Biot-Savart-Laplace Dynamical Systems

Constantin Udrişte and Sorin Udrişte

Abstract §1 recalls known facts about the magnetic field H⃗ produced by the Biot-Savart-Laplace law for a massive conductor D̄. §2 proves that generally the part in extD̄ of a magnetic line is a trajectory of a potential dynamical system of order two (a geodesic of the Riemann-Jacobi structure) and the part in intD̄ is a trajectory of a nonpotential dynamical system of order two (new Lorentz world– force laws) describing new magnetic dynamics. This paragraph presents also some properties of magnetic traps, two significant examples and formulates an open problem. §3 describes the magnetic dynamical systems which can be reduced to 2-dimensional Hamiltonian systems. §4 analyses the magnetic fields which are symmetric or antisymmetric with respect to some symmetries.

Mathematics Subject Classification: 34A26, 35F10, 53C21, 78A25 Keywords: magnetic dynamical systems, potential and nonpotential dy-

namical systems, geodesics, Lorentz laws, symmetries.

1 Biot-Savart-Laplace vector field

Let D be an open connected set of R^{3}, with a piecewise smooth boundary ∂D. Denote by J⃗ a C^{∞} vector field on D̄ = D ⋃ ∂D.

The vector field

H⃗(M) = \frac{1}{4π} ∫_{D} \frac{J⃗(P) × P⃗ M}{P M^{3}} dv_{P} , M ∈ R^{3}

is called the Biot–Savart–Laplace vector field. The name comes from the fact that in case D̄ is a domain in which there exists a current density J⃗(P), P ∈ D̄, then the magnetic field H⃗ generated on R^{3} by the electrical current is approximated by the preceding formula due to J.B.Biot, F.Savart, P.S.Laplace [2]. Remarks. 1) Since the measure (volume) of ∂D is zero, the preceding integral can be considered on D̄ = D ∪ ∂D.

2) The integral defining H⃗(M), M ∈ D̄ is an improper integral of the first type (both of the first and of the second type) if the domain D is bounded (unbounded).

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 125-136 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

3) The Biot-Savart-Laplace vector field H⃗ is of class C^{∞} on R^{3} \ ∂D and of class C^{o} on ∂D.

4) The vector field J⃗ can have zeros on D̄.

The vector field H⃗ is solenoidal. Hence it admits a vector potential

A⃗(M) = \frac{1}{4π} ∫_{D} \frac{J⃗(P)}{P M}dv_{P} .

Indeed,

rotA⃗(M) = ∇_{M} × A⃗(M) = ∇_{M} × \frac{1}{4π} ∫_{D} \frac{J⃗(P)}{P M}dv_{P} =

= \frac{1}{4π} ∫_{D} ∇_{M} × \frac{J⃗(P)}{P M}dv_{P} = −\frac{1}{4π} ∫_{D} J⃗(P) × ∇_{M} \frac{1}{P M} dv_{P} =

= \frac{1}{4π} ∫_{D} \frac{J⃗(P) × P⃗ M}{P M^{3}} dv_{P} = H⃗(M).

On the other hand,

divA⃗(M) = (∇_{M}, A⃗(M)) = \frac{1}{4π} ∫_{D} (∇_{M}, \frac{J⃗(P)}{P M})dv_{P} =

= \frac{1}{4π} ∫_{D} (J⃗(P), ∇_{M} \frac{1}{P M})dv_{P} = −\frac{1}{4π} ∫_{D} (J⃗(P), ∇_{P} \frac{1}{P M})dv_{P} =

= −\frac{1}{4π} ∫_{D} (∇_{P} , \frac{J⃗(P)}{P M})dv_{P} + \frac{1}{4π} ∫_{D} \frac{1}{P M} (∇_{P} , J⃗(P))dv_{P} =

= −\frac{1}{4π} ∫_{∂D} \frac{(n⃗(P), J⃗(P))}{P M}dv_{P} + \frac{1}{4π} ∫_{D} \frac{div J⃗(P)}{P M}dv_{P} ,

where n⃗(P) is the unit normal vector field of the surface ∂D. If J⃗ is a solenoidal vector field (a stationary electrokinetic field), and ∂D is a field surface of J⃗ , i.e., (n⃗(P), J⃗(P)) = 0, then div A⃗(M) = 0, and so A⃗ is a solenoidal vector field.

