Electromagnetic Dynamical Systems

Constantin Udrişte and Aneta Udrişte

Abstract

§1 describes the dynamics induced by the Biot-Savart-Laplace vector field using Hamiltonians and new variants of Lorentz world-force laws on Riemann-Jacobi or Riemann-Jacobi-Lagrange manifolds and points out some open problems. §2 transcribes the Lorentz world-force laws in the first paragraph in the Hamiltonian language using suitable symplectic forms. §3 presents the classical theory of motion of a charged particle in the electromagnetic field in order to show that the classical Lorentz world-force law is different from those introduced in the first paragraph. §4 proves that the dynamics induced by the electric field E⃗ or the magnetic field H⃗ can be described by Hamiltonians and symplectic forms intrisecally connected to the field, obeying to some Lorentz world-force laws on Riemann-Jacobi-Lagrange manifolds whose structure is imposed just by the vector field and by Maxwell equations. §5 analyses the electromagnetic dynamical systems appearing in the relativistic model. The results can be extended to any C^{∞} vector field on a Riemannian manifold.

Mathematics Subject Classification (1991): 53C22, 83C22 Key words: conservative differential systems, Lorentz world-force laws, symplectic forms, Lagrangians, Hamiltonians, energy-stress tensor.

1 Biot-Savart-Laplace dynamical systems

Let U be an open and connected set (domain) of R^{3}, with a piecewise smooth boundary ∂U, and J⃗ be a C^{∞} vector field on Ū = U ∪∂U. The Biot-Savart-Laplace formula

H⃗(m) = \frac{1}{4π} ∫_{U} \frac{J⃗ × \overrightarrow{pm}}{pm^{3}}dv_{p}

defines on R^{3} a vector field H⃗ which is C^{∞} on R^{3} − ∂U and of class C^{0} on the boundary ∂U.

Suppose J⃗ is solenoidal, and ∂U is a field surface of J⃗. If J⃗ is a stationary electrokinetic field (conduction current density), then H⃗ is an approximation of the magnetic field generated by J⃗. The magnetic field H⃗ satisfies the relations

divH⃗ = 0

Balkan Journal of Geometry and Its Applications, Vol.2, No.1, 1997, pp. 129-140 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

rotH⃗(m) = \{ \begin{matrix}0 & for & m ∈ R^{3} \ Ū \\ J⃗(m) & for & m ∈ Ū. \end{matrix}

Obviously the vector field J⃗ can have zeros on Ū. Also the domain U can be replaced by a certain surface or a certain curve. In the case of a curve, J⃗ must be nonzero everywhere.

Let m(x^{1}, x^{2}, x^{3}) be a point of R^{3} and {i⃗_{1}, i⃗_{2}, i⃗_{3}} be the Cartesian frame. The Biot-Savart-Laplace vector field can be express in the form H⃗ = H_{1}i⃗_{1} + H_{2}i⃗_{2} + H_{3}i⃗_{3}. The magnetic line α which starts from the point m_{0}(x_{0}^{1}, x_{0}^{2}, x_{0}^{3}) at the moment t = 0 is the oriented curve

α : (−a, a) → R^{3}, α(t) = (x^{1}(t), x^{2}(t), x^{3}(t))

which satisfies the Cauchy problem

\frac{dx^{i}}{dt} = H_{i}, x^{i}(0) = x_{0}^{i}, i = 1, 2, 3.

The set of all images of the maximal magnetic lines is called the phase portrait of the magnetic field H⃗.

Let f : R^{3} → R, f = \frac{1}{2}(H_{1}^{2} + H_{2}^{2} + H_{3}^{2}) be the energy of the magnetic field H⃗, leaving aside the multiplicative factor μ. The following theorems are true [7]-[11].

1.1. Theorem. Every magnetic line in R^{3} − Ū is a trajectory of a potential dynamical system with 3 degrees of freedom associated to the potential V = −f, namely

(1)  \frac{d^{2}x^{i}}{dt^{2}} = \frac{∂f}{∂x^{i}} , i = 1, 2, 3.

