Stable relative equilibria in the system of superconductive and permanent magnetic dipoles
This paper analytically proves the existence of stable orbital motions in a system of superconductive and permanent magnetic dipoles. As opposed to a system of two permanent magnetic dipoles, that has been studied in the work of I.Tamm and V. Ginzburg in this system we have no <<problem 1/R³&g...
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irk-123456789-1120982017-01-18T03:03:00Z Stable relative equilibria in the system of superconductive and permanent magnetic dipoles Zub, S.S. Вычислительные и модельные системы This paper analytically proves the existence of stable orbital motions in a system of superconductive and permanent magnetic dipoles. As opposed to a system of two permanent magnetic dipoles, that has been studied in the work of I.Tamm and V. Ginzburg in this system we have no <<problem 1/R³>>, because of that the stability of the system becomes possible. Аналітично доведено існування стійких орбітальних рухів у системі, що складається з надпровідного та постійного магнітних диполів. На відміну від системи з двох постійних магнітних диполів, що досліджено І. Таммом та В. Гінзбургом , <<проблема 1/R³>> в даній системі не виникає, і отже стійкість стає можливою. Аналитически доказано существование устойчивых орбитальных движений в системе, состоящей из сверхпроводящего и постоянного магнитных диполей. В отличие от исследованной И. Таммом и В. Гинзбургом системы из двух постоянных магнитных диполей <<проблема 1/R³>> в данной системе не возникает, и устойчивость становится возможной. 2015 Article Stable relative equilibria in the system of superconductive and permanent magnetic dipoles / S.S. Zub // Вопросы атомной науки и техники. — 2015. — № 3. — С. 143-147. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 45.05.+x, 45.20.-d, 45.20.Jj http://dspace.nbuv.gov.ua/handle/123456789/112098 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Вычислительные и модельные системы Вычислительные и модельные системы |
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Вычислительные и модельные системы Вычислительные и модельные системы Zub, S.S. Stable relative equilibria in the system of superconductive and permanent magnetic dipoles Вопросы атомной науки и техники |
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This paper analytically proves the existence of stable orbital motions in a system of superconductive and permanent magnetic dipoles. As opposed to a system of two permanent magnetic dipoles, that has been studied in the work of I.Tamm and V. Ginzburg in this system we have no <<problem 1/R³>>, because of that the stability of the system becomes possible. |
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Zub, S.S. |
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Zub, S.S. |
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Zub, S.S. |
| title |
Stable relative equilibria in the system of superconductive and permanent magnetic dipoles |
| title_short |
Stable relative equilibria in the system of superconductive and permanent magnetic dipoles |
| title_full |
Stable relative equilibria in the system of superconductive and permanent magnetic dipoles |
| title_fullStr |
Stable relative equilibria in the system of superconductive and permanent magnetic dipoles |
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Stable relative equilibria in the system of superconductive and permanent magnetic dipoles |
| title_sort |
stable relative equilibria in the system of superconductive and permanent magnetic dipoles |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2015 |
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Вычислительные и модельные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/112098 |
| citation_txt |
Stable relative equilibria in the system of superconductive and permanent magnetic dipoles / S.S. Zub // Вопросы атомной науки и техники. — 2015. — № 3. — С. 143-147. — Бібліогр.: 12 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT zubss stablerelativeequilibriainthesystemofsuperconductiveandpermanentmagneticdipoles |
| first_indexed |
2025-07-08T03:23:32Z |
| last_indexed |
2025-07-08T03:23:32Z |
| _version_ |
1837047501122699264 |
| fulltext |
COMPUTING AND MODELLING SYSTEMS
STABLE RELATIVE EQUILIBRIA IN THE SYSTEM OF
SUPERCONDUCTIVE AND PERMANENT MAGNETIC
DIPOLES
S.S.Zub
H.S. Skovoroda Kharkiv National Pedagogical University, 61002, Kharkiv, Ukraine
(Received January 27, 2015)
This paper analytically proves the existence of stable orbital motions in a system of superconductive and permanent
magnetic dipoles. As opposed to a system of two permanent magnetic dipoles, that has been studied in the work
of I. Tamm and V.Ginzburg in this system we have no ¾problem 1/R3¿, because of that the stability of the system
becomes possible.
