Electromagnetic least squares waves

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Electromagnetic least squares waves

CONSTANTIN UDRISTE, IONEL TEVY

University Politehnica of Bucharest

Faculty of Applied Sciences

Department of Mathematics-Informatics

Splaiul Independentei 313

060042, BUCHAREST, ROMANIA

[email protected]; [email protected]

FLORIN MUNTEANU, DOREL ZUGRAVESCU

Institute of Geodynamics

Romanian Academy

Dr. Gerota 19-21

020032, BUCHAREST, ROMANIA

[email protected]; [email protected]

Abstract: The aim of this paper is two-fold: (1) to describe the least squares approximations of the solutions

of Maxwell PDEs via their Euler-Lagrange prolongations, (2) to reveal the electromagnetic least squares waves.

Current interest in these areas is driven by the growth in wireless and fiber-optic communications, information

technology, and materials science.

KeyWords: Electromagnetic least squares waves, Udriste-Maxwell Lagrangian, Maxwell PDEs. MathematicsSubject Classification 2010: 49K20, 78M30; 35Q61.

1 Introduction

Herein is described an extraordinary result in sci-

ence: electromagnetic least squares waves, which

are more subtle than those in textbook or text paper

physics. These electromagnetic waves that compose

electromagnetic radiation can be imagined as a self-

propagating transverse oscillating wave of electric andmagnetic fields. Plane diagram: the electric field is in

a vertical plane and the magnetic field in a horizon-

tal plane; the two types of fields in these waves are

always in phase with each other, and no matter how

powerful, have a ratio of electric to magnetic intensity

which is fixed and never varies.

Section 2 recalls the classical wave PDEs derived

from the Maxwell equations. Section 3 gives the

Euler-Lagrange prolongation of the Maxwell PDEs

system and their non-classical wave solutions. Section

4 describes the least squares monopole waves. Sec-

tion 5 addresses an open problem regarding our theory

in the context of differential forms on a Riemannian

manifold. Section 6 underlines the importance of the

least squares Lagrangian and least squares waves.

2 Classical electromagnetic waves

The electromagnetic ingredients are: E= electric fieldstrength, B = magnetic field strength, D = electric dis-placement field, H = magnetic displacement field, =electric charge density, b = magnetic charge density,

15th WSEAS International Conference on Automatic Con-

trol, Modelling & Simulation (ACMOS-13), Brasov, Romania,

June 1-3, 2013

= permeability, = permitivity, c = speed of light infree space. They determine the Maxwell PDEs

curl E+B

t= 0, curl H

D

t= 0

div E = 0, div H= 0, B = H, D = E

on R3 R. The values of and for vacuum satisfy

=1

c2. Taking the curl of the curl, the foregoing

equations give

curl curl E =

tcurl H =

2E

t2,

curl curl H =

tcurl E =

2H

t2.

Now we add the vector identity

( V) = ( V) 2V,

where V is any vector function of space, to create theclassical wave PDEs

E = grad div E curl curl E =1

c22E

t2

H = grad div H curl curl H =1

c22H

t2,

i.e., the classical waves (CW) are solutions of thePDEs

E =1

c22E

t2, H =

1

c22H

t2. (CW)


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2.1 Direct search of the classical waves

The second order PDEs (CW) admit waves solutionsof the form (see also, [2]-[3])

E(r, t) = E0ei(1tk,r), H(r, t) = H0ei(1tk,r),

where t is time (in seconds), 1 is the angular fre-quency (in radians per second), k = (kx, ky, kz) is thewave vector (in radians per meter) and r = (x,y,z)is the position vector. Of course, we accept that the

electric field, the magnetic field, and direction of wave

propagation are all orthogonal, and the wave propa-

gates in the same direction as E0 H0. The electro-magnetic waves are transversal, i.e.,

div E = ik, E = 0, div H = ik, H = 0

E

t= i1E,

2E

t2= 21E,

E

x= ikxE,

2E

x2= k2xE, rot E = iE k

H

t= i1H,

2H

t2= 22H,

H

x= ikxH,

2H

x2= k2xH, rot H = iH k.

Theorem 1 The wave vectork is related to the angu-

lar frequency by ||k|| = 1c = 21, where ||k|| is

the wave number and1 is the wave length.

ProofReplacing into the wave PDEs, we find thesecond order equation

||k||2 1

c221 = 0.

The pair

E(r, t) = E0ei(1tk,r), H(r, t) = H0e

i(1tk,r)

verifies the Maxwell equations if and only if

H0 k = 1E0.

