Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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238
L.
Pantelidis
"It E
lR,
d(.r,y(t))
:::;
d(1:,z) + d(z,y(t)) <
E3
+
E1
<
2E1
<
E.
We
have shown
that
"IE>
O,:J5
>
0:
F
t
(B(.T,5)) C
B(1:,
E),
"It E
lR,
therefore
1:
is
Lyapunov
stable
on
S. 0
One
may want
to
include
the
case
that
:r
has zero spin by replacing
the
semi closed
interval in
the
definition of
fh).n
with a closed interval. However, if v'" = 0
the
proof
of
the
Lemma
breaks down for
the
map
R
cannot
be defined in
that
ca.se.
Now
a.s
long
as t.he
total
spin docs
not
vanish,
Lemma
5.2 immediately applies
t.o
t.he following four
cases:
• Ferromagnet.
(gjl
:::>
0 for all
j,
I).
The
FR
stat.e (all spins parallel) corresponds
to
minimum h.
• Ant.iferromagnet without intra-sublattice interaction (bipart.it.e lattice, sublattices
A
and
B,
gj(A)I(A)
=
gj(B)I(B)
=
0,
gj(A)I(B)
:::;
0).
The
FR
statc
corresponds
t.o
maximulIl h.
• Ferromagnet wit.hout. intra-sublattice interaction
(bipartitc
lattice, sublattices A
and
B,
gj(A)I(A)
=
gj(B)I(B)
=
0,
gj(A)I(B)
:::>
0).
The
AF
state
(all spins parallel in a
sublat.tice, sublattices anti-parallel) corresponds
to
maxilIlUIn h.
• Antiferromagnet
(bipartite
lattice, sublattices A
and
B,
gj(A)I(A)
:::>
0,
gj(B)I(B)
:::>
0,
9j(A)I(B)
:::;
0).
The
AF
stat.e corresponds
to
minimum
h.
Note
that
in all four cases
the
extrema
arc global. Moreover,
they
arc
extrema
on every
leaf
S, i.e., for
random
spin lengths (ferrimagnetie case).
On
the
other
hand, a
LFP
1:,
which
is
a
strict
local
extremum
of
the
Hamiltonian,
may
well be
unstable
if its spin
is
zero. To show this, consider
an
antiferromagnet or
ferromagnet with a finite
bipartite
lattice
without
int.ra-sublattice exchange
and
with
periodic
boundary
conditions.
In
this
ca.se,
there
exists a solution such
that
all the spins
in a sublattice are equal
[36].
The
sublattice
total
spins
VA
and
VB
precess
around
the
direction of V =
VA
+
VB
(Figure
2)
with frequency
Now notice
that
if
VA
f 1)B,
then
as
the
geometric angle between
VA
and
VB
tends
to
7r,
the
angle between V
and
VA
(or
VB)
tends
to
either 0 or 7r. In either case
the
amplitUde
(i.e.,
the
radius) of revolution of each spin t.ends
to
zero in agreement
with
the
fact
that
the
AF
state
with V f 0 must
be
a stable LFP. However, if
VA
=
VB,
t.he angle between
V
and
VA
(or
VB)
tends
to
7r
/2
which corresponds
to
maximum
amplitude
of revolution,
i.e.,
the
spins wander away from
their
nearly equilibrium position.
Of
course,
at
this
limit
the
frequency tends
to
zero. Nevert.heless,
we
have found periodic
orbits
passing
arbitrarily
close
to
t.he
AF
st.at.e on
the
equal-spins leaf (which has V = 0), while also
moving far from this
state
after a long enough time.
We
conclude
that
the
AF
state
on
the
equal-spins leaf
is
unstable.
Qualitative Dynamics
of
Interacting Classical Spins
239
j
/'
-
-I-
II
r :
'~
-
r-
-
~
I
I
lJ
I
j
I
~v
\ \
V
v=O
\ \
\ \
I
1\
1\
\ I
1\
I I
v
iO
II
Figure
2. Solutions for a 4 -spin ring with the two spins in each subiatbee equal.
If
the total spin v of the
AF
state is not
zem
(left), the
AF
state is stable.
