J. Electromagnetic Analysis & Applications, 2011, 3, 7-12
doi:10.4236/jemaa.2011.31002 Published Online January 2011 (http://www.SciRP.org/journal/jemaa)
7
Stokes Representation for the Solutions of
Maxwell-Vlasov
Monica Pîrvan, Constantin Udrişte
University Politehnica of Bucharest, Department of Mathematics I, Splaiul Independeţei, Bucharest, Romania.
Received September 25th, 2010; revised November 12th, 2010; accepted December 15th, 2010.
ABSTRACT
Maxwell-Vlasov PDEs system describes the dynamics of plasma consisting of charged particles with long-range inter-
action. Their solutions can be written using some Stokes potentials. Section 1 presents the experimental devices which
can produce a magnetic trap. Magnetic geometric dynamic provides mathematical tools for describing the magnetic
flow (see [1-7]). Stokes representation for the solutions of PDEs as Maxwell PDEs or Maxwell-Vlasov PDEs are used
analyzing electromagnetic energy in magnetic traps. Section 2 studies Maxwell-Vlasov PDEs system. Stokes represen-
tation of its solutions, using Maximum Principle for a multitime optimal control problem, is obtained. Section 3 dis-
cusses a method for changing a given ODEs system into a geodesic motion under a gyroscopic field of forces (geomet-
ric dynamics). Section 4 proposes a modified form for Maxwell-Vlasov PDEs, by replacing the classical gyroscopic
force with the one appearing in geometric dynamics. Stokes representation for the solutions of modified Max-
well-Vlasov PDEs is also obtained.
Keywords: Maxwell-Vlasov PDEs, Magnetic Traps, Dynamics of Plasma, Gyroscopic Field of Forces, Dynamic
Systems
1. Magnetic Traps
Our theory is appropriate for the study of electromagnetic
energy in magnetic traps. These devices are used for at-
oms or charged particles:
1) magnetic traps, used to trap neutral atoms in a
magnetic field gradient:
2) Penning trap, used to trap charged particles or ions
in a combination of electrostatic potential and uniform
magnetic field;
3) magneto-optical trap (or MOT), a trap using a
magnetic gradient and laser to trap neutral atoms;
4) magnetic tweezers, a trap using a magnetic field to
trap micrometreseized ferromagnetic beads.
The first magnetic trap was realized by M. Ioffe for
plasma confination having in mind the following idea: if
an atom is not too energetic, it can be held in a magnetic
nonzero minimum – a region from which the strength of
the magnetic field grows stronger in every direction [3].
Other modern magnetic traps, analyzed from the perspec-
tive of geometric dynamics in [6,7], are used today in
EDM experiments or for creating an ECR source of MCI.
The ALPHA experiment at CERN [8] seeks to trap an-
tihydrogen atoms inside a magnetic bottle consisting of a
super-conducting octupole magnet and two mirror coils.
We do it with superconducting magnets – mirror coils to
create a minimum in the middle of the trap’s axis, plus an
octupole magnet to create a minimum in the center of the
trap’s radius. To study the energy spectra of antihydrogen
atoms it will be necessary to keep them from blowing
themselves up for much longer than a few thousandths of
a second currently possible – then will have to be de-
tained for at least a few seconds. Luckily, even neutral
atoms have a small magnetic moment; they can be con-
fined by a magnetic field of the right shape and strength.
The current ALPHA plan uses magnetic fields specially
shaped by an octupole magnet as part of the trap. The
design of ALPHA magnetic trap was developed and re-
fined at Berkeley Lab by a large team of Berkeley Lab
and UC Berkeley scientists, visitors and students.
2. Stokes Representations for the Solutions of
Maxwell-Vlasov PDEs System
The ideas of this paper start from the papers [1-20] and
from the conversations with D. Wang about the Vlasov-
Maxwell-Boltzmann PDEs, carried at Siam Conference
on Analysis of Partial Differential Equations, December

Stokes Representation for the Solutions of Maxwell-Vlasov
8
7-10, 2009, Miami, Florida.
We consider the case of collisionless plasma, i.e., the
case of a single species of particles with mass m and
charge e.
Let
    
00000 0
12
,,,,,,,, ,
ΩΩ

Ω
f
ffff f
txvtxvtx txvv

be a hyperparallelipiped in determined by the oppo-
site diagonal points and

7
R

v
000
,,tx ,,
f
ff
txv .
If


,Etx represents a electric field,
1
C


,Btx

1
C
vx
represents a magnetic field and is
plasma particle number density per phase space (
1
C

,,Ltxv


being the velocity), Maxwell-Vlasov PDEs system is
1
,,
11
curl, curl,
div, div0,
LLe L
vEvB
txmcv
BE
EB
ctct c
EρB
 

