Magnetic levitation in orbitron system
This paper devoted to proof the existence of stable quasi-periodic motions of the magnetic dipole that is under the action of the external magnetic field and homogeneous field of gravity. For proof we used energy-momentum method that is the group-theoretic method of Hamiltonian mechanics. Numerical...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Цитувати: | Magnetic levitation in orbitron system / S.S. Zub // Вопросы атомной науки и техники. — 2014. — № 5. — С. 168-176. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860072485599313920 |
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| author | Zub, S.S. |
| author_facet | Zub, S.S. |
| citation_txt | Magnetic levitation in orbitron system / S.S. Zub // Вопросы атомной науки и техники. — 2014. — № 5. — С. 168-176. — Бібліогр.: 14 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | This paper devoted to proof the existence of stable quasi-periodic motions of the magnetic dipole that is under the action of the external magnetic field and homogeneous field of gravity. For proof we used energy-momentum method that is the group-theoretic method of Hamiltonian mechanics. Numerical simulation shows the possibility of realization of stable motions with physically reasonable parameters of the system.
Доказана возможность устойчивых квазипериодических движений магнитного диполя, находящегося под действием как внешнего магнитного поля, так и однородного поля силы тяжести. Для доказательства используются теоретико-групповые методы гамильтоновой механики, а именно: energy-momentum метод. Численное моделирование демонстрирует возможность реализации устойчивых движений при физически разумных параметрах системы.
Доведено можливiсть стiйких квазiперiодичних рухiв магнiтного диполя, що знаходиться пiд дiєю як зовнiшнього магнiтного поля, так i однорiдного поля сили тяжiння. Для доведення використовуються теоретико-групповые методи гамiльтонової механiки, а саме: energy-momentum метод. Чисельне моделювання демонструє можливiсть реалiзацiї стiйких рухiв для фiзично прийнятних параметрах системи.
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COMPUTING AND MODELLING SYSTEMS
MAGNETIC LEVITATION IN ORBITRON SYSTEM
S.S.Zub
Taras Shevchenko National University of Kyiv, 01601, Kyiv, Ukraine
(Received November 8, 2013)
This paper devoted to proof the existence of stable quasi-periodic motions of the magnetic dipole that is under
the action of the external magnetic �eld and homogeneous �eld of gravity. For proof we used energy-momentum
method that is the group-theoretic method of Hamiltonian mechanics. Numerical simulation shows the possibility of
realization of stable motions with physically reasonable parameters of the system.
PACS: 45.05.+x, 45.20.-d, 45.20.Jj
1. INTRODUCTION
In the paper [1] we proved that quasi-periodic mo-
tion of the magnetic dipole in specially constructed
system is exists (i.e. without gravity force). This sys-
tem is called Orbitron, and it was choosen so, to be
the most simplest and the optimal for solely magnetic
task.
As it was shown in further study just that very
feature of the Orbitron is not suitable for the realiza-
tion of levitation, because exactly for this system it is
di�cult to reach compensate the gravity force by the
use of magnetic force while ensuring the stability con-
ditions. The problem can be solved by combining of
the magnetic �eld of Orbitron with the magnetic �eld
of the simplest type for the compensation of gravity
force.
So, the system that was proposed for proof, con-
sists of two magnetic poles that equal by value and
opposite to the sign ±κ that are located on the ∓h
of axis z analogically to Orbitron. Supplementary
magnetic �eld for compensation is axially symmetric
(respect to the axis z) and linearly depends on spatial
coordinates.
We can distinguish three levels of consideration of
the problem.
The �rst level can be called theoretical. On this
level we must prove, in principal, that stability of
motion in the system does not contradict the physical
laws of the rigid body mechanics and magnetostatics.
The second level can be called physical. At this
level of consideration we must sure that the e�ect of
stability occurs at reasonable parameters of the sys-
tem that can be provided, for example, by existing
properties of the materials.
The third level can be called as the level of exper-
imental design. At this level we must o�er a speci�c
scheme of the experiment that demonstrates the cor-
responding example of stable motions.
Our paper involves the investigation only the �rst
two levels. Here it is necessary to mention the works
of V. Kozorez, who has demonstrated of the quasi-
periodic motions of a small permanent magnet in a
magnetic �eld. In that experiment the magnet per-
forms more then one thousands of revolutions (that is
about 6 minutes of �ight). These successful launches
were to rare, because they were not based on an ade-
quate mathematical model, and successful initial con-
ditions in the experiment were reproduced acciden-
tally.
It should be mentioned that neither natural ex-
periment nor numerical experiment does not provide
evidence of the system stability, but can provide the
arguments in favor of the latter.
Therefore, our task is just to give a mathemati-
cally rigorous proof of the system stability based on
the generally accepted theory [2, 3, 4, 1, 5].
2. MATHEMATICAL MODEL
Our model consists of the next elements.
1. Magnetic �eld of the Orbitron.
There are two unlike magnetic poles ±κ on the
axis z that are placed at distance ∓h. So, the mag-
netic �eld of the Orbitron BO has form:{
BO(r) =
∑
ε=±1 Bε(r),
Bε = −µ0
4π εκ
r−εhez
|r−εhez|3 .
(1)
By its structureBO �eld has axial symmetry with
respect to the axis z.
2. The axially symmetrical �eld BL that used for
the compensation is linearly depends on the coordi-
nates.
We can use the result[6] where the decomposition
was given in more general form
BLz = B0 +B′z,
BLx = −1
2B
′x,
BLy = −1
2B
′y ,
(2)
then in the cylindrical system we have{
BLz = B0 +B′z,
BLr = − 1
2B
′r.
(3)
Remark. The total magnetic �eld B(x) that acts
on the dipole is the sum of these �elds, i.e.
