamma \prime
j;n1,...,np
that is \Gamma \prime
j;n1,...,np
is expressed in terms of P \prime
j;n\prime
1,...,n
\prime
p
with n\prime 1 + . . .+ n\prime p < n1 + . . .+ np. It is easily
solved since the real parts of both terms in the square brackets are not zero due to the condition\Biggl(
p\sum
k=1
nk\lambda k
\Biggr) 2
+
p\sum
k=1
nk\lambda
2
k \geq \lambda -
\left( \Biggl( p\sum
k=1
nk
\Biggr) 2
+
p\sum
k=1
nk)
\right) , \lambda - = \mathrm{m}\mathrm{i}\mathrm{n}
j
\lambda 2j , (3.4a)
and the expansion for \varphi j , j = l0 + 1, . . . , l, is found. Now we have to prove its convergence with
the help of the majorant technique.
We will use the majorant inequality f << g which means that in the power expansion for g the
coefficients are nonnegative and exceed absolute values of the coefficients in the power expansion
for f. Let \varphi +
j be the power expansion with the coefficients
\bigm| \bigm| P \prime
j;n1,...,np
\bigm| \bigm| , that is
\varphi j << \varphi +
j .
Let
Xj <<
c3X
2
1 - c1X
= \chi , X \prime
j << \chi , X = x1 + . . .+ xl. (3.5)
Then the rewritten second structure equation yields
p\sum
r,k=1
xrxk\lambda r\lambda k\varphi
+
jxkxr
+
p\sum
k=1
\varphi +
jxk
\lambda 2kxk << \chi +
p\sum
r,k=1
\bigl[
\chi (\chi + 2\lambda +xk)\varphi
+
jxkxr
+
+\varphi +
jxk
(\chi xr\chi + \chi xrxr\lambda + + \lambda +\delta k,r\chi )
\bigr]
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
526 W. I. SKRYPNIK
From (3.4a) we obtain
\lambda -
\left( p\sum
r,k=1
xrxk\varphi
+
jxkxr
+
p\sum
k=1
\varphi +
jxk
xk
\right) <<
p\sum
r,k=1
xrxk\lambda r\lambda k\varphi
+
jxkxr
+
p\sum
k=1
\varphi +
jxk
\lambda 2kxk.
The last two inequalities yield
\lambda -
\left( p\sum
r,k=1
xrxk\varphi
+
jxkxr
+
p\sum
k=1
\varphi +
jxk
xk
\right) << \chi +
p\sum
r,k=1
\bigl[
\chi (\chi + 2\lambda +xk)\varphi
+
jxkxr
+
+ \varphi +
jxk
(\chi xr\chi + \chi xrxr\lambda + + \lambda +\delta k,r\chi )
\bigr]
.
We have also
\varphi +
j << \varphi \ast j ,
where
\lambda -
\left( p\sum
r,k=1
xrxk\varphi \ast jxkxr +
p\sum
k=1
\varphi \ast jxk
xk
\right) = \chi +
p\sum
r,k=1
\bigl[
\chi (\chi + 2\lambda +xk)\varphi \ast jxkxr+
+ \varphi \ast jxk
(\chi xr\chi + \chi xrxr\lambda + + \lambda +\delta k,r\chi )
\bigr]
. (3.6)
Now we have to prove that the solutions of this majorized second structure equation is a holomorphic
function. We seek the solutions of the last equation in the form
\varphi \ast j = \psi (x), \varphi \ast jxkxr = \psi xx, \varphi \ast jxk
= \psi x, x = x1 + . . .+ xp.
The right-hand side of the majorized second structure equation is given by
\chi + p2\chi (\chi \psi xx + \psi x\chi x) + p\lambda +
\bigl[
(x\chi x + \chi )\psi x + 2x\chi \psi xx
\bigr]
.
Taking into account that
\chi x = (1 + p\prime \psi x)\chi
\prime , \chi \prime (y) = \partial \chi (y) =
2c3y
1 - c1y
+
c1c3y
2
(1 - c1y)2
, p\prime = l - p,
we see that the one-variable majorized second structure equation is derived from (3.6) and given by
\lambda - x(x\psi xx + \psi x) =
= \chi + p2\chi
\bigl[
\chi \psi xx + \psi x(1 + p\prime \psi x)\chi
\prime \bigr] + p\lambda +
\Bigl\{ \bigl[
x(1 + p\prime \psi x)\chi
\prime + \chi
\bigr]
\psi x + 2x\chi \psi xx
\Bigr\}
, (3.7)
where \chi , \chi \prime depend on x + p\prime \psi . This equation is equivalent to the recursion relation for the coef-
ficients in the power expansion for \psi (its powers exceed unity) whose coefficients are nonnegative.