1.2. Theorem. Every magnetic line in U is the trajectory of a nonpotential dynamical system with 3 degrees of freedom determined by the potential V = −f and by rot H⃗, namely

(2)  \frac{d^{2}x^{i}}{dt^{2}} = \frac{∂f}{∂x^{i}} + (\frac{∂H_{i}}{∂x^{j}} − \frac{∂H_{j}}{∂x^{i}} ) \frac{dx^{j}}{dt} .

1.3. Theorem.1) The trajectories of the dynamical system (1) are the extremals of the Lagrangian

L = \frac{1}{2}δ_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} + f(x^{1}, x^{2}, x^{3}).

2) The trajectories of the dynamical system (2) are the extremal of the Lagrangian

L = \frac{1}{2}δ_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} − H_{j} \frac{dx^{j}}{dt} + f(x^{1}, x^{2}, x^{3}).

3) The dynamical systems (1) and (2) are conservative, the total energy (Hamiltonian) being in both cases

(3)  H (§^{∞}, §^{∈}, §^{∋}, \frac{⌈§^{∞}}{⌈⊔} , \frac{⌈§^{∈}}{⌈⊔} \frac{⌈§^{∋}}{⌈⊔} ) = \frac{∞}{∈}δ_{⟩|} \frac{⌈§^{⟩}}{⌈⊔} \frac{⌈§^{|}}{⌈⊔} − \{(§^{∞}, §^{∈}, §^{∋}),

where δ_{ij} is the Kronecker symbol.

1.4. Theorem (new variant of Lorentz world-force law). Every nonconstant trajectory of the dynamical system (1) which has the total energy H is a reparametrized geodesic of the Riemann-Jacobi manifold

(ext(Ū) \ E, }_{⟩|} = (H + {)δ_{⟩|}, ⟩, | = ∞, ∈, ∋),

where E is the set of zeros of H⃗.

1.5. Theorem (new variant of Lorentz world-force law). Every nonconstant trajectory of the dynamical system (2) which has the total energy H is a reparametrized horizontal geodesic of the Riemann-Jacobi-Lagrange manifold

(U \ E, }_{⟩|} = (H + {)δ_{⟩|}, N_{|}^{⟩} = −_{|∥}^{⟩}†^{∥} + F^{⟩}_{|}, ⟩, | = ∞, ∈, ∋),

where Γ_{jk}^{i} = Riemannian connection induced by g_{ij},

F_{ij} = \frac{∂H_{j}}{∂x^{i}} − \frac{∂H_{i}}{∂x^{j}} = H_{j,i} − H_{i,j}, F^{i}_{j} = g^{ih}F_{hj}.

Open problems. 1) Obviously the differential systems (1) and (2), which describe the dynamics of some particles sensible to the magnetic field H⃗, have also solutions that are not field lines of H⃗. Till the present time for us is not clear which is the physical meaning of these trajectories though the preceding theorems show that they are not only mathematical entities.

2) The Lorentz law of Theorem 1.5 was obtained using an idea of Kawaguchi-Miron [4]. New variants of Lorentz world-force law associated to the dynamical system (2) can be obtained via the interesting and original ideas of Beil [1], [2], of Crampin [3], or of Obădeanu-Vernic [6].

3) The preceding theory can be extend to any vector field on a Riemannian manifold. In other words every dynamical system of order one can be prolonged to a suitable dynamical system of order two whose trajectories are geodesics of a Lagrangian defined by the velocity vector field (Lagrange structure of first order). In a similar way every dynamical system of order two can be prolonged to a suitable dynamical system of order three whose trajectories are geodesics of a Lagrangian defined by velocity and acceleration vector fields (Lagrange structure of order two). This point of view can create better examples for higher order Lagrange spaces [5].

4) The preceding Jacobi-Riemann-Lagrange structure blows up at equilibrium points. Is it possible to eliminate this deficiency ?