If
v vanishes
(right), the
AF
state is unstable.
Acknow
ledgments
The
author
is
indebted
to
Dr. Harilaos Skiadas
for
helpful discussions.
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The
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A Transmission Line Model for the Spherical
Beltrami Problem
C.
D. Papageorgiou I and
T.
E.
Raptis
2
I Dept. of Electrical & Electronic Engineering
National Technical University of Athens, Greece
(e-mail: chrpapa@central.ntua.gr)
2 Division of Applied Technologies
National Centre for Science and Research
"Demokritos"
Patriarchou
Grigoriou & Neapoleos, Athens, Greece
(e-mail: rtheo@dat.demokritos.gr)
241
Abstract:
We extend a previously introduced model for finding eigenvalues and eigenfunctions
of
PDEs with a certain natural symmetry set based on
an
analysis
of
an
equivalent transmission line
circuit. This was previously applied with success
in
the case of optical fibers [8],
[9]
as
well
as
in the
case of a linear Schroedinger equation
[10],
[II]
and recently in the case of spherical symmetry (Ball
Lightning) [12]. We explore the interpretation
of
eigenvalues as resonances of the corresponding
transmission line model. We use the generic Beltrami problem of non-constant eigen-vorticity in
spherical coordinates
as
a test bed and
we
locate the bound states and the eigen-vorticity functions.
1.
Introduction
The notion
of
a Beltrami field appears first
in
hydrodynamics
in
the studies of Eugenio Beltrami
[I] at the end of the
19
th
century where it has found widespread application in vortical fluid states
with a characteristic helical geometry. Its rediscovery
in
modern electromagnetism took place at the
middle
of
the 20
th
century when Lundquist
[2]
proposed the same model for galactic magnetic fields
where the field appears
to
be parallel
to
its own rotation (neglecting displacement current). Later,
Lust and Schluter
[3]
and Chandrasekhar and Woltjier
[4]
proposed a theoretical justification of
these states
in
terms
of
magnetic plasma equilibrium. Several types of solutions are reviewed
in
[6]
and [7].
Nowadays, the Beltrami equation serves
as
a model of
a)
"Force-Free Fields" for which the magnetic part of the Lorentz force disappears
(J
II
B) in
equilibrated plasma and
b) for parallel
Ell
B components when applied to the vector potential directly (in which case
we
get
E
~
-{()A
,B
~
A(r
,
t)A).
The Beltrami problem
is
defined by at least two and at most three PDEs given by
V
xA
~
A(r,t)A (J a)
(J b)
The second condition is usefull
in
applications
in
electromagnetism when A stands either for the
vector potential or the magnetic field. Applying twice the rotation operator
in
the first
of
(I)
gives
the most general representation
of
the non-linear inhomogeneous Helmholtz operator for Beltrami
flows
as
242
C.
D.
Papageorgiou and
T.
E.
Raptis
V'A-l'(r,t)A=VAXA
(2)
In
the case
of
constant A the inhomogeneous part is
of
course zero. Solutions
of
this class are usually
termed linear
"Trkalian" fields
as
they were first studied extensively by Viktor Trkal [5].
In
the next paragraph
we
turn to equation
(I)
and attempt to locate eigen-modes in terms
of
the
orthogonal basis
of
vector spherical harmonics. In section 2, we perform the expansion and derive
the associated
ODEs and in section 3,
we
introduce the transmission line model for this system. In
section 4,
we
solve the model for the associated angular momentum eigenvalue and
we
approximate
the specific eigen-mode for a
"bound state".
2. Spherical expansion
of
the Beltrami equation
In
what follows we restrict attention to purely radial functions for the eigen-vorticity
ACr).
We
introduce the usual basis
of
Vector Spherical Harmonics as
CY
= r'l"m'
tft
=
rV'l',m.