3 Udriste-Maxwell Lagrangian and

least squares waves

The electric field strength E, the magnetic fieldstrength B, the electric displacement field D, the mag-netic displacement field H, the permeability , thepermitivity , the speed of light c determine the over-

determined Maxwell PDEs system

curl E+ H

t= 0, curl H

E

t= 0,

div E = 0, div H= 0

(8 PDEs with 6 unknowns-components of the electro-

magnetic vector fields E, H) defined on R3 R. The

Maxwell PDEs are symmetric under the exchange ofE and H. More precisely, they are invariant underE H, H E and . This symmetry iscalled the electric-magnetic duality, and the exchange

of electric and magnetic fields is known as the duality

transformation. The electric-magnetic duality simply

implies that a theory that describes a vacuum consist-

ing only of the electric and magnetic fields, E1 and H1respectively, has the same physical interpretation as

another theory that describes a vacuum with the elec-

tric field E2 = H1 and the magnetic field H2 = E1.We underline that in classical physics, a vacuum is an

empty space without any particles.The foregoing Maxwell PDEs system and the Eu-

clidean scalar product (metric) determine the least

squares Udriste-Maxwell Lagrangian (see [1], [5]-

[11])

L =1

2

curl E+ Ht2 +

curl H Et2

+(div E)2 + (div H)2

.

The expression of the Lagrangian

L(x , y , z , t; E; H; Ex, Ey, Ez, Et; Hx, Hy, Hz, Ht)

in coordinates is

L =1

2

(E3y E

2z + H

1t )

2 + (E1z E3x + H

2t )

2

+(E2x E1y + H

3t )

2 + (H3y H2z E

1t )

2

+(H1z H3x E

2t )

2 + (H2x H1y E

3t )

2

+(E1x + E2y + E

3z )

2 + (H1x + H2y + H

3z )

2

.Since the partial derivatives of this Lagrangian are

L

E1x= E1x + E

2y + E

3z ,

L

E1y= (E2x E

1y + H

3t ),

L

E1z= E1zE

3x+H

2t ,

L

E1t= (H3yH

2z E

1t ),

L

H1x= H1x+H

2y +H

3z ,

L

H1y= (H2xH

1y E

3t ),

LH1z= H1z H3xE2t , LH1t

= (E3yE2z +H1t ),

we write easily the Euler-Lagrange PDEs.


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Theorem 2 The Euler-Lagrange PDEs associated to

the Lagrangian L are

E

+2

2E

t2 = (

+

)

t (curl H

),

H + 22H

t2= ( + )

t(curl E). (N CW)

Properties. (1) These PDEs represent the Euler-Lagrange prolongation of the Maxwell PDEs system,

i.e., the solutions of Maxwell PDEs (Maxwell flow)

are also solutions of this second order system.

(2) The PDEs system in the Theorem is sym-

metric (i) under the duality transformation E H,H E, (this symmetry is called the

electric-magnetic duality, and the exchange of elec-tric and magnetic fields in this way is known as the

duality transformation) or (ii) under the duality trans-

formation E H, t t, , which underlinethe reversibility of time.

3.1 Direct search of the non-classical waves

For the electromagnetic PDEs system (N CW)in Theorem 2, we search solutions of wave

type E(r, t) = E0ei(2tp,r) and H(r, t) =

H0

ei(2tp,r), where t is time (in seconds), 2

is

the angular frequency (in radians per second), p =(px, py, pz) is the wave vector (in radians per meter)and r = (x,y,z).

By direct calculations, we get

H

t= i2H,

2H

t2= 22H,

H

x= ipxH,

2H

x2= p2xH, rot H = iH p

E

t = i2E,2E

t2 = 22E,

E

x = ipxE,

2E

x2= p2xE, rot E = iE p.

Replacing in the PDEs system (N CW), we find theconditions

(||p||2 + 222)E0 = ( + )2H0 p,

(||p||2 + 222)H0 = ( + )2E0 p.

It follows that the vectors {E0, H0, p} determine anorthogonal frame. Moreover,

H0 p = E0, E0 p = H0

and hence the essential condition that relates the wave

vector and the angular frequency is

||p||2 + 222 = ( + )2.

The conditions ||p||2 =22

c2and = 2 are

equivalent (classical electromagnetic waves). If =2, then the waves

E(r, t) = E0ei(2tp,r), H(r, t) = H0e

i(2tp,r)

are called non-classical electromagnetic waves. Al-

though the physical significance of these waves is not

enough clear, we are sure that they have a fundamental

role in electromagnetism.

Theorem 3 A solution (E, H) of the PDEs system(N CW) is solution of the Maxwell system if and onlyifE andH are solutions of the PDEs (CW).

Remark The electric field, the magnetic field,and direction of wave propagation are all orthogo-

nal, and the wave propagates in the same direction as

E0H0. The electromagnetic non-classical waves aretransversal, i.e.,

div E = p, E = 0, div H = ip, H = 0.

Corollary If E and H are solutions of theMaxwell system, than the wave vector p is related tothe angular frequency 2 by

||p||2 1

c222 = 0,

c=

2

2,

where ||p|| is the wave number and 2 is the wavelength.

Corollary The classical electromagnetic wavesare solutions of the PDEs system (N CW). Thenon-classical electromagnetic waves do not verify the

PDEs (CW).Corollary A classical wave has the same angular

frequency with a non-classical wave if and only ifthe elliptic relation

2c2||k||2 + ||p||2 = ( + )

holds.