2 Hamiltonian formulation of the Biot-Savart-Laplace dynamical systems

Let us show that the differential systems (1) and (2) can be described like Hamiltonian systems.

Let M be a differentiable manifold and Ω be a 2-form on M. The pair (M, Ω) is called a symplectic manifold if Ω satisfies

1) dΩ = 0 (i.e., Ω is closed), 2) Ω is nondegenerate. Let (M, Ω) be a symplectic manifold and let φ ∈ F(M). Let X_{φ} be the unique

vector field on M satisfying

Ω_{p}(X_{φ}(p), v) = dφ(p) · v, ∀v ∈ T_{p}M.

We call X_{φ} the Hamiltonian vector field of φ. Hamiltonian equations are the differential equations on M given by

\frac{dp}{dt} = X_{φ}(p).

Let T_{t} be the flow of the Hamilton equations, i.e., T_{t}(p) is a field line of X_{φ} starting at p. Then the energy φ is conserved, i.e., φ ∘ T_{t} = φ (conservation of the energy).

2.1. Theorem. The dot ”·” will stand for the derivative with respect to the parameter t. On T R^{3} ≃ R^{6}, the equations of motion of a particle sensible to the Biot-Savart-Laplace magnetic field are Hamiltonian with respect to the total energy (3) and the following symplectic form:

1) If the particle belong to int(R^{3} − U), then the symplectic form is

Ω = δ_{ij}dx^{i} ∧ dẋ^{j}.

2) If the particle belong to U, then the symplectic form is

Ω = δ_{ij}dx^{i} ∧ dẋ^{j} + J,

where the current density J is viewed as a closed 2-form

J = J_{1}dx^{2} ∧ dx^{3} + J_{2}dx^{3} ∧ dx^{1} + J_{3}dx^{1} ∧ dx^{2}

to whom is associated the solenoidal vector field

J⃗ = J_{1}i⃗_{1} + J_{2}i⃗_{2} + J_{3}i⃗_{3}.

Proof. Let H be the function (3) and

Ω = \{ \begin{matrix}δ_{ij}dx^{i} ∧ dẋ^{j} + J & on & U × R^{3} \\ δ_{ij}dx^{i} ∧ dẋ^{j} & on & (extŪ) × R^{3}. \end{matrix}

Denote X_{H} = (u^{1}, u^{2}, u^{3}, u̇^{1}, u̇^{2}, u̇^{3}) and let us discuss the case U × R^{3}. The condition

i_{X_{H}}Ω = dH,

which defines X_{H}, may be written as

u^{1}dẋ^{1} − u̇^{1}dx^{1} + u^{2}dẋ^{2} − u̇^{2}dx^{2} + u^{3}dẋ^{3} − u̇^{3}dx^{3}+
+J_{1}u^{2}dx^{3} − J_{1}u^{3}dx^{2} + J_{2}u^{3}dx^{1} − J_{2}u^{1}dx^{3} + J_{3}u^{1}dx^{2} − J_{3}u^{2}dx^{1} =
= ẋ^{1}dẋ^{1} + ẋ^{2}dẋ^{2} + ẋ^{3}dẋ^{3} − (\frac{∂f}{∂x^{1}} dx^{1} + \frac{∂f}{∂x^{2}} dx^{2} + \frac{∂f}{∂x^{3}} dx^{3}) .

By identification we find

u^{1} = ẋ^{1}, u^{2} = ẋ^{2}, u^{3} = ẋ^{3}

u̇^{1} = \frac{∂f}{∂x^{1}} + J_{2}u^{3} − J_{3}u^{2}, u̇^{2} = \frac{∂f}{∂x^{2}} + J_{3}u^{1} − J_{1}u^{3}, u̇^{3} = \frac{∂f}{∂x^{3}} + J_{1}u^{2} − J_{2}u^{1},

i.e.,

ẍ^{1} = \frac{∂f}{∂x^{1}} + J_{2}ẋ^{3} − J_{3}ẋ^{2}, ẍ^{2} = \frac{∂f}{∂x^{2}} + J_{3}ẋ^{1} − J_{1}ẋ^{3}, ẍ^{3} = \frac{∂f}{∂x^{3}} + J_{1}ẋ^{2} − J_{2}ẋ^{1},

which is the same with the differential system (2).

