Nonclassical Electromagnetic Dynamics

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Nonclassical Electromagnetic Dynamics

CONSTANTIN UDRISTE

University Politehnica of Bucharest

Department of Mathematics

Splaiul Independentei 313

060042 Bucharest

ROMANIA

[email protected]

DOREL ZUGRAVESCU

Institute of Geodynamics

Dr. Gerota 19-21

020032 Bucharest

ROMANIA

[email protected]

FLORIN MUNTEANU

Institute of Geodynamics

Dr. Gerota 19-21

020032 Bucharest

ROMANIA

[email protected]

Abstract: Our paper is concerned with effects of special forces on the motion of particles. 1(2) studies the single-time geometric dynamics induced by electromagnetic vector fields (1-forms) and by the Euclidean structure of the

space. 3 defines the second-order forms or vectors. 4 (5) describes a nonclassical electric (magnetic) dynamics

produced by an electric (magnetic) second-order Lagrangian, via the extremals of the energy functional. 6 gen-eralize this dynamics for a general second-order Lagrangian, having in mind possible applications for dynamical

systems coming from Biomathematics, Economical Mathematics, Industrial Mathematics, etc.

KeyWords: geometric dynamics, second-order vectors, second-order Lagrangian, nonclassical dynamics.

1 Single-time geometric dynamics

induced by electromagnetic vector

fields

Let U be a domain of linear homogeneous isotropic

media in the Riemannian manifold (M = R3, ij).Maxwells equations (coupled PDEs of first order)

div D = , rot H = J + tD, t = timederivative operator

div B = 0, rot E = tBwith the constitutive equations

B = H, D = E,

on R U, reflect the relations between the electro-magnetic fields:

E [V /m] electric field strengthH [A/m] magnetic field strengthJ [A/m2] electric current density [As/V m] permitivity [V s/Am] permeabilityB [T] = [V s/m2] magnetic induction

(magnetic flux density)

D [C/m2] = [As/m2] electric displacement(electric flux density)

Since div B = 0, the vector field B is sourcefree, hence may be expressed as rot of some vectorpotential A, i.e., B = rot A. Then the electric fieldstrength is E = grad V tA.

It is well-known that the motion of a charged par-

ticle in the electromagnetic fields is described by the

ODE system (Lorentz World-Force Law)

md2x

dt2= eE+

dx

dt B ,

x = (x1, x2, x3) U R3,

where m is the mass, and e is the charge of the par-ticle. Of course, these are Euler-Lagrange equations

produced by the Lorentz Lagrangian

L1 =1

2mij

dxi

dt

dxj

dt+ eij

dxi

dtAj eV.

The associated Lorentz Hamiltonian is

H1 =m

2ij

dxi

dt

dxj

dt+ eV.

A similar mechanical motion was discovered by

Popescu [7], accepting the existence of a gravitovor-

tex fieldrepresented by the force G +dx

dt , where

G is the gravitational field and is a vortex determin-ing the gyroscopic part of the force. The papers [5],

[8], [9], [12], [14], [15] confirm the point of view of

Popescu via geometric dynamics.

1.1 Single-time geometric dynamics pro-

duced by vector potential A

To conserve the traditional formulas, we shall referto field lines of the vector potential A using di-mensional homogeneous relations. These curves are

WSEAS TRANSACTIONS on MATHEMATICS Constantin Udriste, Dorel Zugravescu, Florin Munteanu

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Issue 1, Volume 7, January 2008


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solutions of the ODE system

mdx

dt= eA.

This ODE system and the Euclidean metric produce

the least squares Lagrangian

L2 =1

2ij

m

dxi

dt+ eAi

m

dxj

dt+ eAj

=1

2

mdxdt + eA2 .

The Euler-Lagrange equations associated to L2 are

md2xi

dt2= eAj

xi

Ai

xj dxj

dt+

e2

m

fA

xi etA,

where

fA =1

2ijA

iAj

is the energy density produced by A. Equivalently, itappears a single-time geometric dynamics

md2x

dt2= e

dx

dt B +

e2

mfA etA, B = rot A,

a motion in a gyroscopic force [12], [14], [15] or a

B-vortex dynamics. The associated Hamiltonian is

H2 =1

2ij

m

dxi

dt eAi

m

dxj

dt+ eAj

=m2

2ij

dxi

dt

dxj

dt e2fA.