Under the hypothesis div A⃗(M) = 0, we compute

rot H⃗(M) = ∇_{M} × (∇_{M} × A⃗(M)) = ∇_{M}(∇_{M}, A⃗(M)) − (∇_{M}, ∇_{M})A⃗(M) =

= ∇_{M}div A⃗(M) − ∇_{M}^{2} A⃗(M) = −∇_{M}^{2} A⃗(M) = −Δ_{M} A⃗(M),

so that rot H⃗(M) = 0 for M ∈ R^{3} \ D̄, and rot H⃗(M) = J⃗(M) for M ∈ D̄; also, we have

divH⃗(M) = div rotA⃗ = 0.

Consequently, the vector field H⃗ is not irrotational, but the restriction of H⃗ to R^{3} \ D̄ is an irrotational vector field. This restriction admits a local scalar potential. Remarks.

1) We notice that

∇_{M}F (P M) = F′(P M)\frac{P⃗ M}{P M} = −∇_{P} F (P M), ∀F : R → R, derivable.

2) If the point M has the coordinates (x, y, z) and the point P has the coordinates (ξ, η, ζ), then

\frac{∂}{∂x}A⃗(M) = − ∫_{D} \frac{∂}{∂ξ} (\frac{J⃗(P)}{P M})dv_{P} + ∫_{D} \frac{1}{P M} \frac{∂}{∂ξ} J⃗(P)dv_{P}, etc.

3) If J⃗ is a constant vector field on D̄ , then the magnetic field H⃗ generated on R^{3} by J⃗ is a biscalar field, i.e., (H⃗, hboxrotH⃗) = 0.

4) The domain D can be replaced by a surface or a curve. In case of a curve, the current density J⃗ must be nonzero everywhere along the curve.

2 Magnetic lines and surfaces. Magnetic traps

Let H⃗ = H_{x}i⃗ + H_{y}j⃗ + H_{z}k⃗ be a magnetic field defined on R^{3}. Denote by r⃗ = xi⃗ + yj⃗ + zk⃗ the position vector of the point M(x, y, z).

The magnetic line α starting of M_{0}(x_{0}, y_{0}, z_{0}) at moment t = 0 is the oriented curve r⃗ = r⃗(t), t ∈ (−ϵ, ϵ), which satisfies the Cauchy problem

\frac{dr⃗}{dt} = H⃗(r⃗), r⃗(0) = r⃗_{0}.

The magnetic surface ∑ : h(x, y, z) = c relying on a curve β : (a, b) → R^{3} is the solution of the Cauchy problem

(H⃗, ∇h) = 0, h(β(u)) = h(β(0)), ∀u ∈ (a, b).

A magnetic surface is generated by magnetic lines, and, in the absence of symmetries, a magnetic line is an open curve, and sometimes its image is densely in the magnetic surface.

Let U be an open connected set of R^{3} with a piecewise smooth boundary ∂U and T_{t} the flow generated by the magnetic vector field H⃗. The flow T_{t} conserves the spaces volume since div H⃗ = 0. The set U or its closedness Ū is called a trap region of the magnetic field H⃗ (magnetic trap) if T_{t}(Ū) ⊂ Ū, ∀t ≥ 0. Particularly, any T_{t}-invariant set is a trap region. A magnetic trap is characterized by the fact that a magnetic line starting inside cannot leave it (because such a line cannot attend the boundary ∂U). The magnetic line starting in the exterior of the magnetic trap can enter or not inside the trap.

Suppose that the unit normal vector field n⃗ of the surface ∂D is oriented towards extU. If U is a magnetic trap, then on the boundary ∂U we have (n⃗, H⃗) ≤ 0. Conversely, if (n⃗, H⃗) = 0 on ∂U, then ∂U is a magnetic surface; if (n⃗, H⃗) > 0 on ∂U, then R^{3} \ U is a trap region of the magnetic field H⃗, and if (n⃗, H⃗) < 0 on ∂U, then U is a trap region of the magnetic field H⃗ .

Suppose there exist two magnetic traps U_{1} and U_{2} such that their boundaries ∂U_{1}, ∂U_{2} have a common part which is a surface Σ or a curve γ. Then Σ is a magnetic surface, and γ is a magnetic line, respectively.