<I>
= r X
V'l"m
}
and
we
seek an appropriate decomposition
of
the field A and its eigen-vorticity A(r). Then there must
be a unique reference frame for which both equations
(I
a-b) can be satisfied by taking
,
A = I I
[a'm(r)Y"'m
+
b'm(r)tft'm
+c'm(r)<I>,J
(3)
10",0
1I1'=-{
Then condition VA = 0 is satisfied
if
VA -
~
~[(a
+
2a'm
J -
1(1
+ I) b (
J'l'
-0
-
L..J
L...J
1m
/m
r)
/111-
!=()
111=_/
r r
(4)
while the choice
of
the A function is restricted by
VA.
A =
O.
For a purely radial A function it
suffices to take the additional condition that the radial part
of
A is set to zero which is a very
stringent condition on the coefficients
of
(3). We note however that in the more general case we may
allow for violations
of
condition
(I
b).
In
electromagnetic applications
we
may instead assume the
presence
of
static charges satisfying the Lorenz gauge a
,t/J
=
-cV
A =
cxlv
A.
A .
From now on we will deal solely with the first condition
(I
a) for the rotation
of
A.
We now
turn back to the first
of
(I)
for the rotation operator which obtains
I
~
I' {
1(1
+ I) - (
c)
- ( . b a J - }
VxA
=
----c
Y -
i:
+-""-
'l'
+ b
+-""--~
<I>
(5)
1m
fill
1m
/111
/111 /111
!=()
111=-1
r r r r
For the rotation to be parallel to
AA
we
should have
1(1
+
I)
----e,1II
=
All/III
r
C/I/!
+~+AbIIl1
=0
r
b +
him
-
~
-
AC
= 0
1m
fill
r r
(6)
A Transmission Line
Modelfor
Spherical Beltrami Problem 243
Using the first of (6) to the rest two equations and replacing
elm
with e
and
blm
with b results in the
ODE system
(rb)' =
[A
-
1(1
+
l)](rc)
r2
A
(re)'
=
-A(rb)
3. The Transmission Line model
(7)
At this section we introduce
an
isomorphism between the system (7) and a lossless transmission line
via the mapping
v = re
--->
dV
=
-jAI
dr
.
dl
y'
JI
=rb
--->-=--V
dr
jA
(8)
Where we have introduced the propagation constant defined by the relation
Equivalently we may define the transmittances
to
be
Z,
=
-jA
Y,
=-j~
(9)
Then the characteristic impedance Z and the propagation factor y
of
the equivalent transmission line
are defined through the relations:
-Z
rz;
.
.,1,
'~l
=Vy:=Jy=V;:y--l
y=~Z,y,
(10)
From standard transmission line theory
we
have that propagation is defined for
2 I A I
Y
<0
or
r>
~
-yl(l+l)
While damping (evanescent modes) for
,
1.,1,1
r>O
or
r<
~
-yl(l+l)
(11)
(12)
An
element
of
such a transmission line corresponds
to
a lumped-element model of the initial domain
[0,
ooj
of the radius r into a set of mesh points r
o
'
r,
"'"
r"
with each element interpolating into the
arbitrary intervals
[r",r,,+,j,We then take each element
to
correspond to a segment of infinitesimal
width
LY"
=
':,+1
-
,:,
for which the propagation factor y and the characteristic impedance Z can be
244
C.
D. Papageorgiou
and
T.
E.
Raptis
taken approximately constant (defined
as
Yn
and Z
,,
) . For such
an
element we may take the
equivalent relations
for
a homogeneous transmission line
as
follows.
[
V(~,ll
[ cosh(q,,)
I
(~,l
= sinh(q,,) /
Z"
where
we
have put q" =
r"
.
/';.~
,
.
(13)
Thus the value
of
the input impedance at each point
of
the line will be given by the recursive
formula
Z(r
) = V(r,,) = Z Z(r,,+l) +
Z"
tanh(q,,)
" I (r,,) " Z(r,,+
l)
tanh(q,,) +
Z"
(14)
The boundary conditions are defined through the respective
limitsZ(r
~
O),Z(r
~
00)
bas
ed
on
the
assumption that the asymptotic limit of a lossless transmission line corresponds
to
the fixed value of
the characteristic impedance given by
Z" according
to
(14) for r
~
0 and r
~
00.