Remark. Generally, the single-time geometric

dynamics produced by the vector potential A isdifferent from the classical Lorentz World-Force Law

because

L2 mL1 =

1

2 e2

ijAi

Aj

+ meV

and the forcee

mfA tA is not the electric field

E = V tA. In other words the LagrangiansL1 and L2 are not in the same equivalence class ofLagrangians.


duced by magnetic induction B

Since we want to analyze the geometric dynamics us-

ing units of measure, the magnetic flow must be de-scribed by

mdx

dt= B,

where the unit measure for the constant is[kgm3/V s2].

The associated least squares Lagrangian is

L3 = 12

ij

m dxidt

Bi

m dxjdt

Bj

=1

2

mdxdt B2 .

This gives the Euler-Lagrange equations (single-time

magnetic geometric dynamics)

md2x

dt2=

dx

dt rot B +

2

mfB + tB,

where

fB =1

2ijB

iBj =1

2||B||2

is the magnetic energy density. This is in fact a dy-

namics under a gyroscopic force or in J-vortex.The associated Hamiltonian is

H3 =1

2ij

m

dxi

dt Bi

m

dxj

dt+ Bj

=m2

2

ijdxi

dt

dxj

dt

2fB .


duced by electric field E

The electric flow is described by

mdx

dt= E,

where the unit measure of the constant is[kgm2/V s]. It appears the least squares Lagrangian

L4 =1

2ij

m

dxi

dt Ei

m

dxj

dt Ej

=1

2

m dxdt E2 ,

with Euler-Lagrange equations (single-time electric

geometric dynamics)

md2x

dt2=

dx

dt rot E+

2

mfE + tE,

where

fE =1

2ijE

iEj =1

2||E||2


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is the electric energy density; here we have in fact a

dynamics in tB-vortex. The associated Hamiltonianis

H4 =1

2 ij

mdxi

dt Ei

mdxj

dt + Ej

=m2

2ij

dxi

dt

dxj

dt 2fE.

Open problem. As is well-known, charged par-

ticles in the magnetic field of the earth spiral from

pole to pole. Similar motions are also observed in

laboratory plasmas and inferred for electrons in metal

subjected to an external magnetic field. Till now,

these motions where justified by the classical Lorentz

World-Force Law. Can we justify such motions using

the geometric dynamics produced by suitable vectorfields in the sense of present paper and the papers [5],

[7]-[12], [14], [15]?

2 Potentials associated to electro-

magnetic forms

Let U R3 = M be a domain of linear homoge-neous isotropic media. Which mathematical object

shall we select to model the electromagnetic fields;

vector fields or forms? It was shown [1] that, the

magnetic induction B, the electric displacement D,and the electric current density J are all 2-forms; themagnetic field H, and the electric field Eare 1-forms;the electric charge density is a 3-form. The operatord is the exterior derivatives and the operator t is thetime derivative.

In terms of differential forms, the Maxwell equa-

tions on U R can be expressed as

dD = , dH = J + tD

dB = 0, dE = tB.

The constitutive relations are

D = E, B = H,

where the star operator is the Hodge operator, isthe permitivity, and is the scalar permeability.

The local components Ei, i = 1, 2, 3, of the 1-form E are called electric potentials, and the localcomponents Hi, i = 1, 2, 3, of the 1-form Hare calledmagnetic potentials. Since the electric field E, and themagnetic field H are 1-forms [1], in the Sections 4-5we combine our ideas [8]-[16] with the ideas of Emery

[3] and Foster [4], creating a nonclassical electric or

magnetic dynamics. Finally, we generalize the resultsto nonclassical dynamics induced by a second-order

Lagrangian (Section 6).

2.1 Potential associated to electric 1-form E

Let us consider the function V : R U R,(t, x) V(t, x) and the Pfaff equation dV = E or

the equivalent PDE systemV

xi = Ei, i = 1, 2, 3.