Obviously divX_{H} = 0 and hence the flow generated by X_{H} preserves the volume.

3 Classical equations of motion for a charged particle in a stationary electromagnetic field

To avoid some misunderstandings, we recall some well known facts. Let

B = B_{1}dx^{2} ∧ dx^{3} + B_{2}dx^{3} ∧ dx^{1} + B_{3}dx^{1} ∧ dx^{2}

be a closed 2-form on R^{3} and

B⃗ = B_{1}i⃗_{1} + B_{2}i⃗_{2} + B_{3}i⃗_{3}

the associated divergence free vector field. The connection between the magnetic induction B⃗ and the magnetic vector field H⃗ is B⃗ = μ_{0}H⃗. Thinking of B⃗ as a magnetic field, and taking the eletromagnetic field on R^{3} given by the electric field E⃗ and the magnetic field B⃗, the equations of motion for a particle with charge e and mass m in the electromagnetic field are given by the Lorentz force-law

m\frac{dv⃗}{dt} = e(E⃗ + v⃗ × B⃗),

where v⃗ = ẋ^{1}i⃗_{1} + ẋ^{2}i⃗_{2} + ẋ^{3}i⃗_{3} is the field of velocities and the dot ”·” denote the derivative with respect to the parameter t.

Since rot E⃗ = 0 (∂_{t}B⃗ = 0), we can write (locally) E⃗ = gradφ.

On R^{3} × R^{3}, i.e., on (x^{1}, x^{2}, x^{3}, ẋ^{1}, ẋ^{2}, ẋ^{3}) -space, we consider the symplectic form

Ω_{B} = mδ_{ij}dx^{i} ∧ dẋ^{j} − eB

and the Hamiltonian (total energy)

H = \frac{⇕}{∈}δ_{⟩|}{∨ \atop §^{⟩}}{∨ \atop §^{|}} + ⌉φ(§).

Denoting X_{H}(u^{1}, u^{2}, u^{3}) = (u^{1}, u^{2}, u^{3}, u̇^{1}, u̇^{2}, u̇^{3}) the condition of defining X_{H}, i.e.,

i_{X_{H}}Ω_{B} = dH

becomes

m(u^{1}dẋ^{1} − u̇^{1}dx^{1} + u^{2}dẋ^{2} − u̇^{2}dx^{2} + u^{3}dẋ^{3} − u̇^{3}dx^{3})−

−e(B_{1}u^{2}dx^{3} − B_{1}u^{3}dx^{2} + B_{2}u^{3}dx^{1} − B_{2}u^{1}dx^{3} + B_{3}u^{1}dx^{2} − B_{3}u^{2}dx^{1}) =

= m(ẋ^{1}dẋ^{1} + ẋ^{2}dẋ^{2} + ẋ^{3}dẋ^{3}) + e (\frac{∂f}{∂x^{1}} dx^{1} + \frac{∂f}{∂x^{2}} dx^{2} + \frac{∂f}{∂x^{3}} dx^{3}) .

Consequently

u^{1} = ẋ^{1}, u^{2} = ẋ^{2}, u^{3} = ẋ^{3},
mu̇^{1} = e(E_{1} + B_{3}u^{2} − B_{2}u^{3}), mu̇^{2} = e(E_{2} + B_{1}u^{3} − B_{3}u^{1}),
mu̇^{3} = e(E_{3} + B_{2}u^{1} − B_{1}u^{2})

or

mẍ^{1} = e(E_{1} + B_{3}ẋ^{2} − B_{2}ẋ^{3})
mẍ^{2} = e(E_{2} + B_{1}ẋ^{3} − B_{3}ẋ^{1})
mẍ^{3} = e(E_{3} + B_{2}ẋ^{1} − B_{1}ẋ^{2}),

which are the same with classical Lorentz equations. Thus the equations of motion for a charged particle in an electromagnetic field are Hamiltonian, with total energy H and with the symplectic form Ω_{B}.