2.1. Theorem. Let U be an open connected set of R^{3} with a piecewise smooth boundary ∂U. If U is a magnetic trap and Ū = U ⋃ ∂U is compact, then the closed surface ∂U is a magnetic surface and T_{t}(Ū) = Ū. Proof. Suppose that the unit normal vector field n⃗ of the surface ∂U is oriented towards ext U. Since divH⃗ = 0, by Gauss Theorem we obtain ∫_{∂U}(n⃗, H⃗)dσ = 0. If ∂U is not a magnetic surface, i.e., (n⃗, H⃗) ≠ 0, then (n⃗, H⃗) must change sign on the closed surface ∂U and consequently U is not a magnetic trap.

The last assertion of the Theorem is a consequence of the conservation of the volume. Remark. If H⃗ is a magnetic field and Ū is not a compact set, then it is possible that U is a magnetic trap for H⃗ without ∂U be a magnetic surface. For example, the magnetic field H⃗(x, y, z) = i⃗, the region U : x + y + z + 1 > 0, the boundary ∂U : x + y + z + 1 = 0, the unit normal n⃗ = \frac{−i⃗−j⃗−k⃗}{\sqrt{3}} imply (n⃗, H⃗) < 0; hence U is a magnetic trap. Also, this example shows that in the preceding context the T_{t}-invariant set ∩_{t≥0}T_{t}(Ū) can be the void set.

In the following part of this paragraph we refer only to the Biot-Savart-Laplace vector field. Is the region D in the Biot-Savart-Laplace formula a magnetic trap or not ? A possible answer was given in the preceding theorem. The next theorem presents another alternative. 2.2. Theorem. Suppose that the unit normal vector field n⃗ of the surface ∂D is oriented towards extD. Let

φ : D̄ × ∂D → R, φ(P, M) = (n⃗(M), J⃗(P) × P⃗ M).

1) If φ(P, M) = 0, ∀P ∈ D̄, ∀M ∈ ∂D, i.e., the vector fields n⃗(M), J⃗(P), P⃗ M are coplanar on D̄ × ∂D, then ∂D is a magnetic surface, and R^{3} \ D and D are magnetic traps.

2) If φ(P, M) > 0, ∀P ∈ D̄, ∀M ∈ ∂D, then R^{3} \ D is a magnetic trap.

3) If φ(P, M) < 0, ∀P ∈ D̄, ∀M ∈ ∂D, then D is a magnetic trap. Proof. Consequences of the relation

(n⃗(M), H⃗(M)) = ∫_{D} \frac{(n⃗(M), J⃗(P) × P⃗ M)}{P M^{3}}dv_{P} , ∀M ∈ ∂D.

Remark. In the case 1) the vector fields J⃗ and H⃗ are tangent to ∂D. In the cases 2)-3), the vector field J⃗ is tangent to ∂D and the vector field H⃗ is transversal to ∂D.

Let

f : R^{3} → R, f = \frac{μ_{0}}{2} (H_{x}^{2} + H_{y}^{2} + H_{z}^{2})

be the energy of H⃗, where μ_{0} is the absolute magnetic permeability of the medium. 2.3. Theorem. Any magnetic line in int(R^{3} \ D) is a trajectory of a potential dynamical system with three degrees of freedom associated to the potential −fμ_{0}^{−1} on int(R^{3} \ D). Proof. Let α be a magnetic line included in int(R^{3} \ D). Deriving \frac{dr⃗}{dt} = H⃗ along α we find the prolongation

\frac{d^{2}r⃗}{dt^{2}} = μ_{0}^{−1}∇f.

which is a potential dynamical systems of order two. Remark. If α contains at least one point of ∂D, then the previous assertion fails.

Denoting

\frac{dr⃗}{dt} = −v⃗,

the differential system of order two

\frac{d^{2}r⃗}{dt^{2}} = μ_{0}^{−1}∇f

can be transcribed as a Hamiltonian system

\frac{dr⃗}{dt} = −v⃗, \frac{dv⃗}{dt} = −μ_{0}^{−1}∇f,

on the phase space R^{6}, with the Hamiltonian

H(r⃗, v⃗) = \frac{1}{2} ∥ v⃗ ∥^{2} −μ_{0}^{−1}f(r⃗).