PACS: 45.05.+x, 45.20.-d, 45.20.Jj
1. INTRODUCTION
In the system of permanent dipoles there is no static
equilibrium. This result was proved in [1] and that
proof is similar to the well known Earnshaw theo-
rem [2]. But the inner and relative rotations of the
dipoles can act as stabilizing factors, whereas Earn-
shaw's theorem not be applicable to the dynamic sys-
tems. Therefore, the question of the existence of
stable motions of the magnetic dipoles in connec-
tion with the hypothesis about magnetic nature of
the nuclear forces [3] was considered by I. Tamm and
V.Ginzburg in both cases for classical and quantum
theory. And they formulate so called ¾problem 1/R3¿
that relates to the interaction of magnetic dipoles.
This result casts doubt on the possibility of stable
motions in systems of small bodies that interact by
magnetic forces.
In the context of classical electrodynamics the
magnetic dipole is an equivalent of a small loop of
current. It is assumed that the current in loop is
constant. Other hand, the laws of electrodynamics
does not forbid us to consider a small superconduc-
tive loop. A current of such a loop is not constant,
but magnetic �ux is constant (or as the phrase goes
"frozen"). It is perfectly acceptable to call such an
object as a superconductive dipole.
In the paper [1] we derived an expression for the
potential energy of interaction in a system consisting
of permanent magnets and superconducting circuits.
It develops that interaction of permanent and su-
perconductive magnetic dipoles does not fall under
the above-mentioned ¾problem 1/R3¿. This raises
the question of the possibility of stable orbital mo-
tions in such a system.
The modern Hamiltonian formalism based on
group-theoretical methods [4, 5, 6, 7, 8] is an e�ective
tool for studying the stability of magnetic systems
with symmetries [9, 10].
This approach allows us to analytically prove the
existence of stable orbital motions in a system con-
sisting of a superconductive and permanent magnetic
dipoles.
2. MATHEMATICAL MODEL
Let's consider the superconducting dipole as a small
circular loop that �xed in the origin of coordinates
with a normal of γ = e3 that is directed along the
axis of z. Its radius is denoted by rs, total "frozen"
magnetic �ux is Ψ, and its self-inductance is L.
The movable dipole can be described as a circular
loop with radius rp, and a current I that associated
with its magnetic moment µ:
µ = πr2pIν = |µ|ν. (1)
In the dipole approximation, mutual inductance
M of two loops has the form [1]:
M = µ0π
r2sr
2
p
4r5
(
3⟨γ, r⟩⟨ν, r⟩ − r2⟨γ,ν⟩
)
, (2)
where rs, rp � the corresponding radiuses of the cur-
rents loops (s � superconductive current and p � di-
rect current);
γ,ν � the corresponding normals;
r � the radius-vector of a movable dipole.
Potential energy of the interaction of the loops
was derived in [1]:
V =
1
2
(Ψ−MI)2
L
, (3)
where MI, using (1) can be transformed to
MI =
Sµ0|µ|
4πr5
(
3⟨γ, r⟩⟨ν, r⟩ − r2⟨γ,ν⟩
)
, (4)
where S � the area of the superconducting dipole, or
MI =
Sµ0|µ|
4πr3
(3c′c′′ − c′′′) , (4a)
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.143-147.
143
where
r = |r|;
er = r/|r|;
c′ = e3 · er = x3/r;
c′′ = ν · er;
c′′′ = e3 · ν = ν3.
(5)
Then the potential energy has the form
V (r, c′, c′′, c′′′) =
1
2
κ
(
ψ − 3c′c′′ − c′′′
r3
)2
=
=
κ
2
π(r, c′, c′′, c′′′)2, (6)
where
κ = 1
L
(
Sµ0|µ|
4π
)2
;
ψ = 4πΨ
Sµ0|µ| ;
π(r, c′, c′′, c′′′) = ψ − 3c′c′′−c′′′
r3 .
(7)
For the �rst derivatives we have
∂rV = 3κπ0
r30
3c′c′′−c′′′
r0
;
∂c′V|ze = − 3κπ
r3 c
′′;
∂c′′V|ze = −3κπ
r3 c
′;
∂c′′′V = κπ
r30
.
(8)
With regard to the mechanical properties of the
movable magnetic dipole it is a small rigid body
� symmetric top (two principal moments of inertia
I1 = I2 = I⊥), and its mechanical symmetry coin-
cides with the magnetic.
The corresponding Hamiltonian of the system will
be [9]:
H = T + V, T =
1
2M
p2 +
α
2
n2, (9)
where M � mass of permanent dipole; α = 1/I⊥ �
the position of the dipole; p � its momentum; n �
the intrinsic angular momentum of the dipole.