Of course, the complete integrability conditions re-

quire dE = 0 (electrostatic field), which is not alwayssatisfied. In any situation we can introduce the least

squares Lagrangian

L9 =1

2ij

V

xi+ Ei

V

xj+ Ej

=1

2||dV + E||2.

These produce the Euler-Lagrange equation (Poisson

equation)V = div E,

and consequently V must be the electric potential. Fora linear isotropic material, we have D = E, with = div D = div E. We get the potential equationfor a homogeneous material ( = constant)

V =

.

For the charge free space, we have = 0, and then thepotential V satisfies the Laplace equation

V = 0

(harmonic function). The Lagrangian L9 produces theHamiltonian

H9 =1

2(dV E,dV + E),

and the momentum-energy tensor field

Tij =V

xjL

V

xi

L9ij

=V

xj

V

xi+ Ei

L9

ij .

2.2 Potential associated to magnetic 1-form

H

Now, we consider the Pfaff equation d = H or the

PDE system

xi= Hi, i = 1, 2, 3, : R U R,

(t, x) (t, x). The complete integrability condi-tions dH = 0 are satisfied only for particular cases. If

we build the least squares Lagrangian

L10 =1

2ij

xi Hi

xj Hj


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=1

2||d H||2,

then we obtain the Euler-Lagrange equation (Laplace

equation) and consequently must be the magnetic

potential. The Lagrangian L10 produces the Hamilto-

nian H10 =1

2(d H,d + H).

2.3 Potential associated to 1-form potential

A

Since dB = 0, there exists an 1-form potential A sat-isfying B = dA. Now let us consider the Pfaff equa-

tion d = A or the equivalent PDEs system

xi=

Ai, i = 1, 2, 3, : R U R, (t, x) (t, x).

The complete integrability conditions dA = 0 are sat-isfied only for B = 0. If we build the least squaresLagrangian

L11 =1

2ij

xi Ai

xj Aj

=1

2||d A||2,

then we obtain the Euler-Lagrange equation (Laplace

equation) = 0. The Lagrangian L11 gives the

Hamiltonian H11 =1

2

(d A,d + A) and the

momentum-energy tensor field

Tij =

xj

xi Ai

L11

ij .

Open problem. Find interpretations for the ex-

tremals of least squares Lagrangians

L12 =1

2||dA B||2

=1

2

ikjl Aixj

Aj

xi BijAk

xl

Al

xl Bkl

L13 =1

2||dE+ tB||

2 +1

2||dH J tD||

2

+1

2||dD ||2 +

1

2||dB||2,

which are not solutions of Maxwell equations.

Remarks. 1) There are a lot of applications of

previous type in applied sciences. One of the most

important is in the material strength. In problems as-

sociated with the torsion of a cylinder or prism, one

has to investigate the functional

J(z()) =

D

z

x y

2+

z

y+ x

2dxdy

for an extremum. The Euler-Lagrange equation,

2z

x2+

2z

y2= 0, shows that the extremals are har-

monic functions. Of course, J has no global mini-

mum point since the PDE system zx

= y, zy

= x

is not completely integrable.

2) All previous examples in this section are

single-component potentials. Similarly we can intro-

duce the multi-component potentials.

3 Second-Order Forms and Vectors

Now we will concentrate on certain geometric ideas

that are very important in the physical and stochastic

applications. To avoid too much repetition, M willdenote a differentiable manifold of dimension n, andall the functions are of class C.

Let xi = xi(xi

), i, i = 1, . . . , n be a changingof coordinates on M. Then we introduce the symbols

Dii =xi

xi, Diij =

2xi

xixj.

For a differentiable function f : M R we use thesimplified coordinate expression f(xi

) = f(xi(xi

)),the first order derivatives f,i , f,i and the second orderderivatives f,ij , f,ij . These are connected by the rule

(f,i , f,ij) = (f,i, f,ij)

Dii D

iij

0 DiiDjj

. (1)

The pair (first-order partial derivatives, second-

order partial derivatives) possesses a tensorial

change law that the second derivative, by itself, lacks.