Let us put x - 1\psi = \Psi . The function
\Phi (x) = \Psi (x) + 3x\Psi x(x) + x2\Psi xx(x) = \psi x + x\psi xx
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MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 527
has power expansion with nonnegative coefficients. Now we majorize the right-hand side of (3.7) in
such a way that it should depend only on \Phi and x. In order to do this one has to substitute \psi x+x\psi xx
instead of \psi x and x\psi xx in (3.7). For the term in the first square bracket one obtaines
\chi \psi xx + \psi x(1 + p\prime \psi x)\chi
\prime << (\psi x + x\psi xx)(1 + p\prime (\psi x + x\psi xx))\chi
\prime + (\psi x + x\psi xx)x
- 1\chi =
= \Phi
\bigl[
(1 + p\prime \Phi )\chi \prime + x - 1\chi
\bigr]
and the expression in the figure bracket is majorized by\bigl[
x(1 + p\prime \Phi )\chi \prime + \chi
\bigr]
\Phi + 2x\chi \Phi .
The right-hand side of (3.7) contains x as a multiplier since
\chi = x2
c3(1 + p\prime \Psi )2
1 - c1x(1 + p\prime \Psi )
<< x2
c3(1 + p\prime \Phi )2
1 - c1x(1 + p\prime \Phi )
,
\chi \prime = x
2c3(1 + p\prime \Psi )
1 - c1x(1 + p\prime \Psi )
+ x2
c1c3(1 + p\prime \Psi )2
(1 - c1x(1 + p\prime \Psi ))2
<<
<< x
2c3(1 + p\prime \Phi )
1 - c1x(1 + 2p\prime \Phi )
+ x2
c1c3(1 + p\prime \Phi )2
1 - 2c1x(1 + p\prime \Phi )
.
Due to the fact that \chi , \chi \prime are proportional to x2, x, respectively, (3.7) is majorized by the following
rational equation for \Phi \ast :
\Phi <<
xP (x,\Phi )
1 - 3c1x(1 + p\prime \Phi )
, \Phi \ast =
xP (x,\Phi \ast )
1 - 3c1x(1 + p\prime \Phi \ast )
, \Phi << \Phi \ast ,
where P is a polynomial of two complex variables. Here we used the relation
k\prod
j=1
(1 - xj)
- 1 <<
\left( 1 -
k\sum
j=1
xj
\right) - 1
.
The last equation can be rewritten as
F (x,\Phi \ast ) = \Phi - xP \prime (x,\Phi \ast ) = 0,
where P \prime is a polynomial with positive coefficients. That is \partial \ast F (0, 0) \not = 0, where \partial \ast is the
derivative in \Phi \ast . From the holomorphic implicit function theorem [13, 14] it follows that \Phi \ast (x)
is a holomorphic function at the origin with nonnegative coefficients in its power expansion. The
same is true for \psi since it is majorized by x\Phi . Hence the power expansion for \varphi j , j = l0+1, . . . , l,
is a holomorphic function at the origin in all the variables.
Now we have to show that the solution of the first structure equation is also a holomorphic
function. This equation is given by
- \lambda j\varphi j +
p\sum
k=1
\varphi jxk
\lambda kxk = Xj -
p\sum
k=1
Xk\varphi jxk
with (3.4). This equation is reduced to the recursion relation
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
528 W. I. SKRYPNIK\Biggl(
- \lambda j +
p\sum
k=0
nk\lambda k
\Biggr)
Pj;n1,...,np = \Gamma j;n1,...,np
that is \Gamma j;n1,...,np is expressed in terms of Pj;n\prime
1,...,n
\prime
p
with n\prime 1 + . . .+ n\prime p < n1 + . . .+ np. It is easily
solved and the expansion for \varphi j , j = p+1, . . . , l0, is found. Now we have to prove its convergence.