B(x) = BO(x) +BL(x). (4)
168 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.168-176.
3. Let's suggest that magnetic dipole is a small
rigid body with axial symmetry (or symmetric top)
and its magnetic moment also directed along the axis
of symmetry, i.e. symmetries of the magnetic and
mass distributions coincide.
Generally accepted that the con�guration space of
the rigid body is group SE(3). The relevant Hamil-
tonian formalism can be based on the classical sym-
plectic structure that exists on the cotangent bundle
T ∗(SE(3)) [7, 8, 5].
Remark. Most of the relationships that is required
for group-theoretic description of the Hamiltonian
formalism on the T ∗(SE(3)) can be easily derived
from generalized description, that given in [8].
In this study, mostly we rely on results of the
energy-momentum method that were designed for the
Orbitron in [5]. However, as opposed to the men-
tioned work we use the inertial frame but not the
body frame that associated with the body.
So, as it shown in [9] the condition of the magnetic
dipole in the inertial frame can be describe by four
values ((x, A), (p,π)), where xi � the coordinates
of the mass center of the rigid body A � the matrix
of rotation for transformation from space coordinates
(in inertial frame) to the coordinates that connected
with body (in body coordinates), pi � the compo-
nents of momentum of the body, πi � components of
the angular momentum in the inertial frame.
Symplectic and Poisson structures on the
T ∗(SE(3)) are completely equivalent [7, 8]. Let's give
the appropriate Poisson brackets in the inertial frame
{xi, xj} = 0, {xi, pj} = δij ,
{xi, Ajk} = 0, {xi, πj} = 0,
{pi, pj} = 0, {pi, Ajk} = 0,
{pi, πj} = 0, {πi, Ajk} = εijlAlk,
{πi, πj} = εijlπl.
(5)
4. Hamiltonian system is given by expression
h = T (π,p) + V (ν,x) = (6)
=
1
2M
p2+
1
2
απ2+
1
2
β⟨π,ν⟩2−m⟨B(x),ν⟩+Mgx·e3,
where M � the dipole mass, g � free fall acceler-
ation, m � the quantity of the magnetic moment,
α = 1/I⊥,I⊥ = I1 = I2 � moments of inertia of a
symmetric top, β = 1
I3
− 1
I1
, ν = Ae3, e3 = (0, 0, 1)
arithmetic vector (column), and thus, ν � this is the
third column of the matrix A.
Remark. In the Poisson reduction of the symmet-
ric top [9] the components of 1st and 2nd columns
of the matrix of rotation A disappear from our de-
scription. At that time the term of the Hamiltonian
β⟨π,ν⟩2 can be discarded because it is the Casimir
function of our Poisson structure.
However, it is does not give any advantages for
the application of energy-momentum method, so we
do not spend reduction and thus we stay in the frame-
work of the classical Poisson structure (5). Then
the value of ⟨π,ν⟩ is an integral of motion, but not
a Casimir function, and therefore it cannot be dis-
carded.
3. THE TORAL ACTION AND THE
MOMENTUM MAP
There are two symmetries in our case. The �rst of
them is associated with axial symmetry of the mag-
netic �eld and the second one, in fact, associated with
symmetry of the magnetic body.
Let's formalize two symmetries that were de-
scribed above in the form of toral action:
Φ :
(T2 = S1 × S1)× SE(3) −→ SE(3)
((R(ϕ), R(ψ)) , (x, A)) 7−→
(R(ϕ)x, R(ϕ)AR(−ψ)),
(7)
where the matrix R(θ) is the rotation matrix around
the 3rd axis at the angle θ, in particular
R(θ)e3 = e3 . (8)
The action (7) on the group SE(3) as on the con�gu-
ration space continues (Cotangent Lift) by the action
of the same group (of torus) on the cotangent bundle
T ∗(SE(3)) as follows:
Φ(R(ϕ),R(ψ)) ((x, A), (p,π)) = (9)
= (R(ϕ)x, (R(ϕ)AR(−ψ)), (R(ϕ)p, R(ϕ)π)) .
The momentum map that corresponds to this action
is two-component function on the phase space of the
group, i.e. T ∗(SE(3))
J((x, A), (p,π)) = (⟨j, e3⟩,−⟨π, Ae3⟩),
where j � total angular momentum of the body, or
J((x, A), (p,π)) = (⟨π + x × p,e3⟩,−⟨π,ν⟩). (10)
It is easy to check that the Hamiltonian (6) is the
invariant value relative to the action (9), using in
particular (8). From that follows that components of
the moment (10) are the motion integrals.
4. THE ENERGY-MOMENTUM
METHOD
This method is the most transparent way to study
the relative equilibria [10, 7, 5] of Hamiltonian sys-
tem with symmetry and allow both to formalize the
searching of the relative equilibria but also investigate
their stability.
Energy-momentum method inherits the idea of
Lagrange of the �nding the conditional extremum,
but at the same time the algorithm become more sim-
ple because of using group-theoretic approaches. The
most remarkable simpli�cation of research is that it
comes down to the analyze of some structures of lin-
ear algebra in a single point of the relative equilib-
rium.
Remark. Since the research is conducted only at
one point then in our case the strength of the mag-
netic �eld at that point, the component of the Jaco-
bian and the Hessian are the independent variables
169
and in particular this allowed to divide our task into
relatively independent parts.
We expect that the relative equilibrium that were
found based on the energy-momentum method can be
stable with relation to some parameters of mentioned
above elements of the system.
As well as for the Lagrange method the objective
function has been constructing and it given by the
expression:
hξ := h− Jξ = h− ξ1J
1 − ξ2J
2 = (11)
=
1
2M
p2 +
1
2
απ2 +
1
2
β⟨π,ν⟩2
−m⟨B(x),ν⟩+Mgx · e3
−ξ1⟨π + x× p, e3⟩+ ξ2⟨π,ν⟩.