The Hamiltonian flow conserves the phase space volume. The Hamiltonian H is a first integral of the Hamiltonian dynamical system. The theory of Hamiltonian systems shows that 2.4. Theorem. The trajectory of the dynamical system

\frac{d^{2}r⃗}{dt^{2}} = μ_{0}^{−1}∇f, r⃗(0) = r⃗_{0} ∈ R^{3} \ D̄,

having H > −μ_{0}^{−1}f as constant total energy and staying in R^{3} \ D̄ is a reparametrized geodesic of the Riemann–Jacobi metric

g_{ij} = (H + μ_{0}^{−1}f)δ_{ij}, i, j = 1, 2, 3.

Remarks. 1) If the domain D is reduced to a curve γ, then the theorems 2.3 and 2.4 hold true on R^{3} \ γ.

2) Theorem 2.4 shows that the preceding prolongation is a new Lotentz law based on a geometrical structure which incorporates the magnetic field. 2.5. Theorem. Any magnetic line included in D is a trajectory of a nonpotential dynamical system with three degrees of freedom for which the energy

H(r⃗, v⃗) = \frac{1}{2} ∥ v⃗ ∥^{2} −μ_{0}^{−1}f(r⃗)

is conserved. Proof. If r⃗(0) = r⃗_{0} ∈ D and α rests in D, then deriving \frac{dr⃗}{dt} = H⃗ along α we get a prolongation

\frac{d^{2}r⃗}{dt^{2}} = μ_{0}^{−1}∇f + J⃗ × \frac{dr⃗}{dt} ,

which is a nonpotential dynamical system of order two. If we take the scalar product with \frac{dr⃗}{dt}, we find \frac{d}{dt}H = 0, i.e., the energy H is conserved. Remarks.

1) The vector field J⃗ × \frac{dr⃗}{dt} does not produce a dissipation of energy along the solution α since it is orthogonal to the curve α.

2) Recently [8], [9] we have discovered a new geometrical structure showing that the preceding conservative nonpotential dynamical system of order two describes a new Lorentz world–force law (new magnetic dynamics).

3) The trajectories of the preceding conservative (potential or nonpotential) dynamical system of order two divide into three classes:

– the set of original magnetic lines with the energy H = 0;

– a set of trajectories with the energy H = const. < 0,

– a set of trajectories with the energy H = const. > 0.

4) Another prolongation is the nonpotential nonconservative dynamical system

\frac{d^{2}r⃗}{dt^{2}} = μ_{0}^{−1}∇f + J⃗ × H⃗.

Since rot(J⃗ × H⃗) ≠ 0, the vector field J⃗ × H⃗ corresponds to a dissipation of energy along the solutions α which are not orthogonal to J⃗ × H⃗.

5) In both cases the projection of the acceleration \frac{d^{2}r⃗}{dt^{2}} on J⃗ depends only of the projection of ∇f on J⃗. Open problem. What is the physical signification for trajectories of the preceding conservative differential systems, with positive or negative constant energy ?

If the magnetic line α traverses ∂D, then its part contained in D is a trajectory of a nonpotential dynamical system of order two, and its part in R^{3} \ D̄ is a trajectory of a potential dynamical system of order two. Obviously, α can be or not smooth at the traversing point of ∂D, being a field line of the vector field H⃗ which is of class C^{∞} on R^{3} \ ∂D and of class C^{o} on ∂D. Examples.

1) Suppose that D̄ = D ⋃ ∂D is a circular cylinder of radius a carrying a steady current I. We fix the Cartesian frame Oxyz such that Oz is the axis of the cylinder and J⃗ = \frac{I}{πa^{2}} k⃗ is the current density. The electrical current generates the magnetic field [2] (Fig.1) given by

H⃗(M) = \frac{−yi⃗ + xj⃗}{2πa^{2}}I, for M(x, y, z) ∈ D̄,

and

H⃗(M) = \frac{−yi⃗ + xj⃗}{2π(x^{2} + y^{2})}I, forM(x, y, z) ∈ R^{3} \ D̄.