The inequality
1 < n1 + . . .+ np \leq c2
\bigm| \bigm| \bigm| \bigm| \bigm| - \lambda j +
p\sum
k=1
nk\lambda k
\bigm| \bigm| \bigm| \bigm| \bigm| ,
the first structure equation and (3.5) lead to
p\sum
k=1
\varphi +
jxk
xk << c2
\Biggl(
1 +
p\sum
k=1
\varphi +
jxk
\Biggr)
\chi ,
and the majorized first structure equation
p\sum
k=1
\varphi \ast jxk
xk = c2
\Biggl(
1 +
p\sum
k=1
\varphi \ast jxk
\Biggr)
\chi , \varphi +
j << \varphi \ast j ,
with (3.4) added, where \varphi +
j is the power expansion with the coefficients | Pj;n1,...,np | . Taking into
account the previous notation we derive the one-variable majorized first structure equation
x\psi x = c2(1 + \psi x)\chi
which determines the recursion relation for the coefficients of the power expansion for \psi . Here
\chi =
(x+ p\prime \psi )2
1 - c1(x+ p\prime \psi )
.
The power expansion for \psi has nonnegative coefficients. Let us put x - 1\psi = \Psi . Then
\Phi (x) = \Psi (x) + x\Psi x(x) = \psi x.
That is the power expansion for \Phi has nonnegative coefficients and
\Phi <<
c2x(1 + p\prime \Phi )3
1 - c1x(1 + p\prime \Phi )
, \Phi << \Phi \ast .
The final majorized first structure equation is given by
\Phi \ast =
c2x(1 + p\prime \Phi \ast )
3
1 - c1x(1 + p\prime \Phi \ast )
, \Phi << \Phi \ast .
From the holomorphic implicit function theorem it follows that \Phi \ast (x) is a holomorphic function at
the origin with nonnegative coefficients in its power expansion. The same is true for \psi since it is
majorized by x\Phi \ast . Hence the power expansion for \varphi j , j = p+ 1, . . . , l0, is a holomorphic function
at the origin in all the variables. It follows from the first equation in (3.3) that \varphi j has the same
properties as Xj described in the Proposition 2.1 if \lambda j , Xj , X
\prime
j correspond to (2.1). The reality of
X0 follows from the dependence of Xj , X
\prime
j on \varphi j + \varphi j+n0 , j = 1, . . . , n0, and reality of the latter
since \varphi \ast
j = \varphi j+n0 for a positive \sigma j ,which follows from the first equation in (3.3), and reality of both
functions for a nonpositive \sigma j .
Theorem 2.1 is proved.
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MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 529
4. Proof of Theorem 1.2. The simplest example of a mechanical system with an equilibrium is
the d-dimensional system of the three point charges e1 = - e0, e2 = - e0, e3 =
e0
4
> 0 with masses
m1,m2,m3 and the potential energy
U(x(3)) =
1
2
3\sum
j \not =k=1
ejek
| xj - xk|
, (4.1)
where x(3) \in \BbbR 3d, xj =
\bigl(
x1j , . . . , x
d
j
\bigr)
, | x| 2 = (x1)2 + . . .+ (xd)2. Its equilibrium is determined by
x011 = - a, x012 = a, x013 = 0, x0\alpha j = 0, \alpha = 2. This potential energy is a holomorphic function at
a neighborhood of the equilibrium. The case of equal masses of the one-dimensional three charges
was considered in [1], where eigenvalues of U0 were calculated.
Theorem 1.2 is proved with the help of Theorem 1.1 and following theorem.
Theorem 4.1. In the one-dimensional system M - 1U0 has the doubly degenerate zero eigen-
value and the eigenvalue -
\bigl(
m - 1
1 + m - 1
2 + 4m - 1
3
\bigr)
u\prime , u\prime =
e20
4a3
. In the two-dimensional and
three-dimensional systems M - 1U0 has the zero eigenvalue, which is four and six times degene-
rate, respectively, and the eigenvalues -
\bigl(
m - 1
1 +m - 1
2 +4m - 1
3
\bigr)
u\prime , 2 - 1
\bigl(
m - 1
1 +m - 1
2 +4m - 1
3
\bigr)
u\prime the
latter of which is doubly degenerate in the three-dimensional system.