4. Trapped modes and bound states
The form
of
the propagation constant leads
to
the possibility
of
"trapped" waves and their
respective eigen-modes. A necessary condition for trapping is
to
have at least one region r
1
::;
r
::;
r,
where
y2
(r)
< o allowing propagation and two disparate regions r <
r"
r > r
2
with
y2
(r)
> 0 where
only evanescent modes exist.
This situation is reminiscent
of
the bound states of quantum mechanical systems with the
function
11
2
(r)
playing a role analogous
to
that
of
the potential
in
the radial Schroedinger equation.
Evanescent modes are then analogous to the scattered wave functions above a threshold.
On the
other hand, a constant value
of
A allows only a simple root r3 =
~l(l
+
I)
/
Au
beyond which
propagation is allowed up
to
the infinity. We will next evaluate one such example
of
an eigen-
vorticity function supporting a trapped mode.
According
to
a standard technique introduced in [11] and [12] , a method for locating the eigen-
values and compute the associated eigen-mode is based on a simple resonance theory for
transmission lines.
In this viewpoint, we may take the bound state
to
correspond
to
an area where
ordinary resonance takes place through the condition
Z L = Zc between the "capacitive" and
"inductive" elements in the imaginary part
of
the respective complex impedance. This can
effectively be restated
as
(15)
Where
Z
+
(~)
and
Z
-
(~.)
are the input impedances
of
the equivalent transmission line
as
shown
from the point
rc (right and left of it).
Evolution of the above quantity
in
the region [0,00] prescribes also the associated eigen-mode.
In
order
to
locate a certain mode for a given value of angular momentum
we
iterate the mapping (14)
forward
(0
~
r, ) and backward (00
~
1',
.) and expand the interval until
we
find a root of (15).
As
an
example let us make a simple choice of the propagation factor given below
'=
L2(r-~)(r-r2)
h
L=l(l
1)
r 4
2'
were,
+
r + fir2r
(16)
A Transmission Line
Modelfor
Spherical Beltrami Problem 245
The propagating region
is
'i
:0;
r
:0;
r,
and the evanescent regions are r <
~,
r >
r,
U
·
hI·
,.2
L'
,'.
smg t
ere
atlOn
r = 2 -
/L.
It
can be proved that:
r
,12
=
L2
.(~
+r
2
)
r·(r'
+~r,)
(17)
For given integer values
of
I and real positive values
of
rl
using the equation (15) for
rl
and variable
values
of
r2>rl
i.e.
(18)
Quantities
Z+(~)
and
Z·(~)are
calculated through the recursive relation (14). Then the value
of
r2
for which the equation (18) holds can be defined and though it the proper function A(r) by (17) for
which the Beltrami equation
(1
a)
is fulfilled.
In Fig.
1 we show
fori
= I rl=1 and
r2=8.12
the exact form
of
the propagation factor y' and the
eigen·vorticity
,12
that is positive definite everywhere.
1=1
;r1=1
;r2=B.12
3,-----~-----,------._----_,------,_----~
.£
" 2.5
t
o
>
C
I
~
I I I I j
--I----~------T-------I-------I-------~------
I I I I
<lJ
OJ
.ijj
I I
2
--1---r------1------~-------r------r------
\ I I I I I
"
ffi
\ I
\ I I I I j
o 1.5
- - -
--\.
- - T - - - - - - I - - - - - -
-1-
- - - - - - r - - - - - - T - - - -
--
~
c
o
.~
g>
"'-
e
a..
0.5
·0.5'----~-----'------'-----'----~---~
o 2 4 6 B 10
12
radial distance r
Fig.!
In Fig. 2 we show the evolution
of
the Function (25)
of
the sum
of
the imaginary parts for 1= 1 and
rl=l.The
boundary values
of
the impedances are defined for the terminal points left and right
- + -
2d
h
I·Z-',1·
Z·,"",,,/
=Z(drI2),Z
'"'Ii,,/
=Z(R+drl
)
uetot
ereatlOn
=J._=J.
r
r·
(~
+
r,)
. The roots
(r-~)·(r-r,)
are between the poles
of
the Function (18) and can be easily calculated. The first three values
of
r2
are 8.12, 33.5428, 86.8528.