 

 


0,
1
j
,
(1)
where c is the speed of light, j is the current defined by L,
 

2
,
0
Ω
,,
f
vv
jtxeLtxvvdv
and
ρ
is the charge density,
 

2
,
0
Ω
,,
f
vv
ρtxeLtxvdv,

Theorem 1. Let

 000
,,,, ,
Ω
f
ff
txvt xv be the parallelepi-
ped fixed by two diagonal points and

000
,,txv

,,
f
ff
txv and
 

7
,,,, ,,, ,,,txvN txvN txvntxv

be the unit
normal vector of the boundary
 
000
,,,, ,
Ω
f
ff
txvt xv
solutions of Maxwell - Vlasov PDEs (1) admit the Stokes
representation
. The
 
 
 
 

1
,curl, grad,
,,grad,,
1
,curl,grad,
,,grad,,,
v
v
q
Etx ptxαtx
ct
eLtxv γtxv
m
p
Btx qtxβtx
ct
eLtxv γtxv v
mc








(2)
together with the condition

,
1,grad, ,0
x
v
γgrad γv
t
eEvBγtxv
mc



(3)
and the boundary conditions





 

 


  



 



7000
7000
,,, ,,,
,, ,,,,
1,,,Ω,,,, ,0,
,,, ,,,
,, ,,,,
1,, ,,
1,,,Ω,,,, ,0,
1
,, ,
fff
fff
αtx N txvptxN txv
eγtxv LtxvN txv
m
qtxntxvtx vtxv
c
βtx NtxvqtxN txv
eγtxv LtxvN txv
m
γtxvLtxvv N
mc
ptxntxvtx vtxv
c
eγtxvEv BN
mc
c













 

7000
,,,, Ω,,,, ,0,
fff
γtxvntxvtx vtxv


(4)
where






,, ,, ,, ,, ,,ptx qtx αtx βtx γtxv

are
Stokes potentials.
Proof. We consider the following multitime optimal
control problem

 
 



,
000
Ω
,,,, ,
22
2
1
max ,2
(,, )
fff
Etxvtxv
IE
Et xB t xvdxdvdt
  


,
(5)
constrained by Maxwell-Vlasov PDE system (1), where
the electric field
1
C


,Etx is a control vector func-
tion and the magnetic field and the veloc-
ity v are state vector functions,
1
C

,Btx

 
00 0
,,,
f
ff
Bt xBBtxB
Let






1, 3
,,
ii
ptxp tx ,
 
1, 3i, ,,qtx qtx


,tx  ,


,tx  ,


,,txv  be functions, consid-
ered as co-state variables (Lagrange multipliers) and the
Lagrange function
1
C


 

 
  
    

1
22
2
12 3
123
L,,,,,,,,,
1,,
2
1
,,curl,
11
,,curl,,
,div,,,div,
,,
1
,.
txvEBpq
EtxBtx v
B
ptx Etx
ct
E
qtxBtx jtx
ct c
txEtxtxtx Btx
LLLL
txvv vv
tx xx
eL
EBv
mc v



 















Stokes Representation for the Solutions of Maxwell-Vlasov9

.
The optimal control problem (5), with constraints (1),
becomes

 
 



,, ,, ,
000
,
00 0
max ,
L,,,,,,,,, ,
,,,
txvt x v
fff
E
ff f
IE
txvEBp qdxdvdt
Bt xBBtxB


  


(6)
Supposing that optimal problem (5), with constraints
(1) admits an interior optimal solution , we con-
sider the variation , where

,Etx

,htx  
 
,, ,Etx Etx 
0

 and is a arbitrary vector function.
We define

,htx

,,Btx

1
C


and


,,,Ltxv

the correspon-
dent state vector function, respectively the plasma parti-
cle number density for the variation

,,Etx


. For
h

 , we construct the Lagrangian

,
1,,ltxv

by
the formula
 

 

 
  
  
    



22
2
1
L,,,,,,,,,,,,
,,,,,, ,,
1(,,,,)
2
1
,,,,curl,,
11
,,,,curl,,,
,div,,,,div,,
,,,,,
Etx Btx Ltxvptx
qtxtxtx txv
EtxBtx v
B
ptxtxEtx
ct
E
qtxtxBtxjtx
ct c
txEtxtxtx Btx
L
txvtxv v
t
 





 

 











 
  