Then equations of motion have the form [11, 9]:
ṙ = 1
M p;
ṗ = −∂rV er − 1
r (∂c′V P
e
⊥(e3) + ∂c′′V P
e
⊥(ν));
ν̇ = α(n× ν);
ṅ = ∂c′′V (er × ν) + ∂c′′′V (e3 × ν),
(10)
where the operator P e
⊥ � projection onto the plane
that perpendicular to the vector e⃗r, i.e. P e
⊥(e3) =
e3 − c′er and P
e
⊥(ν) = ν − c′′er.
3. RELATIVE EQUILIBRIA
Group-theoretic methods Hamiltonian dynamics
have proven e�ective in many problems of mechan-
ics [4, 5, 6, 8] and, in particular, in the study of the
stability of the magnetic dynamical systems [9, 10].
There are a number of theorems [7, 8], which give
us the conditions of stability of relative equilibria, i.e.
such trajectories of the Hamiltonian system that are
also the orbits of the one-parameter subgroups of the
invariance group of the system under study [4, 7].
Modern Hamiltonian formalism developed in two
basic versions: symplectic manifolds and Poisson
manifolds. Appropriate tools to investigate the sta-
bility of relative equilibria are available in both ap-
proaches.
The system has axial symmetry about z axis, i.e.
invariant with respect to the subgroup S1 of the ro-
tation group SO(3) and, additionally, has a mirror
symmetry with respect to the plane z = 0.
Thus, the system under study has the same set
of symmetries, the same set of dynamic variables to
describe the state, and the same kinetic energy as for
the Orbitron system in work [9].
Therefore, to describe the system under study is
suitable Hamiltonian formalism on the basis of the
Poisson structure (see [9] ), and the di�erence is in
the form of potential energy of the system.
The main tool for studying the stability of rela-
tive equilibria in our case, as in the above-mentioned
work, will be the Theorem 4.8. [8], which is valid for
dynamical systems with symmetry in general case of
Poisson manifolds.
An important advantage of this theorem is that
investigation of the function space of the trajectories
replaced on the investigation of the �nite-dimensional
vector space of the variations of dynamic variables in
the supporting point of the trajectory (relative equi-
libria). At this case the scheme of stability investi-
gation is broadly similar to the study of conditional
extremum of the function using Lagrange multipliers
method.
As it was already mentioned, the invariance group
of the Orbitron is S1. Each one-parameter subgroup
of this group will be characterized by the own angular
velocity ω0 = ω0e3. The rate of change of any phys-
ical quantity v of our problem along the orbit of this
subgroup will be given by the formula v̇ = ω0 × v.
Therefore, for a relative equilibria such relations
must be satis�ed
ṙ = ω0(e3 × r);
ṗ = ω0(e3 × p);
ν̇ = ω0(e3 × ν);
ṅ = ω0(e3 × n).
(1)
We will show that there is a dynamic orbit, for
which performed these relations. Taking into con-
sideration the mirror symmetry of the problem let's
consider the orbit that located in the plane z = 0.
We also assume that ν ∥ e3 and n ∥ e3. Then along
all this trajectory c
′
= c
′′
= 0, c
′′′
= ±1 and from
(8)�2 followed that ∂c′V = ∂c′′V = 0.
3rd and 4th equation of system (10)�2 thus per-
formed identically, and the 1st and 2nd are reduced
to second-order equation of the form:
M r̈ +
(
∂rV
r
)
|r=r0
r = 0. (2)
144
For the conditions (∂rV )|r=r0 > 0 the equation
(2) has a solution and corresponds to motion along a
circle of radius r0 at a rate that is determined by the
relation (
∂rV
r
)
|r=r0
= ω2
0M. (3)
Thus, we can conclude that for this orbit, indeed
was proved that it is a relative equilibrium.
4. CHOICE OF THE SUPPORTING
POINT
Let's set the point on an orbit of relative equilibrium
ze =
x0 = r0e1;
p0 = p0e2;
ν0 = ν0e3, ν0 = ±1.
n0 = n0e3;
(1)
In this point
c′ = 0;
c′′ = 0;
c′′′ = ν0;
(2)
∂rV|ze = −3κν0π0
r40
;
∂c′V|ze = 0;
∂c′′V|ze = 0;
∂c′′′V|ze = κπ0
r30
;
π0 = ψ + ν0
r30
(3)
and the following expressions for the di�erentials of
the arguments of the function V
dr = dx1;
dc′ = d⟨e,e3⟩ = 1
r (dx
3 − c′dx1);
dc′′ = dν1 + 1
r (ν0dx
3 − c′′dx1);
dc′′′ = dν3 .