Here ξ1 and ξ2 are the Lagrangian coe�cients that
have dimensions of the frequency.
As will be shown soon in the relative equilibrium
the magnetic dipole describes a circle in the xy plane
with frequency ξ1. Interpretation of frequency ξ2
somewhat more complicated but it associates with
the intrinsic angular velocity of rotation of the body.
As well as in Lagrange method the �rst variation
of the objective function (11) determines the relative
equilibrium in the task. Also, will call the conditions
of the ful�llment of the δhξ = 0 as necessary condi-
tions of equilibrium. Su�cient conditions is obtained
from the analysis of the second variation δ2hξ [10, 5].
This analysis is the multi-step, and it is this will be
the content of this article.
5. NECESSARY CONDITIONS OF THE
RELATIVE EQUILIBRIUM
Let v ∈ T((x,A),(p,π))(T
∗(SE(3))), i.e. in represen-
tation of the right trivialization it is corresponds
to the inertial frame [11, 8, 5], and has form:
((δx, δA), (δp, δπ)). Then for variations (i.e. for
derivatives in the direction v, see [5]) that constitute
hξ we have
δvT (π,p) = dT (π,p) · v =
= α⟨π, δπ⟩+ 1
M
⟨p, δp⟩+β⟨π,ν⟩(⟨ν, δπ⟩+⟨ν×π, δA⟩)
δvV (A,x) = dV (A,x) · v =
= −m⟨DB(x)[ν], δx⟩+Mg⟨e3, δx⟩−m⟨ν×B(x), δA⟩
δvJ
ξ (x, A,p,π) = dJξ (x, A,p,π) · v =
= ξ1(⟨e3, δπ⟩+ ⟨p× e3, δx⟩ − ⟨x× e3, δp⟩)−
−ξ2(⟨ν, δπ⟩+ ⟨ν × π, δA⟩)
i.e.
δ̃vT (π,p) = α⟨π, δπ⟩+ 1
M ⟨p, δp⟩;
δvV (A,x) = −m⟨DB(x)[ν], δx⟩+
+Mg⟨e3, δx⟩ −m⟨ν ×B(x), δA⟩;
δ̃vJ
ξ (x,ν,p,π) = dJξ (x,ν,p,π) · v
= ξ1[⟨e3, δπ⟩+ ⟨p× e3, δx⟩−
−⟨x× e3, δp⟩]− ξ̃2[⟨ν, δπ⟩+
+⟨ν × π, δA⟩]
, (12)
where the wave-symbol means the regrouping the
terms in the �rst variation of the associated Hamil-
tonian � the summand with β is taken from �rst row
and is placed into the third row (see (13)), of course,
this does not a�ect on the procedure of the compu-
tation and also on it result.{
ν = Ae3;
ξ̃2 = ξ2 + β⟨π,ν⟩.
(13)
From formula (12), collecting the coe�cients of
the independent variations, we get the following nec-
essary conditions:
δA : Ae3 × [ξ2 + β⟨π, Ae3⟩)π −mB(x)] = 0 ;
δx : ξ1 ·(e3×p)−mDB(x)T [Ae3]+Mge3 = 0 ;
δπ : απ − ξ1e3 + (ξ2 + β⟨π, Ae3⟩)Ae3 = 0 ;
δp :
1
M
p− ξ1(e3 × x) = 0 .
See also [5], but remember that in this article was
used coordinate system that associates with body.
In view of (13) the necessary conditions of equi-
librium take the form:
1
M p− ξ1(e3 × x) = 0;
ξ1(e3 × p)−mDB(x)T [ν] +Mge3 = 0;
απ − ξ1e3 + ξ̃2ν = 0;
ν × (ξ̃2π −mB(x)) = 0.
(14)
In conformity with the Lagrange approach the
equations are determine not only the dynamic vari-
ables in the equilibrium, but also the Lagrange mul-
tipliers ξ1 and ξ̃2.
Remark. In the purely magnetic case [1, 5] for
all relative equilibria that were investigated, the con-
stant ξ2 remains free. Some restrictions on this value
appeared only at the stage of studying the stability
of the relative equilibria.
The �rst two equations (14) do not contain any
variables that characterize the proper rotation of the
body, nor the values of the magnetic �eld (only the
components of the Jacobian). The Lagrange mul-
tiplier ξ1 is determined only from these equations.
First equation in (14) can be written as ẋ = ξ1(e3 ×
x). This means that vector x rotates around the axis
e3 with angular velocity ξ1. Accordingly, the vector
p = M ẋ is also rotated around this axis with the
same angular velocity.
Remark. Thus we have shown that the magnetic
dipole is moved around the circle in the plane or-
thogonal to the axis z. Farther on the search of the
relative equilibria we will restrict by the condition of
z = 0.
170
By combining of �rst two equations of (14) we
write the conditions of the relative equilibrium in the
form:
mDB(x)T [ν] = −Mξ21x⊥ +Mge3. (15)
Equation (15) and condition ν2 = 1 is fully de�ne
the components of ν and the value ξ1 independently
of the other variables of the system.
Let's describe the relative positions for physical
vectors taking into consideration (14) and axial sym-
metry of the magnetic �eld. First of all, from the ax-
ial symmetry of the �eld we have B ∈ span {e3,x}.
From third equation
π =
ξ1
α
e3 −
ξ̃2
α
ν −→ π ∈ span {e3,ν} , (16)
and from the fourth we have
ν ∥ (ξ̃2π −mB(x)) −→ ξ̃2π −mB(x) = kν. (17)
Substituting in (17) the expression for π from (16)
we obtain
mB(x) =
ξ1ξ̃2
α
e3 −
(
k +
ξ̃2
α
)
ν −→ (18)
B ∈ span {e3,ν} .