This pair was used in the classical works like contact

element or like jet.

If a curve is given by the parametric equations

xi = xi(t), t I, then the preceding diffeomorphismmodifies the pair (x, x x)T as follows

xi

= xiDi

i , xi = xiDi

i + xixjDi

ij ,

xi

xi

xj

=

Dii Diij

0 Di

i Dj

j

xi

xixj

.

The pair (acceleration, square of velocity) is sug-

gested by the equations of geodesics

xk(t) + kij(x(t))xi(t)xj(t) = 0.

Consequently, when first and second derivatives

come into play together, then matrices of blocks such


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as

K = Dii D

iij

0 D

i

i

D

j

j , K

1 =

Di

i Di

ij

0 Di

i D

j

j

are useful for changing the components of pairs of ob-

jects.

Definition. Any pair (i, ij) admitting thechanging law (1), where (i) is an 1-form and (ij) issymmetric, will be called a second-order form on M.A second-order vector field on M is a second-orderdifferential operator, with no constant term.

A second-order form can be written as =id

2xi + ijdxi dxj , where the components i and

ij = ji are smooth functions. A second-order vector

field writes Xf = ij

Dijf + i

Dif, where the com-ponents ij = ji and i are smooth functions. Thetheory of second-order vectors or forms appears in

Emery [3], with applications in stochastic problems,

and in Foster [4], suggesting new point of view about

the fields theory. We use these ideas to define mean-

ingful second-order Lagrangians (kinetic potentials)

and to study their extremals.

4 Dynamics induced by second-

order electric form

Let Ei be the electric potentials. The usual deriva-tive Ei,j may be decomposed into skew-symmetricand symmetric parts,

Ei,j =1

2(Ei,j Ej,i) +

1

2(Ei,j + Ej,i).

The skew-symmetric part (vortex)

mij =1

2(Ei,j Ej,i)

is called Maxwell tensor field giving the opposite ofthe time derivative of magnetic induction. The sym-

metric part1

2(Ei,j + Ej,i)

is not an ordinary tensor, but the pair

(Ei,1

2(Ei,j + Ej,i))

is a second-order object [3]-[4].

Let ij be a general object such that (Ei, ij) is a

second-order object. The difference

(0, ij 1

2(Ei,j + Ej,i))

is a second-order object determined by gij = ij 1

2(Ei,j + Ej,i). Ifij is symmetric, then gij is a sym-

metric tensor field; if we add det(gij) = 0, then gijcan be used as a semi-Riemann metric. In this context

the equality

(Ei, ij) = (Ei,1

2(Ei,j + Ej,i)) + (0, gij)

shows that the valuable objects

1

2(Ei,j + Ej,i)

come from electricity and mate with gravitational po-

tentials gij . Consequently they have to be electrograv-itational potentials.

The preceding potentials determine an electricenergy second-order Lagrangian,

Lel = Ei(x(t), t)d2xi

dt2(t)

+1

2(Ei,j + Ej,i)(x(t), t)

dxi

dt(t)

dxj

dt(t).

If the most general energy second-order Lagrangian

is given by

Lge = Ei(x(t), t) d

2

x

i

dt2 (t)+ij(x(t), t) dx

i

dt (t) dx

j

dt (t),

and the gravitational energy first-order Lagrangian is

Lg = gij(x(t), t)dxi

dt(t)

dxj

dt(t),

then Lge = Lel + Lg.To simplify, let us take E = E(x). Since the dis-

tribution generated by the electric 1-form E is givenby the Pfaff equation Ei(x)dx

i = 0, the electric en-ergy Lagrangian is zero along integral curves of this

distribution. Also the general energy second-order La-grangian for E = E(x), ij = ij(x) determines theenergy functional

ba

Ei(x(t))

d2xi

dt2+ ij(x(t))

dxi

dt(t)

dxj

dt(t)

dt.

(2)We denote

ijk =1

2(kj,i + ki,j ij,k)

(the Christoffel symbols ofij),

Eijk =1

2(Ek,ij + Ej,ik Ei,jk) .