(4)
5. NECESSARY CONDITION OF
STABILITY AND LAGRANGIAN
COEFFICIENTS
As motion integrals we will take
j3 = x1p2 − x2p1 + n3;
C1 = 1
2ν
2;
C2 = ⟨ν,n⟩;
(1)
where 1st line represents a 3rd conserved quantity of
a body total angular momentum, and the other two
are Casimir functions of the system.
E�ciency function (adjoined Hamiltonian) looks
like
H̃ = T + V − ω0j3 + λ1C1 + λ2C2 . (3)
The necessary condition of stability in theorem
4.8. [8] requires the di�erential of e�ciency function
to be equal to zero in a supporting point.
Write out the correspondent di�erentials in a sup-
porting point ze{
(dT )|ze = p0
M dp2 + αn0dn3;
(dV )|ze = ∂rV dx1 + ∂c′′′V dν3;
(4)
(dj3)|ze = p0dx
1 + r0dp2 + dn3;
(dC1)|ze = ν0dν3;
(dC2)|ze = (n0dν3 + ν0dn3).
(5)
Collecting the di�erentials of e�ciency function,
we get
dH̃|ze = (∂rV|ze − ω0p0)dx
1 +
( p0
M
− ω0r0
)
dp2 (6)
+(∂c′′′V|ze +λ1ν0+λ2n0)dν
3+(αn0−ω0+λ2ν0)dn3.
Equating dH̃|ze = 0, we derive the following ex-
pression for Lagrange multipliers
p0/M = ω0r0;
ω0p0 = ∂rV|ze ;
λ2 = ν0ω0 − αν0n0;
λ1 = −ν0∂c′′′V|ze − λ2ν0n0 =
−ν0∂c′′′V|ze + n0(αn0 − ω0).
(7)
The 1st equation in (7) is an ordinary relationship
between linear and angular velocity during circular
orbital motion.
The 2nd equation in (7) represents the equality of
centrifugal (on the left) and centripetal (on the right)
forces.
From this two expressions we get the relationship
for angular velocity, namely:
Mω2
0 =
1
r0
∂rV|ze = −3κν0π0
r50
= −3ν0
r20
∂c′′′V|ze . (8)
6. ALLOWABLE VARIATIONS
For the application of the su�cient condition of sta-
bility in the theorem 4.8. in [8] it is necessary to
extract a linear subspace of allowable variations.
Let's consider the variations of the dynamic vari-
ables annihilating the di�erentials in formula (5)�5.
From the 2nd line in (5)�5 it follows, that δν3 = 0,
then it ensues from the 3rd line, that δn3 = 0.
Thus, we obtain
δν3 = 0;
δn3 = 0;
δp2 = −p0
r0
δx1.
(1)
Hence it ensues that the variations in the form
δx1, δx2, δx3; δp1, δp3; δν1, δν2; δn1, δn2 (2)
can be considered as independent variations, further-
more, we must exclude from this subspace the direc-
tion which is tangent to the orbit.
It ensues from formula (1)�3, that this direction
(in ze point) is determined as
δx = r0e2;
δp = −p0e1;
δν = 0;
δn = 0.
(3)
145
In order to eliminate the variation (3), we impose
another additional condition on variations, and then
we get the constraints
δν3 = 0;
δn3 = 0;
δp1 = p0
r0
δx2;
δp2 = −p0
r0
δx1,
(4)
and an independent set of variations will be
δx1, δx2, δx3; δp3; δν1, δν2; δn1, δn2. (5)
7. BASIC QUADRATIC FORM
Su�cient condition for a minimum consists in
positive de�niteness of quadratic form of type
d2H̃|ze (δz, δz), where variation vector δz must be ex-
pressed through independent variations (5)�6 taking
into account the constraints (4)�6. Quadratic form
de�ned in independent variations we denote by Q.
Calculations of the e�ciency function hessian (ad-
joined Hamiltonian) and basic quadratic form in in-
dependent variations were performed in Maple.
After insigni�cant transposition of columns (and
corresponding lines with the same number) the ma-
trix of basic quadratic form acquires a form
Q11 0 0 0 0 0 0 0
0 Q22 0 0 0 0 0 0
0 0 Q44 0 0 0 0 0
0 0 0 Q33 Q35 0 0 0
0 0 0 Q35 Q55 Q57 0 0
0 0 0 0 Q57 Q77 0 0
0 0 0 0 0 0 Q66 Q68
0 0 0 0 0 0 Q68 Q88
.
(1)
Let's write out diagonal elements from matrix of
quadratic form Q:
Q11 = 3κ
r50
(
ν0π0 +
3
r30
)
;
Q22 = 4Mω2
0 ;
Q33 = 3Mω2
0 ;
Q44 = 1
M
;
Q55 = n0 (αn0 − ω0) +
1
3
Mr20ω
2
0 ;
Q66 = Q55;
Q77 = Q88 = α.