Proof. We find eigenvalues of U0 at first for the one-dimensional case. In our calculations of
partial derivatives of the potential energy we will use the two equalities for x \in \BbbR and x \in \BbbR d,
respectively,
\partial
\partial x1
| x1 - x2| - k = - k x1 - x2
| x1 - x2| k+2
,
\partial
\partial x\alpha
\bigl( \sqrt{}
| x| 2 + b2
\bigr) - k
= - k x\alpha \bigl( \sqrt{}
| x| 2 + b2
\bigr) k+2
which gives
\partial
\partial xj
U(x(3)) = - ej
3\sum
k=1,k \not =j
ek
xj - xk
| xj - xk| 3
,
that is
\partial
\partial x1
U(x(3)) = - e20
x1 - x2
| x1 - x2| 3
+ e0e3
x1 - x3
| x1 - x3| 3
,
\partial
\partial x2
U(x(3)) = - e20
x2 - x1
| x1 - x2| 3
+ e0e3
x2 - x3
| x2 - x3| 3
,
\partial
\partial x3
U(x(3)) = e0e3[
x3 - x1
| x1 - x3| 3
+
x3 - x2
| x2 - x3| 3
].
The equality
\partial
\partial 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
\partial
\partial xj
U(x(3)) = 0, j = 1, 2.
The second derivatives of the potential energy are calculated as follows:
\partial U(x(3))
\partial xj\partial xk
=
\partial U(x(3))
\partial xk\partial xj
= - 2ejek| xj - xk| - 3, k \not = j,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
530 W. I. SKRYPNIK
\partial 2
\partial x2j
U(x(3)) = 2ej
3\sum
k=1,k \not =j
ek| xj - xk| - 3.
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\prime , U0
3,1 = U0
1,3 = U0
2,3 = U0
3,2 = 2u\prime ,
U1,1 = U2,2 = - u\prime , U3,3 = - 4u\prime .
That is
U0 = - u\prime
\left(
1 1 q
1 1 q
q q q2
\right) = u\prime U \prime , q = - 2. (4.2)
Let us put
M \prime
0(\lambda , q) =
\left(
k1 - \lambda k1 qk1
k2 k2 - \lambda qk2
qk3 qk3 q2k3 - \lambda
\right) , kj = m - 1
j .
Then
M - 1U0 - \lambda I = - u\prime M \prime
0
\biggl(
- \lambda
u\prime
, q
\biggr)
, - \mathrm{D}\mathrm{e}\mathrm{t}(M - 1U0 - \lambda I) = u\prime 3\mathrm{D}\mathrm{e}\mathrm{t}M \prime
0
\biggl(
- \lambda
u\prime
, q
\biggr)
, q = - 2,
and making the expansion of the determinant in the elements of the first row of M \prime
0 we obtain
\mathrm{D}\mathrm{e}\mathrm{t}M \prime
0(\lambda , q) =
= (k1 - \lambda )
\bigl[
(k2 - \lambda )(q2k3 - \lambda ) - q2k2k3
\bigr]
-
- k1
\bigl[
k2(q
2k3 - \lambda ) - q2k2k3] + qk1[qk2k3 - qk3(k2 - \lambda )
\bigr]
=
= (k1 - \lambda )
\bigl[
\lambda 2 - \lambda (k2 + q2k3)
\bigr]
+ \lambda k1k2 + \lambda q2k1k3 =
= \lambda
\bigl[
(k1 - \lambda )(\lambda - q2k3 - k2) + k1k2 + q2k1k3
\bigr]
.
Hence
\mathrm{D}\mathrm{e}\mathrm{t}M \prime
0(\lambda , q) = \lambda 2( - \lambda + k1 + k2 + q2k3)
and
\mathrm{D}\mathrm{e}\mathrm{t}(M - 1U0 - \lambda I) = - \lambda 2
\bigl[
\lambda + (m - 1
1 +m - 1
2 + 4m - 1
3 )u\prime
\bigr]
.
The last formula proves the theorem for the one-dimensional case.
Let us consider the two-dimensional case. For the first partial derivatives of the planar potential
energy we have
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MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 531
\partial
\partial x\alpha 1
U(x(3)) = - e20
x\alpha 1 - x\alpha 2
| x1 - x2| 3
+ e0e3
x\alpha 1 - x\alpha 3
| x1 - x3| 3
,
\partial
\partial x\alpha 2
U(x(3)) = - e20
x\alpha 2 - x\alpha 1
| x1 - x2| 3
+ e0e3
x\alpha 2 - x\alpha 3
| x2 - x3| 3
,
\partial
\partial x\alpha 3
U(x(3)) = e0e3
\biggl[
x\alpha 3 - x\alpha 1
| x1 - x3| 3
+
x\alpha 3 - x\alpha 2
| x2 - x3| 3
\biggr]
.