246
C.
D. Papageorgiou and
T.
E.
Raptis
1~1
,rl~1
,the roots
of
the Function are between the poles of the Function
300
~
~oo
---I----~----~-----I----~----~----~---
'"
"
c::
"g>
.~
-300
---I-------j----,-----I-------,-------j----.,---
'0
, , ,
---I----'----~-----I----~----~----~---
,----
c::
-400
0
"-B
, "
c::
::>
-500
LL
I I I I I I I I
--~----~----~----~---------~----~---
~----
-600
20
40 60
80
100 120 140 160
180
eigen
values of
r2
as roots
of
the Function
Fig. 2
In
appendix I a matlab program
of
the Function( IS) is shown.
5. Beltrami solutions through the Lorentz group
From the previous analysis a fundamental observation occurs based on the hyperbolic relation
connecting the triplet
{y,
It,
LI
r}
which can be written
as
Going to the transformed variables
x =
y'(r)
y
=lt2(r)
T =
j/
r
(26)
(27)
leads to the Lorentzian manifold M(2,1) with the usual Minkowski metric
of
signature (1,1,-1) that
has an invariant
"null" length (virtual photon path)
s2=X'+Y'-(LT)2=O
(2S)
and a "speed-of-light" c = L.
This means that given a set
of
"core" functions (Yo,lt
o
) and an angular momentum value L we
can produce
an
infinite family
of
deformed solutions which are connected by both proper and
improper Lorentz transforms. Thus, for the choice represented by (17)
we
may find a continuous
A Transmission Line
Modelfor
Spherical Beltrami Problem 247
trajectory passing through different families of functions
{r",
A"
) by a direct application
of
the
generic linear transformation (X
,T)~(X',T'):
(rv,A,,)~(rv+'
,A
v
+,) where the matrix
A(L)
is parametrized according to the four conjugacy classes of the full Lorentz group (elliptic,
hyperbolic, parabolic and Loxodromic) thus leading
to
different families of solutions.
6.
Conclusions
We showed that a general class
of
solutions of the spherical Beltrami problem can be found
in
terms
of vector spherical harmonics of which the resulting system is equivalent with a generic loss less
transmission line. When applied
to
a specific choice of eigen-vorticity function it allows finding the
set of parameters necessary for each mode
to
exist. We have given explicitly
an
example of such a
function for which trapped modes can exist in a region. We believe that this phenomenon deserves
further examination especially with recourse to electromagnetic applications.
References
[1]
E.
Beltrami, Rend. Inst. Lombardo Acad. Sci. Lett. 22 (1889) 121-131
[2]
S.
Lundquist, Ark. Fys. 2 (1951)
361
[3]
R.
Lust, A. Schluter,
Z.
Astrophys. 34 (1954) 263
[4]
S.
Chandrasekhar,
P.
C.
Kendall, Astrophys.
J.
126 (1957) 126
[5]
A.
Lakhtakia, Czechoslovak
J.
Phys. 44(2) (1994)
89
- 96
[6]
H.
Zaghloul,
O.
Barajas, Am.
J.
Phys. 58(8) (1990) 783-788
[7]
G.
E.
Marsh, "Force-Free Magnetic Fields", World Sci. (1996)
[8]
C.D. Papageorgiou,
10
Kanellopoulos,
J.
Phys.
A:
Math. Gen.
15
(1982) 2569-2580.
[9]
c.
D.
Papageorgiou, A.
D.
Raptis,
Compo
Phys. Com. 43 (1987) 325 - 328
[10]
c.
D.
Papageorgiou et aI., J.
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App. Math. 29 (1990) 61-67
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Papageorgiou et al.,J.
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Phys. 88(2) (1990) 477-483.
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D.
Papageorgiou,
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E.
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