4 Electromagnetic dynamical systems

The physical-mathematical objects of the electromagnetism are:

U ⊂ R^{3} = domain of linear homogeneous isotropic media,

t = the time,

E⃗ = the electric vector field (electric intensity),

H⃗ = the magnetic vector field (magnetizing force),

B⃗ = the magnetic flux density (magnetic induction),

D⃗ = the electric displacement (electric induction),

J⃗ = the electric current density (conduction current density),

ρ = the electric charge density,

∂_{t} = the time derivative operator,

μ = the scalar permeability,

ε = the permitivity. The previous fields are defined on U × R and satisfy the Maxwell equations

div D⃗ = ρ, rot H⃗ = J⃗ + ∂_{t}D⃗
div B⃗ = 0, rot E⃗ = −∂_{t}B⃗,

the associated constitutive equations relating the fields being

B⃗ = μH⃗, D⃗ = εE⃗.

Let E⃗ = E_{1}i⃗_{1} + E_{2}i⃗_{2} + E_{3}i⃗_{3} be the electric vector field on the domain U × R, i.e., E⃗ = E⃗(x, t). The electric line α which starts at the moment s = 0 from the point m_{0}(x_{0}^{1}, x_{0}^{2}, x_{0}^{3}) is the oriented curve

α : (−a, a) → U, α(s) = (x^{1}(s), x^{2}(s), x^{3}(s)),

the solution of the Cauchy problem

\frac{dx^{i}}{ds} = E_{i}, x^{i}(0) = x_{0}^{i}, i = 1, 2, 3.

The set of all images of maximal electric lines is called the phase portrait of the electric field E⃗. Obviously the parameter s is different from the time parameter t. The time parameter t can produce bifurcation in the equilibrium set of E⃗ or Hopf bifurcation of the flow of E⃗. The coincidence between the parameters t and s remains open though we can use the ideas of the paper [6] in order to study the case s = t.

Let f : U → R, f = \frac{1}{2}(E_{1}^{2} + E_{2}^{2} + E_{3}^{2}) be the energy of E⃗, leaving aside the multiplicative factor ε.

4.1. Theorem. Every electric line is the trajectory of a nonpotential dynamical system with 3 degrees of freedom determined by the potential V = −f and by rot E⃗, namely

(4)  \frac{d^{2}x^{i}}{ds^{2}} = \frac{∂f}{∂x^{i}} + ∂_{t}B_{k} \frac{dx^{j}}{ds} − ∂_{t}B_{j} \frac{dx^{k}}{ds} ,

{i, j, k} being a cyclic permutation of {1, 2, 3}.

Proof. Deriving \frac{dx^{i}}{ds} = E_{i} along a solution α, using rot E⃗ = −∂_{t}B⃗ and replacing

\frac{dx^{i}}{ds} only in terms which permit to recover ∇f we find the prolongation

\frac{d^{2}x^{i}}{ds^{2}} = \frac{∂f}{∂x^{i}} + (\frac{∂E_{i}}{∂x^{j}} − \frac{∂E_{j}}{∂x^{i}} ) \frac{dx^{j}}{ds} ,

which is the same with (4).

In this context we can prove the following propositions.

4.2. Theorem. The dynamical system (4) is conservative, the total energy being

(5)  H = \frac{∞}{∈}δ_{⟩|} \frac{⌈§^{⟩}}{⌈∫} \frac{⌈§^{|}}{⌈∫} − \{(§^{∞}, §^{∈}, §^{∋}).