The connection between classical and non-

classical waves is normal, because here the Euler-

Lagrange equations are prolongations of the Maxwell

equations. The main observations are: (1) the clas-

sical wave equations separate the electric field and

magnetic field, (2) the non-classical wave equations(Euler-Lagrange equations) combine the electric field

and magnetic field.


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4 Least squares monopole waves

On the physical source space R3 R, we use ingredi-ents from the electromagnetic target space. Their list

include: E= electric field strength, B = magnetic fieldstrength, D = electric displacement field, H = mag-netic displacement field, j = density of electric cur-rent, = electric charge density, b = magnetic chargedensity, = permeability, = permitivity, c = speed oflight.

For relaxation phenomena, the vectorial inter-

nal variables and monopole existence, the extended

Maxwell PDEs (8 first order PDEs with 12 unknowns

= components of the vector fields E , D , H, B) are

curl E+1

c

B

t

= 0, curl H1

c

D

t

=1

c

j,

div D = , div B = b,

with constitutive laws of materials (9 conditions thatcan be approximated by 9 almost algebraic equations)D = E, B = H, j = E. The least squares La-grangian determined by the extended Maxwell PDEs

and by the Euclidean metric (Udriste-Maxwell La-

grangian) is

2L =

curl E+1

c

B

t

2

+

curl H 1c Dt 1cj2

+(divD)2+(divBb)2

+(D E)2 + (B H)2 + (j E)2.

In coordinates, the Lagrangian

L(x , y , z , t; E , H, B, D , j; Ex, Hx, Bx, Dx, jx,etc)

can be written

2L = (E3y

E2z

+1

cB1

t) + (E1

z E3

x+

1

cB2

t)

+(E2x E1y +

1

cB3t ) + (H

3y H

2z

1

cD1t

1

cj1)2

+(H1z H3x

1

cD2t

1

cj2)2

+(H2x H1y

1

cD3t

1

cj3)2 + (D1x + D

2y + D

3z )

2

+(B1x + B2y + B

3z b)

2 + (D1 E1)2 + (D2 E2)2

+(D E3)2 + (B1 H1)2 + (B2 H2)2

+(B3 H3)2 + (j1 E1)2

+(j2 E2)2 + (j3 E3)2.

The partial derivatives of this Lagrangian are

L

E1= (j1 E),

L

E1x= 0,

L

E1t= 0;

L

E1y= (E2xE

1y+

1

cB3t ),

L

E1z= (E1zE

3x+

1

cB2t );

L

B1y= 0,

L

B1z= 0;

L

B1= B1 H1;

L

B1x= B1x+B

2y+B

3zb,

L

B1t=

1

c(E3yE

2z+

1

cB1t );

L

D1 = (D1 E1),

L

D1x= D1x + D2y + D3z ;

L

D1t=

1

c(H3y H

1z

1

cD1t

1

cj1),

L

D1y= 0;

L

D1z= 0,

L

H1= (B1 H1),

L

H1x= 0;

L

H1t= 0,

L

H1y= (H2x H

1y

1

cD3t

1

cj3);

L

H1z= H1z H

3x

1

cD2t

1

cj2;

L

j1=

1

c(H3y H

1z

1

cD1t

1

cj1) + j1 E1;

L

j1x= 0,

L

j1t= 0,

L

j1y= 0,

L

j1z= 0.

Theorem 4 The Euler-Lagrange prolongations of the

extended Maxwell PDEs are

E grad(div E) 1

c

tcurl B + (j E) = 0,

B Hgrad(divB b)1

c

tcurlE

1

c2Btt = 0,

H grad(div H) +1

c

tcurl D +

1

ccurl j

+(B H) = 0,

j E 1

ccurl H +

1

c2Dt +

1

c2j = 0.


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5 Open problems

Let us formulate the Udriste-Maxwell theory in terms

of differential forms [10]. In this sense, it is well-

known that E, H are differential 1-forms, J,D,B aredifferential 2-forms, and b are differential 3-forms,and the star operator from D = E, B = His the Hodge operator. If d is the exterior derivativeoperator, and t is the time derivative operator, thenthe extendedMaxwells equations for static media are

coupled PDEs of first order,

dE = tB, dH= J + tD, dD = , dB = b

defined on R3 R. Furthermore, some real problemsrequire to replace the Euclidean manifold (R3, ij)with a Riemannian manifold

(R3, g

ij). In this context

the least squares Lagrangian can be written

2L = ||dE+ tB||2 + ||dH J tD||

2

+||dD ||2 + ||dB b||2.

Find interpretations for the extremals of L which arenot solutions of Maxwell equations.

6 Conclusions

This paper studies the extremals of the least squaresLagrangian associated to Maxwell PDEs. Between

these extremals we have the least squares (non-

classical) waves, but also other solutions whose inter-

pretation is an open problem.

Acknowledgements: Partially supported by Uni-

versity Politehnica of Bucharest, by Institute of Geo-

dynamics Sabba S. Stefanescu of the Romanian

Academy and by Academy of Romanian Scientists.

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