The Oz-axis consists of zeros of H⃗. The nonconstant magnetic lines are circles with the centers on Oz and situated in planes orthogonal to Oz. The boundary ∂D is a magnetic surface, though H⃗ is only continuous on ∂D. The regions D and R^{3} \ D̄ are magnetic traps. Even in this simple case we have no answer for the preceding open problem.

2) Suppose that D̄ = D ∪ ∂D is an infinite prism of rectangular section [−a, a] × [−b, b] carrying a steady current I. We choose a Cartesian frame Oxyz such that Oz is the axis of the prism and J⃗ = \frac{I}{4ab}k⃗ is the current density (Fig.2). The magnetic field generated by J⃗ on R^{3} has the components (H_{x}, H_{y}, 0), where [2]

H_{x} = −\frac{I}{2ab}[(y − b)(arctan \frac{x − a}{y − b} − arctan \frac{x + a}{y − b}) − (y + b)(arctan \frac{x − a}{y + b}−

− arctan \frac{x + a}{y + b}) − \frac{x + a}{2} ln \frac{(x + a)^{2} + (y − b)^{2}}{(x + a)^{2} + (y + b)^{2}} + \frac{x − a}{2} ln \frac{(x − a)^{2} + (y − b)^{2}}{(x − a)^{2} + (y + b)^{2}} ],

H_{y} = −\frac{I}{2ab}[(x − a)(arctan \frac{y + b}{x − a} − arctan \frac{y − b}{x − a}) − (x + a)(arctan \frac{y + b}{x + a}−

arctan \frac{y − b}{x + a}) − \frac{y − b}{2} ln \frac{(x − a)^{2} + (y − b)^{2}}{(x + a)^{2} + (y − b)^{2}} + \frac{y + b}{2} ln \frac{(x − a)^{2} + (y + b)^{2}}{(x + a)^{2} + (y + b)^{2}} ].

The field H⃗ is defined on D and R^{3} \ D by the same formulas and can be extended by continuity on ∂D. The boundary ∂D is piecewise smooth, and is not a magnetic surface. The field H⃗ is continuous on ∂D. If S : R^{3} → R^{3} is the symmetry with respect to Oz and H⃗ is regarded as H⃗ : R^{3} → R^{3} , then H⃗ ∘ S = S ∘ H⃗ .

The Oz–axis consists of zeros of H⃗. The field lines are closed curves (symmetrical with respect to Oz) included in the family of curves

∫_{x_{o}}^{x} H_{y}(x, y)dx − ∫_{y_{o}}^{y} H_{x}(x_{o}, y)dy = b_{k}, z = c.

Computing the integrals, the first equation becomes 132 C.Udrişte and S.Udrişte

\frac{(x − a)^{2} + (y + b)^{2}}{2} arctan \frac{y + b}{x − a} + (y + b)^{2} arctan \frac{x − a}{y + b}−

−\frac{(x − a)^{2} + (y − b)^{2}}{2} arctan \frac{y − b}{x − a} − (y − b)^{2} arctan \frac{x − a}{y − b}+

+\frac{(x + a)^{2} + (y − b)^{2}}{2} arctan \frac{y − b}{x + a} + (y − b)^{2} arctan \frac{x + a}{y − b}−

−\frac{(x + a)^{2} + (y + b)^{2}}{2} arctan \frac{y + b}{x + a} − (y + b)^{2} arctan \frac{x + a}{y + b}−

−\frac{(x − a)(y − b)}{2} ln((x − a)^{2} + (y − b)^{2}) + \frac{(x + a)(y − b)}{2} ln((x + a)^{2} + (y − b)^{2})+

+\frac{(x − a)(y + b)}{2} ln((x − a)^{2} + (y + b)^{2}) − \frac{(x + a)(y + b)}{2} ln((x + a)^{2} + (y + b)^{2})+

+\frac{(y + b)^{2}}{2} (arctan \frac{y + b}{x_{o} + a} + arctan \frac{x_{o} + a}{y + b} − arctan \frac{y + b}{x_{o} − a} − arctan \frac{x_{o} − a}{y + b})+

\frac{(y − b)^{2}}{2} (arctan \frac{y − b}{x_{o} − a} + arctan \frac{x_{o} − a}{y − b} − arctan \frac{y − b}{x_{o} + a} − arctan \frac{x_{o} + a}{y − b}) = c_{k}.