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Theorem. The extremals of the energy functional

(2) are described by the Euler-Lagrange ODEs

gkid2xi

dt2+ (ijk Eijk)

dxi

dt

dxj

dt= 0,

x(a) = xa, x(b) = xb.

(geodesics with respect to an Otsuki connection [6]).

Proof. Using the second-order Lagrangian

L = Eid2xi

dt2+ ij

dxi

dt

dxj

dt,

the Euler-Lagrange equations

Lxk d

dt

Ldxkdt

+d2

dt2

Ld2xkdt2

= 0

transcribe like the equations in the theorem.

To a Lagrangian there may corresponds a field

theory. Consequently we obtain a field theory hav-

ing as basis the general electrogravitational potentials.

The pure gravitational potentials are given by

gij = ij 1

2(Ei,j + Ej,i).

We define ijk = ijkEijk . It is verified that ijk =gijk + mijk , where gijk are the Christoffel symbols of

gij , and mijk = mij,k + mik,j is the symmetrizedderivative of the Maxwell tensor

mij =1

2(Ei,j Ej,i).

Corollary. The extremals of the energy functional

(1) are described by the Euler-Lagrange ODEs

gkid2xi

dt2+ (gkji + mkji)

dxi

dt

dxj

dt= 0,

x(a) = xa, x(b) = xb.


5 Dynamics induced by second-

order magnetic form

Let Hi be the magnetic potentials. The usual deriva-tive Hi,j may be decomposed into skew-symmetricand symmetric parts,

Hi,j =1

2(Hi,j Hj,i) +

1

2(Hi,j + Hj,i),

where

Mij =1

2(Hi,j Hj,i)

is the Maxwell tensor field(vortex) giving the sum be-

tween the electric current density and the time deriva-

tive of the electric displacement. The pair

Hi, 1

2(Hi,j + Hj,i)

is a second-order object. If (Hi, ij) is a generalsecond-order object, then the difference

gij = ij 1

2(Hi,j + Hj,i)

represents the gravitational potentials (a metric) pro-

vided that ij is symmetric, gij is a (0,2) tensor fieldand det(gij) = 0. Consequently the valuable ob-

jects 12 (Hi,j+Hj,i), which come from magnetism and

mate, have to be magnetogravitational potentials.

The preceding potentials produce the following

energy Lagrangians:

1) magnetic energy second-order Lagrangian,

Lma = Hi(x(t), t)d2xi

dt2(t)

+1

2(Hi,j + Hj,i)(x(t), t)

dxi

dt(t)

dxj

dt(t);

2) gravitational energy first-order Lagrangian,


dt(t)

dxj

dt(t);

3) general energy second-order Lagrangian,

Lge = Hi(x(t), t)d2xi

dt2+ ij(x(t), t)

dxi

dt(t)

dxj

dt(t).

These satisfy the relation Lge = Lma + Lg.To simplify, let us take H = H(x). Since the dis-

tribution generated by the magnetic 1-form H is givenby the Pfaff equation Hi(x)dxi = 0, the magnetic en-

ergy Lagrangian is zero along integral curves of this

distribution. Also the general energy second-order La-

grangian for H = H(x), ij = ij(x) determines theenergy functional

ba

Hi(x(t))

d2xi

dt2(t) + ij(x(t))

dxi

dt(t)

dxj

dt(t)

dt

(3)determined by a second-order Lagrangian which is

linear in acceleration.

We denote

ijk =1

2(kj,i + ki,j ij,k)


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Hijk =1

2(Hk,ij + Hj,ik Hi,jk) .

Theorem. The extremals of the energy functional

(3) are solutions of the Euler-Lagrange ODEs

gkid2xi

dt2+ (ijk Hijk)

dxi

dt

dxj

dt= 0,

x(a) = xa, x(b) = xb.


To an energy Lagrangian there may corresponds

a field theory. Consequently we obtain a field theory

having as basis the general magnetogravitational po-

tentials. The pure gravitational potentials are (compo-nents of a Riemann or semi-Riemann metric)

gij = ij 1

2(Hi,j + Hj,i).