The last equality is zero at the equilibrium - x11 = x12 = a, x23 = x13 = x21 = x22 = 0. The first two
give the equilibrium relation e3 =
e0
4
. The second derivatives of the potential energy are given by
\partial U(x(3))
\partial x\alpha 1\partial x
\beta
2
=
\partial U(x(3))
\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,
\partial U(x(3))
\partial x\alpha k\partial x
\beta
3
=
\partial U(x(3))
\partial x\beta 3\partial x
\alpha
k
= - e0e3
\Biggl[
\delta \alpha ,\beta
| xk - x3| 3
- 3
(x\alpha k - x\alpha 3 )(x
\beta
k - x\beta 3 )
| xk - x3| 5
\Biggr]
, k, \alpha , \beta = 1, 2,
\partial 2U(x(3))
\partial x\beta j \partial x
\alpha
j
= 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]
+
+e0e3
\Biggl[
\delta \alpha ,\beta
| xj - x3| 3
- 3
(x\alpha j - x\alpha 3 )(x
\beta
j - x\beta 3 )
| xj - x3| 5
\Biggr]
, j, \alpha , \beta = 1, 2,
\partial 2U(x(3))
\partial x\beta 3\partial x
\alpha
3
= e0e3
\Biggl[
\delta \alpha ,\beta
| x1 - x3| 3
- 3
(x\alpha 1 - x\alpha 3 )(x
\beta
1 - x\beta 3 )
| x1 - x3| 5
+
+
\delta \alpha ,\beta
| x2 - x3| 3
- 3
(x\alpha 2 - x\alpha 3 )(x
\beta
2 - x\beta 3 )
| x2 - x3| 5
\Biggr]
.
For the matrix of the second derivatives at the equilibrium we derive
U0
1,\alpha ;1,\beta = U0
2,\alpha ;2,\beta = e20
\biggl[
\delta \alpha ,\beta
\biggl(
- 1
(2a)3
+
1
4a3
\biggr)
+ 3\delta \alpha ,1\delta \beta ,1
\biggl(
1
(2a)3
- 1
4a3
\biggr) \biggr]
=
=
e20
(2a)3
\delta \alpha ,\beta (1 - 3\delta \alpha ,1\delta \beta ,1) = 4 - 1U0
3,\alpha ;3,\beta ,
U0
1,\alpha ;2,\beta = U0
2,\beta ;1,\alpha =
e20
(2a)3
\delta \alpha ,\beta (1 - 3\delta \alpha ,1\delta \beta ,1),
U0
k,\alpha ;3,\beta = U0
3,\beta ;k,\alpha = - e20
4a3
\delta \alpha ,\beta (1 - 3\delta \alpha ,1\delta \beta ,1), k, \alpha , \beta = 1, 2.
Let’s introduce the numeration
(1, 1) = 1, (2, 1) = 2, (3, 1) = 3, (1, 2) = 4, (2, 2) = 5, (3, 2) = 6,
m4 = m1, m5 = m2, m6 = m3,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
532 W. I. SKRYPNIK
where the first and second numbers in the round brackets correspond to the lower and upper indices
of variables. Then U0
j,k = U0
k,j = 0, j \leq 3, k \geq 4, and
U0
1,1 = U0
2,2 = 4 - 1U0
3,3 = - 2c, U0
1,2 = - 2c, U0
1,3 = U0
2,3 = 4c, c =
u\prime
2
=
e20
(2a)3
,
U0
4,4 = U0
5,5 = 4 - 1U0
6,6 = c, U0
4,5 = c, U0
4,6 = U0
5,6 = - 2c.
This means
U0 = 2cU \prime \oplus - cU \prime ,
where U \prime is given by (4.2).
Let M \prime \prime =M \prime \oplus M \prime and M \prime be the 3\times 3 diagonal matrix with the elements m1, m2, m3. Then
M \prime \prime - 1U0 - \lambda I = - 2cM \prime
0
\biggl(
- \lambda
2c
, - 2
\biggr)
\oplus cM \prime
0
\biggl(
\lambda
c
, - 2
\biggr)
,
\mathrm{D}\mathrm{e}\mathrm{t}(M \prime \prime - 1U0 - \lambda I) = - 23c6\mathrm{D}\mathrm{e}\mathrm{t}M \prime
0
\biggl(
- \lambda
2c
, - 2
\biggr)
\mathrm{D}\mathrm{e}\mathrm{t}M \prime
0
\biggl(
\lambda
c
, - 2
\biggr)
.