The parameters b_{k}, c_{k} are constants depending on the regions k = 1, 2, …, 9 in the Fig.2, containing the fixed point (x_{0}, y_{0}). Also, we must have in mind that

arctan u + arctan \frac{1}{u} = (sign u)\frac{π}{2} .

Some of these field lines traverse the boundary of the rectangle [−a, a] × [−b, b].

Let us consider Runge-Kutta approximations of Cauchy problems

\frac{dx}{dt} = H_{x}, \frac{dy}{dt} = H_{y}, \frac{dz}{dt} = 0,

x(0) = x_{0}, y(0) = 0, z(0) = 0,

obtained by us using a personal computer, refering to the case a = 1, b = 2, the step p = 0, 02 and the following table

\frac{Initial points x_{0} | 0.6 | 0.9 | 1.2 | 1.5 | 1.8 | 2.00 | 2.5}{Number of Iterations | 220 | 220 | 220 | 275 | 350 | 520 | 900}

We remark that Runge–Kutta approximations of magnetic lines through points corresponding to x_{0} ∈ (0; 0.9) have a shape similar to those of an ellipse, and the curvature of the magnetic lines corresponding to x_{0} ∈ (1.2; 2.5) has non-constant sign (Fig.3).

Generally, the sign of the curvature of a magnetic line α coincides with the sign of the function

H_{x}^{2} \frac{∂H_{y}}{∂x} + H_{x}H_{y}(\frac{∂H_{y}}{∂y} − \frac{∂H_{x}}{∂x}) − H_{y}^{2} \frac{∂H_{x}}{∂y} = H_{x}^{2} \frac{∂H_{y}}{∂x} + 2H_{x}H_{y} \frac{∂H_{y}}{∂y} − H_{y}^{2} \frac{∂H_{x}}{∂y}

along α(I), but this sign cannot be constant throughout the plane xOy.

The shape of the preceding magnetic lines around the points on xOz(yOz) becomes obvious if we take into account that component H_{x}(x, y) vanishes for y = 0 (the component H_{y}(x, y) vanishes for x = 0).

In fact, if α(t) = (x(t), y(t), 0), t ∈ R is the maximal solution of the preceding Cauchy problem, then x_{0} ≠ 0 is an extremum value of the function x(t), t ∈ R.

3 Magnetic dynamical systems which are bidimensional Hamiltonian systems

In this paragraph we shall generalize an idea of [4]. Let H⃗ = H_{x}i⃗ + H_{y}j⃗ + H_{z}k⃗ be the magnetic field of Biot-Savart-Laplace and

\frac{dx}{dt} = H_{x}, \frac{dy}{dt} = H_{y}, \frac{dz}{dt} = H_{z}

the associated magnetic dynamical system. Suppose H_{z} = 0, H_{x} = H_{x}(x, y), H_{y} = H_{y}(x, y). Since

0 = div H⃗ = \frac{∂H_{x}}{∂x} + \frac{∂H_{y}}{∂y} ,

there exists a function H : R^{2} → R such that

\frac{∂H}{∂y} = H_{x}, \frac{∂H}{∂x} = −H_{y}

and hence the magnetic dynamical system reduces to the bidimensional Hamiltonian system

\frac{dx}{dt} = \frac{∂H}{∂y}, \frac{dy}{dt} = −\frac{∂H}{∂x}.

(see the examples of §2).

Suppose that H_{z} is not zero function and that

\frac{H_{x}}{H_{z}} , \frac{H_{y}}{H_{z}}

are functions of x and y only. The magnetic dynamical system transcribes

\frac{dx}{dz} = \frac{H_{x}}{H_{z}} , \frac{dy}{dz} = \frac{H_{y}}{H_{z}} .

If there exists H : R^{2} → R such that

\frac{∂H}{∂y} = \frac{H_{x}}{H_{z}} , \frac{∂H}{∂x} = −\frac{H_{y}}{H_{z}} ,

then

0 = \frac{∂}{∂x}(\frac{H_{x}}{H_{z}}) + \frac{∂}{∂y} (\frac{H_{y}}{H_{z}}) = div \frac{H⃗}{H_{z}} = \frac{1}{H_{z}} div H⃗ + (H⃗, grad \frac{1}{H_{z}}) =

= −\frac{(H⃗, grad H_{z})}{H_{z}^{2}}

and hence H_{z} is either a constant function or a first integral of the magnetic dynamical system (i.e., H_{z}(x, y, z) = c are field surfaces of H⃗).