We introduce ijk = ijk Hijk . It is verified the re-lation ijk = gijk +Mijk , where gijk are the Christof-fel symbols of gij , and Mijk = Mij,k + Mik,j is thesymmetrized derivative of the Maxwell tensor

Mij =1

2

(Hi,j Hj,i).

Corollary. The extremals of the energy functional (3)

are solutions of the Euler-Lagrange ODEs

gkid2xi

dt2+ (gkji + Mkji)

dxi

dt

dxj

dt= 0,

x(a) = xa, x(b) = xb.


6 Dynamics induced by second-order general form

Now we want to extend the preceding explanations

since they can be applied to dynamical systems com-

ing from Biomathematics, Economical Mathematics,

Industrial Mathematics etc.

Let i be given potentials (given form). The usualpartial derivative i,j may be decomposed as

i,j =1

2(i,j j,i) +

1

2(i,j + j,i),

where

Mij =1

2(i,j j,i)

is a Maxwell tensor field(vortex). The pair

i,

1

2(i,j + j,i)

is a second-order form. If (i, ij) is a generalsecond-order form, then we suppose that the differ-

ence

gij = ij 1

2(i,j + j,i)

represents the components of a metric, i.e., ij is sym-metric, gij is a (0,2) tensor field and det(gij) = 0.

The preceding potentials produce the following

energy Lagrangians:

1) potential-produced energy Lagrangian,

Lpp = i(x(t), t) d2

xi

dt2(t)

+1

2(i,j + j,i)(x(t), t)

dxi

dt(t)

dxj

dt(t);

2) gravitational energy Lagrangian,


dt(t)

dxj

dt(t);

3) general energy Lagrangian,

Lge = i(x(t), t) d2

xi

dt2+ i,j(x(t), t) dx

i

dtdx

j

dt;

which verify

Lge = Lpp + Lg.

The Pfaff equation i(x)dxi = 0, i = 1, . . . , n

defines a distribution on M. The valuable objects

1

2(i,j + j,i)

are the components of the second fundamental formof that distribution [6]. The potential-produced energy

Lagrangian is zero along integral curves of the distri-

bution generated by the given 1-form = (i(x)).In the autonomous case, the general energy La-

grangian produces the energy functional

ba

i(x(t))

d2xi

dt2(t) + ij(x(t))

dxi

dt(t)

dxj

dt(t)

dt.

(4)This energy functional is associated to a particular La-

grangian L of order two.

We denote

ijk =1

2(kj,i + ki,j ij,k)


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ijk =1

2(k,ij + j,ik i,jk) .

Theorem. The extremals of the energy functional(4) are solutions of the Euler-Lagrange ODEs

gkid2xi

dt2+ (ijk ijk)

dxi

dt

dxj

dt= 0,

x(a) = xa, x(b) = xb


To an energy Lagrangian there may correspond a

field theory. We introduce ijk = ijk ijk . Aftersome computations we find ijk = gijk +mijk , wheregijk are the Christoffel symbols ofgij , and

mijk =Mij,k +Mik,j

is the symmetrized derivative of the Maxwell tensor

fieldM.

Corollary. The extremals of the energy functional

(4) are solutions of the Euler-Lagrange ODEs

gkid2xi

dt2+ (gkji + mkji)

dxi

dt

dxj

dt= 0,

x(a) = xa, x(b) = xb


Open problems: (1) Find the linear connectionsin the sense of Crampin [2] associated to the preced-

ing second-order ODEs. (2) Analyze the second varia-

tions of the preceding energy functionals. (2) Analyze

the symmetries of the preceding second order differ-

ential systems (see also [16]). (3) Find practical inter-

pretations for motions known as geometric dynamics,

gravitovortex motion and second-order force motion

(see also [5]).

Acknowledgements: Partially supported by

Grant CNCSIS 86/ 2007 and by 15-th Italian-

Romanian Executive Programme of S&T Co-

operation for 2006-2008, University Politehnica ofBucharest.

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ISSN: 1109-276939


Nonclassical Electromagnetic Dynamics

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