(2)
For matrix Q to be positive de�nite it is foremost
necessary that all diagonal elements of the matrix
are positive.
The positive de�nite of Q44, Q77, Q88 are scienter
positive.
The positive de�nite of Q22 and Q33 elements en-
sured by the physically obvious requirement:
Mω2
0 = −3κν0π0
r50
−→ ν0π0 < 0. (3)
From (2) (i.e. Q11 > 0 , Q55 = Q66 > 0) and (3)
we have {
− 3
r30
< ν0π0 < 0;
n0 (αn0 − ω0) +
1
3Mr20ω
2
0 > 0.
(4)
Let's consider non-diagonal elements.
First consider lower-right-hand block[
Q66 Q68
Q68 Q8,8
]
, (5)
where
Q68 = ν0 (ω0 − αn0) , (6)
then the conditions of positive de�niteness of (5) will
be
Q88Q66 −Q2
68 = ω0(αn0 − ω0) +
1
3
Mr20ω
2
0 > 0. (7)
Now consider the central 3× 3-block, where
Q35 =Mν0r0ω
2
0 , (8)
Q57 = −ν0(αn0 − ω0), (9)
then the condition of positive de�niteness of central
block are as follows{
Q33Q55 −Q2
35 = 3Mn0ω
2
0(αn0 − ω0) > 0;
Q33Q55Q77 −Q33Q
2
57 −Q2
35Q77 = 3Mω3
0(αn0 − ω0) > 0.
(10)
So, considering the π0 = ψ + ν0
r30
we have the fol-
lowing non-trivial conditions for positive de�niteness
of the matrix Q
−3 < (1 + ν0r
3
0ψ) < 0;
n0(αn0 − ω0) > 0;
ω0(αn0 − ω0) > 0.
(11)
Thus for the stability of a given relative equilib-
rium in our system the geometric conditions must be
carried out
−3 < (1 + ν0r
3
0ψ) < 0 (12)
and the dynamical conditions{
n0(αn0 − ω0) > 0;
ω0(αn0 − ω0) > 0.
(13)
The last conditions can also be written as{
sign(n0) = sign(ω0);
α|n0| > |ω0|;
(14)
or, given the n0 = I3Ω0, where Ω0 � the frequency
of self-rotation of the movable body{
sign(Ω0) = sign(ω0);
|Ω0| > I⊥
I3
|ω0|.
(15)
146
8. ESTIMATION OF PHYSICAL
PARAMETERS
Let's show that we can choose the system parameters
that correspond to the available material and techno-
logical capabilities, and in addition does not violate
the assumption of dipole nature and justice of quasi-
stationary approximation.
Let's consider a small ring Nb3Sn of radius R =
0, 005 (m) with the radius of the wire r = 0, 0005
(m), and disc permanent magnet NdFeB (with den-
sity ρ = 7, 4 · 103 (kg/m3) with residual induction
Br = 0, 25 (T)) with diameter D = 0, 014 (m) and
height h = 0, 006 (m) (M = 0, 0068 (kg) � mass).
These bodies are characterized by a set of mag-
netic parameters. For the ring this is self-inductance
that can be estimated from the expression [12] (5-1)
p.207:
L = µ0R
(
ln
(
8R
r
)
− 7
4
+
1
8
r2 ln
(
8R
r
)
+ 1
3
R2
)
,
for the permanent magnet this is a magnetic moment
µ = 0, 18 (A m2).
Let's freeze the �ow through the ring (i.e., trans-
late it into a superconducting state) when the per-
manent magnet located in the equatorial plane at
a distance of rK = 0, 05 (m) (disc of the magnet
located mirror-symmetrically relative to the equato-
rial plane and with magnetic moment directed down,
i.e. ν0 = −1). Then move the magnet on a dis-
tance r0 ≃ 0, 059 (m) and provide it initial velocity
such that the angular velocity of the orbital motion
was ω0 ≃ 0, 0152 (rad/s) and also small self-rotation
(such that condition (15)�7 will met).
Then such motion will be stable, because all sta-
bility conditions �7 are met.
The smallness of the angular velocity is dictated
due smallness of the centripetal magnetic force, be-
cause in fact a magnetic dipole interacts with itself
through the magnetic �ow induced in the supercon-
ducting ring (at a relatively large distance). It is easy
to deduce that with decreasing of the all size of the
system in k times its circular frequency of rotation
also increased in the same times.
References
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