1
23
23
,,,
,,,,,,
1
,,,,,,,.
Ltxv
x
LL
vtxvvtxv
xx
eL
EtxBtxv txv
mc v


 







 

and the integral function
 
 
,,, ,,
000
1,,,
txvtx v
fff
I
ltx vdxdvdt 


It is necessary for

I

to satisfy the condition
,
 
0II  

h

  . If


1,7
,, ,,
i
i
ntxvn txv 

7
,,NNn, then

 
,,, ,,
000
3 21
1
32
3 12
12
131
1
3 12
23
221
1
'
1
1
txvtx v
fff
p pq
IE
2
3
x
xct x
p pq eLh E
vmxxc tx
q pp eLh E
mvxxtc x














 
 

 


 
 

 





,,, ,,
000
3
3
3 12
1
311
12
322
23
31
1
132 3
1
1
txvt x v
fff
eL hdxdvdt
mv
q pq
Bct xxx
Bp ee
Lv LvB
mc vmc vct
qqee
LvL
xxxmcv mc














 

 





 





 
 


 
,,, ,,
000
2
3
1
312
32
213 1
3
1
2
123
12 3
2132 2
1
1
1
txvt x v
fff
B
v
v
pqq e
BL
ctxxx mcv
B eL vdxdvdt
mc v
vvv
tx xx
evEBv Bv
mvc

v




 







 

 
 









 

 








 
,,, ,,
000
3
21331
2
32113
3
12 34
32
7
11
12 35
31
1
1
1
1
txvtx v
fff
eEBvBv
mv c
eL
EBvB vdxdvdt
mvc
e
npn pnLn
m
qn h
c
e
pnn pnLn
mc






























 

 

,,, ,,
000
7
22
1236 7
21 3
123 5
32 3
67 12 1
213 1
46
312
1
1
1
txvtx v
fff
qn h
e
pnpnnLnqn h
mc
e
nqnqn Lvn
mc
B
eLvnpnqnnq n
mc c
ee
LvnLv npn
mcmc c
 









 




 








3



72
1234 5
112 1
74
3
313232
5
213 13
11
B
ee
qn pnnLvnLvn
mc mc
Be
pnEBv vBn
cmc
eEBv vB n
m

 








 



 

  

 



Stokes Representation for the Solutions of Maxwell-Vlasov
10

67
32121
1
.
eL
EBvvBncnd
mc














To impose the condition , for any vector
function , we need a definition for the Lagrangian
multipliers via certain PDEs ([9-10]). We obtain the
Stokes representation (2), the condition (3) and the
boundary conditions (4).

'0 0I

,htx
3. Geodesic Motion in a Gyroscopic Field of
Forces
Let

,
M
g be a Riemannian manifold and X be a
vector field on M. We consider the flow

dx
X
x
dt 
In order to change this dynamical system into a geode-
zic motion under a gyroscopic field of forces on a double
dimension space, we build the quadratic Hamiltonian
[11-12]
 
2
1
H, 2
x
yyfx
where

2,ygyy

represents the Riemannian kinetic
energy and
  


2
11
,
22
f
xgXx XxXx
is the Riemannian energy density of the vector field X.
The Hamiltonian is conserved along the trajecto-
ries of the dynamics induced by the flow



dx dtXx
in the sense of the following two rules:
1) if is Levy-Civita connection of the Riemannian
manidold


,

M
g with the components i
j
k
G, then we
differentiate the first order differential equation

xdxdt X with respect to t and obtain
2
2
ii jk
i
jk
dxd xdxdx
G
dt dtdtdt dt


2) on the other hand, the right hand member becomes

ij
ii i
jj
dxdx i
X
XF
dtdt dt

 f
k
where iiih
jj kjh
F
Xgg X 
ii
is the external distin-
guished tensor field that characterizes the helicity of the
vector field X, and

h k
kj h
j
f
ggXX are the
contravariant components of the conservative force
f
.
We obtain a single-time geometric dynamics [11-12]
2
2
ijkj
ii
jk j
d xdxdxdx
GF
dt dtdt
dt 
i
f
The gyroscopic force iji
j
F
dx dtf consists in the
gyroscopic term ij
j
F
dxdt and the conservative term
i
f

.
Theorem 2. Every solution of second order ODEs
system is a horizontal geodesic of the Riemannian-La-
grangian manifold