4.3. Theorem. On T R^{3} ≃ R^{6}, the equations (4) of motion of a particle sensible to the electric field E⃗ are Hamiltonian with respect to the total energy (5) and the symplectic form

Ω = δ_{ij}dx^{i} ∧ dẋ^{j} − ∂_{t}B,

where the magnetic induction is viewed as a closed 2-form

B = B_{1}dx^{2} ∧ dx^{3} + B_{2}dx^{3} ∧ dx^{1} + B_{3}dx^{1} ∧ dx^{2}

to whom is associated the solenoidal vector field B⃗ = B_{1}i⃗_{1} + B_{2}i⃗_{2} + B_{3}i⃗_{3}, and the dot ”·” denotes the derivative with respect to the parameter s.

4.4. Theorem (a new version of the Lorentz world force-law). Every nonconstant trajectory of the dynamical system (4) which has the total energy H is a reparametrized horizontal geodesic of the Riemann-Jacobi-Lagrange manifold

(U \ E, }_{⟩|} = (H + {)δ_{⟩|}, N_{|}^{⟩} = −_{|∥}^{⟩}†^{∥} + F^{⟩}_{|}, ⟩, |, ∥ = ∞, ∈, ∋),

where

Γ_{jk}^{i} = Riemannian connection induced by g_{ij},

F_{ij} = \frac{∂E_{j}}{∂x^{i}} − \frac{∂E_{i}}{∂x^{j}} = E_{j,i} − E_{i,j}, F^{i}_{j} = g^{ih}F_{hj}.

Remarks. 1) The vector field \frac{dα⃗}{ds} × ∂_{t}B⃗ does not produce a dissipation of energy along the electric line α since it is orthogonal to α.

2) Other prolongation on U of the dynamical system \frac{dx^{i}}{ds} = E_{i}, i = 1, 2, 3, is the nonconservative dynamical system of order two

\frac{d^{2}x^{i}}{ds^{2}} = \frac{∂f}{∂x^{i}} + E_{j}∂_{t}B_{k} − E_{k}∂_{t}B_{j}, \{i, j, k\} = permutation of \{1, 2, 3\}.

3) The flow generated by X_{H} conserves the volume.

4) For the magnetic lines (the field lines of H⃗) one obtains similar results. The difference is that the associated symplectic form contains the closed 2-form J + ∂_{t}D associated to the solenoidal vector field J⃗ + ∂_{t}D⃗.

Open problem. Find the properties of the field lines of the Poynting vector field

S⃗ = E⃗ × H⃗.

5 Electromagnetic dynamical systems in the relativistic model

Let M be a connected 4-dimensional differentiable manifold and g a Lorentz metric on M. The pair (M, g) is called Lorentz manifold.

5.1. Definition. A spacetime (M, g, ∇) is a connected 4-dimensional, oriented, and time-oriented Lorentz manifold (M, g) together with its Levi-Civita connection ∇.

Let F be the electromagnetic field like a 2-form on M, and J be the chargecurrent density of the matter model M. The relativistic model will be denoted either by (M, M, F) or by (M, F, J).

5.2. Definition. (M, M, F) or (M, F, J) verifies Maxwell equations iff:

1) F is closed, i.e., dF = 0;

2) div F̂ = J, where F̂ is the (1,1)-tensor field physically equivalent to F via the Lorentz metric g.

As a consequence of 1), locally, there exists a 1-form η such that F = dη. We denote by ξ the vector field physically equivalent to η via the Lorentz metric g.

Obviously, J is a solenoidal vector field, i.e.,

div J = div div F̂ = 0.

Usually, the authors study the influence of spacetime M and of the matter model M on the electromagnetic field F.

Let F_{ij} be the componets of F. Then dF = 0 is equivalent to

F_{ij,k} + F_{jk,i} + F_{ki,j} = 0

and

divF̂ = J

is equivalent to

F^{j}_{i,j} = −J_{i}.

If M and J are given ab initio, and the influence of F on M and on M is neglected, then the Maxwell equations become conditions determining F. Here we use Maxwell equations to obtain information about the dynamical systems generated by η and J.