Note that the 2nd term in this case can not be zero
because the total magnetic �eld (4) in the plane z = 0
will be with the orthogonal component e3. Then from
aforesaid it follows that
B ∈ span {e3,x} ,
ν ∈ span {e3,x} ,
π ∈ span {e3,x} .
(19)
This means that in the relative equilibrium all
physical vectors of the task except p lie in a plane of
{e3,x}.
As it was already mentioned, the advantage of the
energy-momentum method of study of stability of the
relative equilibrium that it can be reduce to the anal-
yses of the system of equations in any point of the
relative equilibrium. Having relation (19) we �x the
plane of physical vectors zx. Then, in particular
x0 = r0e1,
p0 = p0e2 =Mξ1r0e2,
ν0 = ν1e1 + ν3e3,
π0 = π1e1 + π3e3 .
(20)
For a given ξ1 this allows us write the solution of the
last two equations (14) as follows:
π1 = m(ν3B1−ν1B3)
ξ1
,
π3 = ξ1
α + ν3
ν1
m(ν3B1−ν1B3)
ξ1
,
ξ̃2 = −αm(ν3B1−ν1B3)
ξ1ν1
.
(21)
In this case, as it was noted above (see (15)) val-
ues of ξ1, ν1, ν3 are completely independent of the
π1, π3, ξ̃2.
6. DETERMINATION OF DYNAMIC
VARIABLES IN THE RELATIVE
EQUILIBRIUM
In the previous section we have formulated the equa-
tion (15) that together with the condition ν2 = 1
allows you to �nd not yet certain values ν1, ν3 and
the Lagrange multiplier ξ1.
These equations with regard (20) can be presented
as the system:
ν1Br,r + ν3Br,z = −M
m ξ
2
1r0;
ν1Bz,r + ν3Bz,z =
M
m g;
ν21 + ν23 = 1 .
(22)
Note that in consequence of the Maxwell equations
for the constant external �eld without source of �eld
in the area of dipole motion the following relation-
ships are performed (see in the cylindrical coordi-
nates): {
Br,z = Bz,r,
Br,r +Bz,z = −Br
r0
.
(23)
Remark. Note that in z = 0 plane the Orbitron's
�eld by reason of mirror symmetry has only vertical
component and BOz,z = 0, thus Bz,z = B′ (see (1,2)).
Work in
ζ2 =
rξ21
g ,
λ = mB′
Mg ,
σ = −Br,z
B′ .
(24)
Let's present the matrix (22) in the form[
Br,z Bz,r
Br,z Bz,z
]
= B′
[
− 1
2 −σ
−σ 1
]
. (25)
Then (22) can be written as[
−1
2 −σ
−σ 1
] [
ν1
ν3
]
=
1
λ
[
−ζ2
1
]
(26)
and solution in the formν
1 = 1
λ
ζ2−σ
σ2+ 1
2
,
ν3 = 1
λ
σζ2+ 1
2
σ2+ 1
2
.
(27)
Quadratic equation for ζ2 is obtained, as before, from
the third equation (22) and has the following proper
solution for the case of σ > 0
ζ2 =
σ(1 + ϱ)
1 + σ2
+
σ2 − ϱ
1 + σ2
√
λ2(1 + σ2)− 1 . (28)
The relations (20),(24),(27),(28),(21) give us the
value of the all signi�cant physical quantities in the
chosen �xed point of the relative equilibrium.
171
7. THE SECOND VARIATION OF THE
ASSOCIATED HAMILTONIAN
As mentioned above, the research of stability is based
on the analysis of the second variation of the associ-
ated Hamiltonian that is conducted in several stages.
On the �rst step we calculate the second variation.
7.1. Calculating of the second variation of
the potential energy
We have (see (12))
δvV (A,x) = −m⟨DBT (x)[ν], δx⟩+Mg⟨e3, δx⟩
(29)
−m⟨ν ×B(x), δA⟩;
δvδvV (A,x) = (30)
= −mδv[⟨DBT (x)[ν], δx⟩+ ⟨ν ×B(x), δA⟩] .
It is evident that tensor notation will be the most
suitable mathematical apparatus for the analysis of
the variations
⟨DBT (x)[ν], δx⟩ = νiBi,kδx
k , (31)
⟨ν ×B(x), δA⟩ = ⟨B(x), δA× ν⟩ = (32)
= νiεiksBkδAs,
where ε � the Levi-Civita symbol.
Then
δv⟨DBT (x)[ν], δx⟩ = Bk,rδνkδx
r +Bi,klδx
kδxl =
= Bk,r(εksiδAsνi)δx
r + νiBi,klδx
kδxl =
= νiεiksBk,rδx
rδAs + νiBi,klδx
kδxl ,
i.e.
δv⟨DBT (x)[ν], δx⟩ = (33)
= νiεiksBk,rδx
rδAs + νiBi,klδx
kδxl .
Moreover
δv⟨ν ×B(x), δA⟩ = δv(νiεiksBkδAs) =
= νiεiksBk,rδxrδAs + εiksBkδAsδνi =
= νiεiksBk,rδxrδAs + εiksBkδAsεirtδArνt =
= νiεiksBk,rδxrδAs+(δkrδst−δktδsr)BkδAsδArνt =
= νiεiksBk,rδxrδAs+BrνsδArδAs−νkBkδArδAr ,
i.e.
δv⟨ν ×B(x), δA⟩ = (34)
= νiεiksBk,rδxrδAs+BrνsδArδAs−νkBkδArδAr.