1
, 1, iiki
j
jk j
RMgNGy F
4. Stokes Representation for the Solutions of
Modified Maxwell-Vlasov PDEs System
We consider next that electric field E and magnetic field
B do not depend explicitly on the variable time t. Re-
placing the classical gyroscopic force 1EcvB from
Maxwell-Vlasov PDE’s (1) with the gyroscopic force
ij
j
i
F
vf

 from geometric dynamics, for the vectorial
field
X
E

and Riemannian metric ij ij g

, and be-
cause curl E) = 0, we obtain the modified Maxwell-
Vlasov PDEs
12 3
123
112 23 3
0,
LLL
vv v
xxx
effffff
mxv xvxv




 

 

 

(7)
where f is density of electric energy, 2 12
f
E.
Theorem 3. Let


00
,, ,

f
f
x
vxv

be the parallelepiped
defined by diagonal points

00
,

x
v and

,

f
f
x
v and


,N


nN be the normal unit vector of the boundary

 00
,, ,
f
f
x
vxv

. If



p x

ii
px,





ii
xqx q,






,,
x
xx 
are Stokes potentials, with grad 0
v


and


,, ,
00
0
xvx v
ff



, then the solutions of modified
Maxwell-Vlasov PDEs (7) admit Stokes representation



  
curl gradEpx  

x (8)



  
curl gradBqx  

x (9)
with condition grad ,0
vv


, and boundary conditions






,, ,
00
,, ,
00
,, ,
00
0,
0,
,0
xvx v
ff
xvx v
ff
xvx v
ff
eE
NpNLN
mx
NqN
ef
vN N
mx







.

 

 






(10)
Proof. We consider the following optimal control
problem



 




,, ,
00
22
2
1
2xvx v
ff
I
max
EEExB xvdxdv

 



Stokes Representation for the Solutions of Maxwell-Vlasov11
subject to modified Maxwell-Vlasov PDE’s (7), where
  
1,3
ii
ExEx  is the electric control vector
function,
1
C
 

1,3
ii
BxB x  and

1,3
ii
vv are
state vector functions,
1
C


Bx 
00
B,

f
f
BBx ,
,

00
vx v

f
f
vx v.
Analogous with the proof of Theorem 1, necessary op-
timal conditions are equivalent with
12 3
123
213 2
122 3
13
31
curl gradgrad0,
curl grad0,
grad,grad ,grad0,
0, 0, 0,
0, 0,
0,
v
xv
eE
EpL
mx
Bp
e
vf
m
EE E
vvv
EE EE
vvvv
EE
vv







 

 





 




satisfying the boundary conditions








 
 




,, ,
00
,, ,
00
,, ,
00
,, ,
00
,, ,
00
,, ,
00
,, ,
00
4
1
5
2
6
3
45
21
56
32
0,
0,
,
0,
0,
0,
0,
xvx v
ff
xvx v
ff
xvx v
ff
xvx v
ff
xvx v
ff
xvx v
ff
xvx v
ff
eE
NpNLN
mx
NqN
ef
vN N
mx
ELn
ELn
ELn
LEn En
LEn En

















 

 













0 ,

 
,, ,
00
,, ,
00
64
13
0,
0,
xvx v
ff
xvx v
ff
LEnEn 




where

,nNN

is the normal unit vector of the
boundary


00
,, ,
f
f
x
vxv
 ,

4,5
i
i
Nn

,


1,3
j
j
Nn


.
Since grad 0
v

 and


,, ,
00
0
xvx v
ff

  from the
hypotheses, it results Stokes representation (8), (9), for
the solutions of Maxwell-Vlasov modified PDEs (7) to-
gether with the boundary conditions (10).
5. Conclusions
The present paper studies the electro-magnetic dynamic
systems from the perspective of optimal control theory
(see also [5,11,18]). Multi-time optimal control problems
when the functional is represented by the density of the
electromagnetic energy, subject to Maxwell PDE’s or
modified Maxwell-Vlasov PDE’s (classical gyroscopic
force 1EcvB

 from Maxwell-Vlasov PDE’s is re-
placed with the gyroscopic force ij i
j
F
vf from
geometric dynamics, for the vectorial field X = E and
Riemannian metric ij ij
g