From this equality and (4.3) we derive
- \mathrm{D}\mathrm{e}\mathrm{t}(M - 1U0 - \lambda I) = \lambda 4
\biggl[
- \lambda +
\bigl(
m - 1
1 +m - 1
2 + 4m - 1
3
\bigr) u\prime
2
\biggr] \bigl[
\lambda + (m - 1
1 +m - 1
2 + 4m - 1
3 )u\prime
\bigr]
.
This concludes the proof for the two-dimensional case.
Let’s consider the 3-dimensional case. Then all the formulas concerning partial derivatives of
the potential energy of this sections will be true adding the condition \alpha , \beta = 1, 2, 3. Let’s use the
following numeration of the variables indices:
(1, 3) = 7, (2, 3) = 8, (3, 3) = 9, m7 = m1, m8 = m2, m9 = m3.
It is not difficult to see that U0
j,k = U0
k,j = 0, j \leq 6, k \geq 7, and U0
7,7 = c1, U
0
8,8 = c1, U
0
9,9 = 4c1,
U0
7,8 = c1, U
0
7,9 = U0
8,9 = - 2c1. Hence
U0 = U \prime \prime \oplus - c1U \prime ,
where U \prime \prime coincides with the planar U0. Moreover
M =M \prime \prime \oplus M \prime , M - 1U0 =M \prime \prime - 1U \prime \prime \oplus - c1M \prime - 1U \prime .
As a result
- \mathrm{D}\mathrm{e}\mathrm{t}(M - 1U0 - \lambda I) = \lambda 6
\biggl[
- \lambda + (m - 1
1 +m - 1
2 + 4m - 1
3 )
u\prime
2
\biggr] 2 \bigl[
\lambda + (m - 1
1 +m - 1
2 + 4m - 1
3 )u\prime
\bigr]
.
Theorem 4.1 is proved.
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MECHANICAL SYSTEMS WITH SINGULAR EQUILIBRIA AND THE COULOMB DYNAMICS . . . 533
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Received 21.09.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
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| id | umjimathkievua-article-1573 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:21Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7d/8db153129c164748cbaa9dfceda7777d.pdf |
| spelling | umjimathkievua-article-15732019-12-05T09:19:04Z Mechanical systems with singular equilibria and the Coulomb dynamics of three charges Механiчнi системи з сингулярними рiвновагами та кулонiвська динамiка трьох зарядiв Skrypnik, W. I. Скрипник, В. І. We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three point charges with singular equilibrium in a line. Розглядаються механiчнi системи, матрицi других похiдних потенцiальних енергiй яких у рiвновазi мають нульовi власнi значення. Припускається, що їхнi потенцiальнi енергiї є голоморфними функцiями в цих сингулярних рiвновагах. Для таких систем доведено iснування власних обмежених для додатного часу розв’язкiв ньютонiвських рiвнянь руху, якi збiгаються до рiвноваги в границi нескiнченного часу. Цi результати застосовуються до кулонiвських систем трьох зарядiв iз сингулярною рiвновагою на прямiй. Institute of Mathematics, NAS of Ukraine 2018-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1573 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 4 (2018); 519-533 Український математичний журнал; Том 70 № 4 (2018); 519-533 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1573/555 Copyright (c) 2018 Skrypnik W. I. |
| spellingShingle | Skrypnik, W. I. Скрипник, В. І. Mechanical systems with singular equilibria and the Coulomb dynamics of three charges |
| title | Mechanical systems with singular equilibria and the Coulomb dynamics
of three charges |
| title_alt | Механiчнi системи з сингулярними рiвновагами
та кулонiвська динамiка трьох зарядiв |
| title_full | Mechanical systems with singular equilibria and the Coulomb dynamics
of three charges |
| title_fullStr | Mechanical systems with singular equilibria and the Coulomb dynamics
of three charges |
| title_full_unstemmed | Mechanical systems with singular equilibria and the Coulomb dynamics
of three charges |
| title_short | Mechanical systems with singular equilibria and the Coulomb dynamics
of three charges |
| title_sort | mechanical systems with singular equilibria and the coulomb dynamics
of three charges |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1573 |
| work_keys_str_mv | AT skrypnikwi mechanicalsystemswithsingularequilibriaandthecoulombdynamicsofthreecharges AT skripnikví mechanicalsystemswithsingularequilibriaandthecoulombdynamicsofthreecharges AT skrypnikwi mehaničnisistemizsingulârnimirivnovagamitakulonivsʹkadinamikatrʹohzarâdiv AT skripnikví mehaničnisistemizsingulârnimirivnovagamitakulonivsʹkadinamikatrʹohzarâdiv |