So
δvδvV (A,x) =
= −mδv[⟨DBT (x)[ν], δx⟩+ ⟨ν×B(x), δA⟩] = (35)
= −m (2νiεiksBk,rδxrδAs + νiBi,klδxkδxl+
+BrνsδArδAs − νkBkδArδAr) .
7.2. Calculation of the second variation of
the associated Hamiltonian
δvδvh
ξ = (36)
δvδvT (π,p) + δvδvV (ν,x)− δvδvJ
ξ (x,ν,p,π) =
= δv δ̃vT (π,p) + δvδvV (ν,x)− δv δ̃vJ
ξ (x,ν,p,π) .
We have the relation
δvν = δA× ν. (37)
For second variation of the kinetic energy, we ob-
tain
δv δ̃vT (π,p) = α⟨δπ, δπ⟩+ 1
M
⟨δp, δp⟩ . (38)
The variation of the potential energy is given in
(35)
δvδvV (A,x) =
= −mδv[⟨DBT (x)[ν], δx⟩+ ⟨ν ×B(x), δA⟩]; (39)
= −m (2νiεiksBk,rδxrδAs + νiBi,klδxkδxl+
+BrνsδArδAs − νkBkδArδAr) .
For calculation of δv δ̃vJ
ξ the following relations
will be useful
δv[⟨p×e3, δx⟩−⟨x×e3, δp⟩] = ⟨δp×e3, δx⟩−⟨δx×e3, δp⟩
= 2⟨e3, δx× δp⟩ ,
δv⟨ν, δπ⟩ = ⟨δν, δπ⟩ = ⟨δA×ν, δπ⟩ = ⟨ν, δπ×δA⟩ ,
δv⟨ν × π, δA⟩ = ⟨δν × π, δA⟩+ ⟨ν × δπ, δA⟩
= ⟨(δA× ν)× π, δA⟩+ ⟨ν × δπ, δA⟩ ,
⟨(δA× ν)× π, δA⟩ = ⟨π, δA× (δA× ν)⟩
= ⟨π, ⟨ν, δA⟩δA− ⟨δA, δA⟩ν⟩ =
= ⟨ν, δA⟩⟨π, δA⟩ − ⟨π,ν⟩⟨δA, δA⟩ ,
⟨ν × δπ, δA⟩ = ⟨ν, δπ × δA⟩ ,
Thus
δv δ̃vJ
ξ =
= 2ξ1⟨e3, δx×δp⟩−δv ξ̃2[⟨ν, δπ⟩+⟨ν×π, δA⟩] (40)
−ξ̃2[2⟨ν, δπ×δA⟩+⟨ν, δA⟩⟨π, δA⟩−⟨π,ν⟩⟨δA, δA⟩] ,
The value of ξ̃2 in (13) formally depends not only
on the Lagrange multiplier ξ2 but also on the dynamic
variables. Therefore, it would seem it is necessary to
consider the variation of this value, but in the energy-
momentum method this form of δ2hξ is analyzed not
for all variations, but only on such that are tangent
to the submanifolds of moment level, i.e.
TzJ · v = 0 −→ δv ξ̃2 = 0 (41)
Thus, in all calculations of stability conditions can be
assumed that ξ̃2 is constant. Then
δv δ̃vJ
ξ = 2ξ1⟨e3, δx× δp⟩ . (42)
−ξ̃2[2⟨ν, δπ×δA⟩+⟨ν, δA⟩⟨π, δA⟩−⟨π,ν⟩⟨δA, δA⟩] .
172
Substituting in (36) the expressions (38), (39) and
(42) we �nally obtain
δvδvh
ξ =
δv δ̃vT (π,p)+δvδvV (ν,x)−δv δ̃vJξ (x,ν,p,π) (43)
= α⟨δπ, δπ⟩+ 1
M
⟨δp, δp⟩
−m (2νiεiksBk,rδxrδAs + νiBi,klδxkδxl+
+BrνsδArδAs − νkBkδArδAr)
−2ξ1⟨e3, δx×δp⟩+ξ̃2[2⟨ν, δπ×δA⟩+⟨ν, δA⟩⟨π, δA⟩
−⟨π,ν⟩⟨δA, δA⟩] .
8. ADMISSIBLE VARIATIONS
Stability investigation of Hamiltonian system with
symmetry in accordance with energy-momentum
method reduces to analyse of the quadratic form Q of
variations. The positive de�niteness of Q will mean
the stability of equilibrium.
Form Q is derivative of form δvδvh
ξ (see (43)) that
called the it initial. In order to obtain the values of
reduced form Q we must use the value of the initial
form not for all possible variations of the dynamical
variables, but only for variations with constraints.
These constraints are divided into two types.
The �rst type of the constraints were already men-
tioned in (see (41)). The variations that are satisfying
to the constraints will be tangent to submanifolds of
the moment level and this is the geometric sense of
the constraints.
The second type of constraints restricts the varia-
tions space in such way that no one of the variations
of this subspace will not be tangent to the orbit of the
toral action for the selected �xed point of the relative
equilibrium. Thus, the second type of constraints will
give us the variations space transverse with respect
to the space that is tangent to the orbit at that point.
Let z0 � point of the relative equilibrium, and v1, v2
� the basis vectors that are tangent to the orbit of
the toral action at that point. Then, span{v1, v2} �
it is the tangent to the orbit space in the point z0.
Let
KerTz0J =W ⊕ span{v1, v2} .
ThenW is a subspace of the admissible variations.
Remark. In contrast to the constraints of the �rst
type the constraints of the second type can be chosen
rather arbitrarily, but keeping only the transversality
condition.
Consider the constraints of the �rst type
⟨e3, δπ⟩+ ⟨p× e3, δx⟩
−⟨x× e3, δp⟩ = 0;
⟨ν, δπ⟩+ ⟨ν × π, δA⟩ = 0 .