) reliefy electromagnetic
evolutions from Stokes representations. The study of
electromagnetic energy in magnetic traps justify our re-
search.
6. Acknowledgements
We have benefited greatly from conversations with Prof.
Dr. V. Prepeliţă and Prof. Dr. I. Ţevy.
REFERENCES
[1] C. Udrişte, “Geometric Dynamics,” Kluwer Academic
Publishers, Article Southeast Asian Bulletin of Mathe-
matics, Springer-Verlag, New York, Vol. 24, 2000, pp.
313-322.
[2] C. Udrişte, “Nonclassical Lagrangian Dynamics and Po-
tential Maps,” Proceedings of the Conference on Mathe-
matics in Honour of Profesor Radu Roşca at the Ocassion
of His Ninetieth Birthday, Belgium, 11-16 December
1999.
[3] C. Udrişte and M. Postolache, “Atlas of Magnetic Geo-
metric Dynamics,” Geometry Balkan Press, Bucharest,
2001.
[4] C. Udrişte, “Tools of Geometric Dynamics,” Buletinul
Institutului de Geodinamică, Academia Româna, Vol. 14,
No. 4, 2003, pp. 1-26.
[5] C. Udrişte, “Maxwell Geometric Dynamics,” European
Computing Conference, Athens, Greece, 24-26 September
2007, pp. 1597-1609.
[6] C. Udrişte and M. Pîrvan, “Magnetic Dynamics around a
Configuration of Two Square AntiHelmholtz Coils,” Pro-
ceedings of 7th WSEAS International Conference Nonlin-
ear Analysis, Non-linear Systems and Chaos (NOLASC
08), Corfu, Greece, 26-28 October 2008, pp. 222-227.
[7] C. Udrişte and M. Pîrvan, “Magnetic Geometric Dynam-
ics around Tornado Trap,” Proceedings of the 10th
WSEAS International Co nference Mathematical Compu-
tational Mathematical Science and Engineering
(MACMESE 08), Bucharest, 7-9 November 2008, pp.
312-317.
[8] P. Preuss, “Angels, Demons, and Antihydrogen: The Real
Science of Anti-Atoms,” 2009. http://www.symmetry
magzine.org/breaking/2009/05/07/angels-demons-and-ant
ihydrogen-the-rel-science-of-anti-atoms/
[9] C. Udrişte and L. Matei, “Teorii Lagrange-Hamilton,” (in

Stokes Representation for the Solutions of Maxwell-Vlasov
12
Romanian) Monographs and Textbooks 8, Geometry
Balkan Press, Bucharest, 2008.
[10] C. Udrişte, “Simplified Multitime Maximum Principle,”
Balkan Journal of Geometry and Its Applications, Vol. 14,
No. 1, 2009, pp. 102-119.
[11] C. Udrişte, M. Ferrara and D. Opriş, “Economic Geomet-
ric Dynamics,” Monographs and Textbooks 6, Geometry
Balkan Press, Bucharest, 2004.
[12] C. Udrişte, “Geodesic Motion in a Gyroscopic Field of
Forces,” Tensor, N. S., Vol. 66, No. 3, 2005, pp. 215-228.
[13] H. Cendra, D. D. Holm, M. J. W. Hoyle and J. E. Mars-
den, “The Maxwell-Vlasov Equations in Euler-Poincaré
form,” Journal of Mathematical Physics, Vol. 39, No. 6,
1998, pp. 3138-3157. doi:10.1063/1.532244
[14] V. Ciancio and C. Udrişte, “Ioffe-Ştefănescu Magnetic
Trap,” Revue Roumaine de Sciences Techniques-Serie
Electrotechnique et Energetique, Vol. 49, No. 2, 2004, pp.
157-176.
[15] Y. Guo, “Global Weak Solutions of the Vlasov-Maxwell
System with Boundary Conditions,” Communications in
Mathematical Physics, Vol. 154, No. 2, 1993, pp. 245-
263. doi:10.1007/BF02096997
[16] E. Marsden and T. S. Raţiu, “Intro-Duction to Mechanics
and Symmetry,” 2nd Edition, Springer, New York, 1999.
[17] C. Neagu and C. Udrişte, “From PDE Systems and Met-
rics to Geometric Multi-time Field Theories,” Seminarul
de Mecanică, Sisteme Dinamice Diferenţiale, 79, Univer-
sitatea de Vest din Timişoara, 2001.
[18] M. Pîrvan and C. Udrişte, “Optimal Control of Electro-
magnetic Energy,” Balkan Journal of Geometry and Its
Applications, Vol. 15, No. 1, 2010, pp. 131-141.
[19] C. Udrişte, “Nonholonomic Aapproach of Multitime
Maximum Principle,” Balkan Journal of Geometry and Its
Applications, Vol. 14, No. 2, 2009, pp. 101-116.
[20] D. Wang and X. Hu, “Weak Stability and Hydrodynamic
Limit of Vlasov-Maxwell-Bolzmann Equations,” SIAM
Conference on Analysis of Partial Differential Equations,
Miami, Florida, 7-10 December 2009, p. 78.