Examples. 1) Constant magnetic field. Set E = 0, and F = 2Bdx^{3} ∧ dx^{1} is an electromagnetic field on the Minkowski space (R^{4}, g); B is a scalar field on R^{4}, and the electric field E in covariant constant (parallel, inertial) reference frame ∂_{4} is everywhere zero. The condition dF = 0 is equivalent to ∂_{4}B = 0 = ∂_{2}B. The condition divF̂ = 0 (zero source J) gives ∂_{3}B = 0 = ∂_{1}B. Consequently B = constant.

2) Waves. Let (R^{4}, g) be a Minkowski space. Near the origin of 3-space are some electric charges that move back and forth in the ∂_{1} direction of 3-space. An electromagnetic field is generated. In the observation region (”wave zero”), this field can be described as

F = 2(f ∘ ϕ)dϕ ∧ dx^{1},

where f : R → R is C^{∞}, and ϕ = (x^{3} − x^{4}) : R^{4} → R. The set (R^{4}, F, 0) verifies the Maxwell equations

dF = f′ ∘ ϕ dϕ ∧ dϕ ∧ dx^{1} = 0
F̂ = 2(f ∘ ϕ)(∂_{3} + ∂_{4}) ∧ ∂_{1} =

= the (2,0)-tensor field physically equivalent to F via the Lorentz metric,

divF̂ = ∂_{1}(f ∘ ϕ) − (∂_{3} + ∂_{4})(f ∘ ϕ) = 0.

F is called a plane, linearly polarized electromagnetic wave on Minkowski space.

The stress-energy tensor T of an electromagnetic field F on M is defined as a (0,2)-tensor field on M of components

T_{ij} = F_{im}F_{j}^{m} − \frac{1}{4}g_{ij}F^{mn}F_{mn}.

5.3. Theorem. Let T̂ be the (2,0)-tensor field physically equivalent to T via the Lorentz metric g.

1) T̂ is symmetric and trace T̂ = 0.

2) T̂(ω, ω) ≥ 0 for every causal 1-form ω.

3) If (M, F, J) verifies Maxwell equations, then divT̂ = −F̂J.

Using the components F_{ij} of F, the condition divT̂ = −F̂J is equivalent to

T^{ij}_{,j} = −F^{i}_{m}J^{m}.

Remark. The stress-energy tensor T unifies and replaces the classical energy

density \frac{1}{2}(ε∥E⃗∥^{2} + μ∥H⃗∥^{2}), Poynting vector field S⃗ = E⃗ × H⃗ and Maxwell stress tensor field of components

t^{αβ} = −(εE^{α}E^{β} + μH^{α}H^{β} − \frac{1}{2}δ^{αβ}(ε∥E⃗∥^{2} + μ∥H⃗∥^{2})).

We consider the vector field ξ of components ξ^{i}, i = 1, 2, 3, 4, physically equivalent to the 1-form η via the Lorentz metric g. The energy associated to ξ is f : M →

R, f = \frac{1}{2}g(ξ, ξ). Obviously

f = \frac{1}{2}g_{ij}ξ^{i}ξ^{j} = \frac{1}{2}g^{ij}η_{i}η_{j}.

The field line α of ξ which starts from the point (x_{0}^{1}, x_{0}^{2}, x_{0}^{3}, x_{0}^{4}) at the moment s = 0 is the oriented curve α : (−a, a) → M, α(s) = (x^{1}(s), x^{2}(s), x^{3}(s), x^{4}(s)) which satisfies the Cauchy problem

\frac{dx^{i}}{ds} = ξ^{i}, x^{i}(0) = x_{0}^{i}, i = 1, 2, 3, 4.

Since ξ is an irrotational vector field the following theorem is true.

5.4. Theorem. Every field line of ξ is a trajectory of a potential dynamical system with 4 degrees of freedom associated to the potential V = −f.

We can obtain automatically a new version of the Lorentz world-force law determined by ξ and g.