(44)
At the point z0 these relations can be written in terms
of coordinates{
δπ3 + p0δx1 + r0δp2 = 0,
ν1δπ1 + ν3δπ3 + (ν3π1 − ν1π3)δA2 = 0 .
(45)
Consider the constraints of the second type{
⟨ν, δA⟩ = 0,
δp1 = p0
r0
δx2
(46)
moreover the �rst constraint �prohibits� the rotation
around the axis of the body symmetry, and the sec-
ond one around the axis z symmetry of the system.
Let's write via the coordinates{
δp1 − p0
r0
δx2 = 0,
ν1δA1 + ν3δA3 = 0 .
(47)
The relations (45) and (47) allow us to exclude the
following variations: δp1, δA3, δπ3, δp2
δp1 = p0
r0
δx2,
δA3 = −ν1
ν3
δA1,
δπ3 = −ν1δπ1+(ν3π1−ν1π3)δA2
ν3
,
δp2 = ν1δπ1+(ν3π1−ν1π3)δA2−p0ν3δx1
r0ν3
.
(48)
Note that in accordance with the formula (27) ν3 ̸= 0.
9. THE MATRIX OF THE REDUCED
QUADRATIC FORM
Let's call via Q the reduced quadratic form of the
initial form Q. In the previous section the algorithm
of the matrix calculation described in detail. It uses
the components of Jacobian and Hessian of magnetic
�eld at the point x0 = r0e1.
One can show that in this point the corresponding
matrices have the form:
DB|z=0 =
−1
2B
′ 0 Bz,r
0 −1
2B
′ 0
Bz,r 0 B′
(49)
and the components of Hessian
D2B1|z=0 =
0 0 Bz,rr
0 0 0
Bz,rr 0 0
, (50)
D2B2|z=0 =
0 0 0
0 0 1
rBz,r
0 1
rBz,r 0
, (51)
D2B3|z=0 = (52)Bz,rr 0 0
0 1
rBz,r 0
0 0 −
(
Bz,rr +
1
rBz,r
)
.
Because of complexity of computation of the
quadratic form computation was conducted in Maple.
Let adopt the following order of independent vari-
ations: δπ2, δx2, δA1, δπ1, δx3, δx1, δA2. Note that
the variable δp3 give us the trivial strictly positive
contribution δp23/M . Therefore, this contribution is
not considered in the next steps.
The resulting matrix has a block structure:
Q =
[
Q1 0
0 Q2
]
, (53)
173
where Q1 � matrix 3× 3, and Q2 � matrix 4× 4.
Matrix Q1 with the next order of the variations
δπ2, δx2, δA1 have the form:
Q1 =
α 0 − ξ̃2
ν3
0 3Mξ21 +
mν3Bz,r
r0
−mBz,z
2ν3
− ξ̃2
ν3
−mBz,z
2ν3
⟨ν,mB−ξ̃2π⟩
ν2
3
.
(54)
For matrix Q2 we have
Q2 =
1
ν2
3
(
α+
ν2
1
Mr20
)
0 − 2ξ1ν1
r0ν3
Q47
0 mν3
(
Bz,rr +
Bz,r
r0
)
−mν1Bz,rr Q57
−2ξ1ν1
r0ν3
−mν1Bz,rr 3Mξ21 −mν3Bz,rr Q67
Q47 Q57 Q67 Q77
, (55)
Q47 = ξ̃2
ν3
− ξ1
(
1 + 1
αMr20
)(
ν2
1
ν2
3
)
;
Q57 = m(−ν3Bz,r + ν1Bz,z);
Q67 = m(ν1Bz,r +
1
2ν3Bz,z) + 2
ξ21
r0α
ν1
ν3
;
Q77 =
ξ21
α
(
1 + 1
αMr20
)
ν2
1
ν2
3
+⟨ν,mB − ξ̃2π⟩.
(56)
10. THE CONDITIONS OF POSITIVE
DEFINITENESS OF THE QUADRATIC
FORM Q
Because of block structure of matrix Q the study of
the positive de�niteness can be devided on the two
separate studies of the matrices Q1 and Q2.
Let's use the Sylvester's criterion [12]. For this
purpose we must check the determinant of the upper-
left submatrices of our matrix on the positive de�nite-
ness. For example, for Q1 it will be the next sequence
of submatrices Q
[1]
1 , Q
[2]
1 , Q
[3]
1 :
Q
[1]
1 = [α], Q
[2]
1 =
[
α 0
0 3Mξ21 +
mν3Bz,r
r0
]
, (57)
Q
[3]
1 = Q1
and so, the relation must be performed
det(Q
[1]
1 ) > 0, det(Q
[2]
1 ) > 0, det(Q1) > 0. (58)
Accordingly for Q2 we have
det(Q
[1]
2 ) > 0, det(Q
[2]
2 ) > 0, (59)
det(Q
[3]
2 ) > 0, det(Q2) > 0.
Remark. Note that more weaker, but necessary
conditions of the positive de�niteness of the matrix
Q is the positivity of its diagonal elements.
Consider the matrix Q1. The �rst condition (58)
is trivial, since wittingly α > 0. Then the second
condition can be written as
3Mξ21 +
mν3Bz,r
r0
> 0. (60)
The third condition can be represented in the form(
3Mξ21 +
mν3Bz,r
r0
)(
⟨ν,mB − ξ2π⟩ −
ξ22
α
)
−
−1
4
m2B2
z,z > 0. (61)
Second multiplier in (61) by using (21) can be sim-
pli�ed to
⟨ν,mB − ξ2π⟩ −
ξ22
α
=
mB1
ν1
. (62)
It is obvious that from (61) with regard to (62) (orig-
inally m > 0) and it follows that
mB1
ν1
> 0 −→ B1
ν1
> 0. (63)
This condition physically means that the mag-
netic moment seeks to orientate oneself by the �eld.
It should be noted that in the supporting point of the
relative equilibrium, where z = 0, only �eld BL has
a component of B1 that is di�erent from zero.
Then
mB1
ν1
= −mB
′r0
2ν1
> 0. (64)
Dividing (61) on the positive value (64) and after
multiplying result on r0 we obtain
3Mξ21r0 +mν3Bz,r +
1
2
mν1B
′ > 0, (65)
remind (see (1,2)) that for z = 0 we have Bz,z = B′.
The last term in (65) due to (64) is less than zero
i.e. condition (65) is a more stringent condition than
(60). Thus, the condition of (65) is the condition of
the positive de�niteness of the matrix Q1.
Let's consider the condition of the positive de�-
niteness of the matrix Q2.
The �rst condition of (59) holds trivially.
It is interesting to consider the remaining diagonal
elements of the matrix.
Condition Q77 is easily derived from (62),(63).
Condition det(Q
[2]
2 ) > 0 is reduced to the positiv-
ity of the second of diagonal element
mν3
(
Bz,rr +
Bz,r
r0
)
= −mν3Bz,zz > 0. (66)
174
This simple condition is purely geometrical and not
dependent of the others.
After rather cumbersome transformations the
condition det(Q
[3]
2 ) > 0 can be written in the form
3Mξ21 −
1
ν23
1 +
ν2
1
αMr20
1− ν2
1
3αMr20
× (67)
(
1 + ν21
Bz,r
r0Bz,zz
)
mν3Bz,rr > 0;
When ν1 → 0 this condition moves to the condi-
tion of positivity of the 3rd of diagonal element of the
matrix Q2.
For condition det(Q2) > 0 we can get a few mem-
bers of the Laurent expansion in the small parameter
ν1, but even in this approximation the expressions are
too cumbersome for the analysis in analytical repre-
sentation not to mention about exact expression for
det(Q2). Therefore, not seems advisable to trying
infer the conditions of positive de�niteness in analyt-
ical representation. Especially because the expression
includes the variables ξ1, ν1, ν3, ξ̃2 that are not ele-
mentary.
The main aim of this paper is to prove of stability
of the relative equilibrium of the Hamiltonian system,
it mean that exists of physically stable orbital motion
of the magnetic dipole under the mutual action of ax-
ially symmetric magnetic �eld and gravity �eld (the
levitation of the Orbitron).
Conditions (58) and (59) are the analytic condi-
tions of stability of the levitating Orbitron. Thus, we
solve the task on the �rst (or theoretical) level. To
solve the task on the second (or physically) level it is
su�cient show the reasonable parameters of the sys-
tem, for which the conditions of stability executed.
Of course, if there is one such set, then there are in-
�nitely many such sets.
As will be shown in the next section such param-
eters are exist.
11. NUMERICAL SIMULATION
The determinants (58),(59) were programmed.
Maple allows to return results both in the symbolic
form and as the numerical values of the functions.
Guided by the formulas (63),(65)�(67) and tak-
ing into account the physical arguments, the suit-
able model parameters have been found. For these
parameters all condition (58),(59) are performed nu-
merically.
Initial conditions, magnetic materials and geo-
metrical parameters were selected.
The magnetic poles and tablets (the magnetic
dipole) were made ofNd−Fe−B and have the follow-
ing physical characteristics: ρ = 7.4 · 103(kg/m3) �
density of the material and Br = 0.25(T ) � residual
induction.
Then the magnetic �charge� of the Orbitron's
poles required to ensure the stability will be κ =
351.5625(A ·m), and the distance between the poles
is L = 2h = 0.1(m).
Linear �eld for compensation of the gravity force
is characterized by two parameters of our model{
B′ = 0.35723477320570427127 (T/m),
B0 = 2.985 (T ).
(68)
Movable magnet (dipole) selects in the form of
disk with diameter d = 0.014(m) and height l =
0.006(m). Its parameters are as follows: the magnetic
moment µ = 0.18375(A · m2), M = 0.00683484(kg)
with mass and α = 1/I⊥ = 0.9594(1/(kg·m2)), where
I⊥ = I1 = I2 the moment of inertia of the body.
For these parameters of our model in the point
of the researched relative equilibrium total magnetic
�eld will be{
B1 = −0.17861738660285213564 (T ),
B3 = 2.9898002901596414059 (T ),
(69)
So for the orbit with r0 = 1.5(m) and h =
0.075(m) we obtain the following parameters for the
stable relative equilibrium (see also (20)):
ξ1 = 6.6142 (rad/s),
ξ̃2 = −138.8134577 (rad/s),
ν1 = −0.059625567564610698431,
ν3 = 0.99822081309327449053,
π1 = −0.86270609223278 10−6(kg m2/s),
π3 = 0.15132393025362 10−4(kg m2/s).
(70)
With these parameters of the system all the con-
ditions of stability (58),( 59) are satis�ed. It seems
that these values appear to be quite reasonable for
the experimental implementation.
In addition the numerical simulation of our model
conducted base on the ODE [13, 1, 14] that describe
the dipole motion in the external magnetic and grav-
itational �eld.
˙⃗x = p⃗/M,
˙⃗p = ∇(µ⃗ · B⃗)−Mge⃗3,
˙⃗µ = (π⃗ × µ⃗)/I⊥, µ⃗ = mν⃗,
˙⃗π = µ⃗× B⃗.