Now we consider the vector field J of components J^{i}, i = 1, 2, 3, 4. The energy

associated to J is φ : M → R, φ = \frac{1}{2}g(J, J), and the field line of J which start from

the point (x_{0}^{1}, x_{0}^{2}, x_{0}^{3}, x_{0}^{4}) at the moment s = 0 is the oriented curve α : (−a, a) → M, α(s) = (x^{1}(s), x^{2}(s), x^{3}(s), x^{4}(s)) which satisfies the Cauchy problem

\frac{dx^{i}}{ds} = J^{i}, x^{i}(0) = x_{0}^{i}, i = 1, 2, 3, 4.

We can obtain easily the prolongation of this dynamical system to a conservative differential system of order two and hence a new variant of the Lorentz world- force law induced by J and g. The flow generated by J conserves the volume because J is a solenoidal vector field.

Suppose that J has no zero on M. Then J = ∥J∥J_{0}, ∥J_{0}∥ = 1 and the restriction of the energy φ to a field line α(s), s ∈ I of J_{0} (s being here the curvilinear abscissa) is well determined by the restriction of divJ_{0} to that line. Indeed, denoting l = ∥J∥ ∘ α, m = divJ_{0} and taking into account that

0 = divJ = D_{J_{0}}∥J∥ + ∥J∥divJ_{0},

we find

\frac{dl}{ds} = −lm.

Consequently

l(s) = l_{0} exp (− ∫_{s_{0}}^{s} m(t)dt) , l(s_{0}) = l_{0}.

If m is nowhere zero, the field line α cannot be closed (the field lines of J_{0} are reparametrizations of the field lines of J).

Acknowledgements. I would like to thank Acad.Prof.Dr. Andrei Ţugulea for helpful comments and useful suggestions regarding electromagnetism and matter. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

References

[1] R.G.Beil, Comparison of unified field theories, Tensor N.S., 56, 2(1995), 175-183.

[2] R.G.Beil, Notes on a new Finsler metric function, Balkan Journal of Geometry and Its Applications, 2, 1(1997), 1-6.

[3] F.M.Crampin, A linear connection associated with any second order differential equation field, In: Eds. L.Tamassy, J.Szenthe, New Developments in Geometry, 77-86, Kluwer Academic Publishers, 1996.

[4] T.Kawaguchi, R.Miron, A Lagrangian model for gravitation and electromagnetism, Tensor N.S., 48(1989), 153-168.

[5] R.Miron, The geometry of higher-order Lagrange spaces, Fundamental Theories of Physics, 82, Kluwer Academic Publishers, 1996.

[6] V.Obădeanu, C.Vernic, Systemes dynamiques sur des espaces de Riemann, Balkan Journal of Geometry and Its Applications, 2, 1 (1997), 73-82.

[7] S.Ştefănescu, C.Udrişte, Magnetic field lines around filiform electrical circuits of right angle type, Scientific Bulletin, Applied Mathematics and Physics, Politehnica University of Bucharest, 55, 1-2 (1993), 3-18.

[8] A.Udrişte, C.Udrişte, Dynamics induced by a magnetic field, Conference on Differential Geometry, Budapest, July 27-30, 1996 (will appear in Proceedings at Kluwer Academic Publishers).

[9] C.Udrişte, Convex functions and optimization methods on Riemannian manifolds, Mathematics and Its Applications, 297, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994.

[10] C.Udrişte, Electromagnetic dynamical systems as Hamilton-Poisson systems, Workshop on Differential Geometry, Global Analysis, Lie Algebras, Aristotle University of Thessaloniki, June 25-29, 1997.

[11] C.Udrişte, S.Udrişte, Biot-Savart-Laplace dynamical systems, Balkan Journal of Geometry and Its Applications, 1,2 (1996), 125-136.

Politehnica University of Bucharest Department of Mathematics I Splaiul Independentei 313 77206 Bucharest, Romania e-mail:udriste@mathem.pub.ro