(71)
For the relative equilibrium with parameters (68)
� (70) we calculated the sheaf of the trajectories for
the movable magnetic dipole. Random initial val-
ues were simulated in the neighborhood of the rela-
tive equilibrium so that deviation from the relative
equilibrium was less then one percent from the cor-
responding value dynamical variable. Totally 100
tosses was done. And Monte Carlo method con�rms
the stability of the all trajectories (by ten full turns).
Note that in the case of unstable motion of the mag-
netic dipole it falls down or �ys away, usually, on the
�rst turn of the orbit.
As it was mentioned above these tests is not in-
tended to prove system stability because it is proved
analytically and that is the main core of this work.
But very important that the test is based on (71) and
completely independent of group-theoretical methods
of Hamiltonian mechanics that were applied above.
175
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// Physica. 1999, v.D 126(1), p. 1-17.
ÌÀÃÍÈÒÍÀß ËÅÂÈÒÀÖÈß Â ÑÈÑÒÅÌÅ ÎÐÁÈÒÐÎÍ
Ñ.Ñ.Çóá
Äîêàçàíà âîçìîæíîñòü óñòîé÷èâûõ êâàçèïåðèîäè÷åñêèõ äâèæåíèé ìàãíèòíîãî äèïîëÿ, íàõîäÿùå-
ãîñÿ ïîä äåéñòâèåì êàê âíåøíåãî ìàãíèòíîãî ïîëÿ, òàê è îäíîðîäíîãî ïîëÿ ñèëû òÿæåñòè. Äëÿ äî-
êàçàòåëüñòâà èñïîëüçóþòñÿ òåîðåòèêî-ãðóïïîâûå ìåòîäû ãàìèëüòîíîâîé ìåõàíèêè, à èìåííî: energy-
momentum ìåòîä. ×èñëåííîå ìîäåëèðîâàíèå äåìîíñòðèðóåò âîçìîæíîñòü ðåàëèçàöèè óñòîé÷èâûõ äâè-
æåíèé ïðè ôèçè÷åñêè ðàçóìíûõ ïàðàìåòðàõ ñèñòåìû.
ÌÀÃÍIÒÍÀ ËÅÂIÒÀÖIß Â ÑÈÑÒÅÌI ÎÐÁIÒÐÎÍ
Ñ.Ñ.Çóá
Äîâåäåíî ìîæëèâiñòü ñòiéêèõ êâàçiïåðiîäè÷íèõ ðóõiâ ìàãíiòíîãî äèïîëÿ, ùî çíàõîäèòüñÿ ïiä äi¹þ
ÿê çîâíiøíüîãî ìàãíiòíîãî ïîëÿ, òàê i îäíîðiäíîãî ïîëÿ ñèëè òÿæiííÿ. Äëÿ äîâåäåííÿ âèêîðèñòîâó-
þòüñÿ òåîðåòèêî-ãðóïïîâûå ìåòîäè ãàìiëüòîíîâî¨ ìåõàíiêè, à ñàìå: energy-momentum ìåòîä. ×èñåëüíå
ìîäåëþâàííÿ äåìîíñòðó¹ ìîæëèâiñòü ðåàëiçàöi¨ ñòiéêèõ ðóõiâ äëÿ ôiçè÷íî ïðèéíÿòíèõ ïàðàìåòðàõ
ñèñòåìè.
176
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| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Zub, S.S. 2015-04-18T13:29:50Z 2015-04-18T13:29:50Z 2014 Magnetic levitation in orbitron system / S.S. Zub // Вопросы атомной науки и техники. — 2014. — № 5. — С. 168-176. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 45.05.+x, 45.20.-d, 45.20.Jj https://nasplib.isofts.kiev.ua/handle/123456789/80480 This paper devoted to proof the existence of stable quasi-periodic motions of the magnetic dipole that is under the action of the external magnetic field and homogeneous field of gravity. For proof we used energy-momentum method that is the group-theoretic method of Hamiltonian mechanics. Numerical simulation shows the possibility of realization of stable motions with physically reasonable parameters of the system. Доказана возможность устойчивых квазипериодических движений магнитного диполя, находящегося под действием как внешнего магнитного поля, так и однородного поля силы тяжести. Для доказательства используются теоретико-групповые методы гамильтоновой механики, а именно: energy-momentum метод. Численное моделирование демонстрирует возможность реализации устойчивых движений при физически разумных параметрах системы. Доведено можливiсть стiйких квазiперiодичних рухiв магнiтного диполя, що знаходиться пiд дiєю як зовнiшнього магнiтного поля, так i однорiдного поля сили тяжiння. Для доведення використовуються теоретико-групповые методи гамiльтонової механiки, а саме: energy-momentum метод. Чисельне моделювання демонструє можливiсть реалiзацiї стiйких рухiв для фiзично прийнятних параметрах системи. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Вычислительные и модельные системы Magnetic levitation in orbitron system Магнитная левитация в системе орбитрон Магнiтна левiтацiя в системi орбiтрон Article published earlier |
| spellingShingle | Magnetic levitation in orbitron system Zub, S.S. Вычислительные и модельные системы |
| title | Magnetic levitation in orbitron system |
| title_alt | Магнитная левитация в системе орбитрон Магнiтна левiтацiя в системi орбiтрон |
| title_full | Magnetic levitation in orbitron system |
| title_fullStr | Magnetic levitation in orbitron system |
| title_full_unstemmed | Magnetic levitation in orbitron system |
| title_short | Magnetic levitation in orbitron system |
| title_sort | magnetic levitation in orbitron system |
| topic | Вычислительные и модельные системы |
| topic_facet | Вычислительные и модельные системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80480 |
| work_keys_str_mv | AT zubss magneticlevitationinorbitronsystem AT zubss magnitnaâlevitaciâvsistemeorbitron AT zubss magnitnalevitaciâvsistemiorbitron |