Conversely, if H_{z} is either a first integral of the magnetic dynamical system or a constant function and the ratios

\frac{H_{x}}{H_{z}} , \frac{H_{y}}{H_{z}}

are functions of x, y only, then the Hamiltonian H does exist. Consequently the following theorem is true. 3.1. Theorem. Suppose that none of the components (H_{x}, H_{y}, H_{z}) is the function zero. The corresponding magnetic dynamical system is reducible to a bidimensional Hamiltonian system if and only if at least one of the components (H_{x}, H_{y}, H_{z}) is either a constant or a first integral of the magnetic dynamical system and the ratios of the other two by this component depend effectively only of the variables who index them. Remark. Suppose that the magnetic dynamical system reduces to a Hamiltonian system on xOy. Then the magnetic lines are geodesics of a Riemannian-Jacobi structure which is conformal to the Euclidean structure on R^{2}.

4 Symmetric and antisymmetric magnetic fields

The examples in §2 suggest the following considerations.

Let S : R^{3} → R^{3} be a symmetry and H⃗ be the Biot–Savart–Laplace field regarded as a function of the type H⃗ : R^{3} → R^{3}.

If H⃗ ∘ S = S ∘ H⃗, then the magnetic field H⃗ is called symmetric. If H⃗ ∘ S = −S ∘ H⃗, then the magnetic field H⃗ is called antisymmetric.

4.1. Theorem. Let D, J⃗, H⃗ the mathematical entities from the formula of Biot-Savart-Laplace, and S the symmetry of R^{3} with respect to the origin. Suppose S(D) = D.

1) If J⃗ ∘ S = S ∘ J⃗, then H⃗ ∘ S = −S ∘ H⃗.

2) If J⃗ ∘ S = −S ∘ J⃗, then H⃗ ∘ S = S ∘ H⃗. Proof. 1) We take into account the change of variables in the triple integral and the action of an orthogonal transformation upon the vector product. Since S(x, y, z) = (−x, −y, −z), succesively we have

H⃗∘S(M) = ∫_{S(D)} \frac{J⃗ ∘ S(P) × S⃗(P)S(M)}{S(P)S(M)^{3}}dv_{S(P)} = ∫_{D} \frac{S ∘ J⃗(P) × S ∘ P⃗ M}{P M^{3}} dv_{P} =

= (sign S)S ∘ ∫_{D} \frac{J⃗(P) × P⃗ M}{P M^{3}} dv_{P} = −S ∘ H⃗(M).

4.2. Theorem. Let D, J⃗, H⃗ the mathematical entities from the formula of Biot-Savart-Laplace, and S be the symmetry of R^{3} with respect to Oz. Suppose S(D) = D.

1) If J⃗ ∘ S = S ∘ J⃗ ,then H⃗ ∘ S = S ∘ H⃗.

2) If J⃗ ∘ S = −S ∘ J⃗ ,then H⃗ ∘ S = −S ∘ H⃗. Proof.1) The symmetry with respect to Oz is S(x, y, z) = (−x, −y, z). It follows

H⃗ ∘ S(M) = ∫_{S(D)} \frac{J⃗ ∘ S(P) × S ∘ P⃗ M}{S(P)S(M)^{3}} dv_{S(P)} = ∫_{D} \frac{S ∘ J⃗(P) × S ∘ P⃗ M}{P M^{3}} dv_{P} =

= ∫_{D}(signS)S ∘ \frac{J⃗(P) × P⃗ M}{P M^{3}} dv_{P} = S ∘ ∫_{D} \frac{J⃗(P) × P⃗ M}{P M^{3}} dv_{P} = S ∘ H⃗(M).

Remark. The theory in this paragraph can be extended to any isometry. Aknowledgements. Both authors would like to thank Profs. V. Balan, Gh. Frăţiloiu, Fl. Hănţilă, C. Mocanu from Politehnica University of Bucharest and E. Petrişor from Technical University of Timişoara, and the referees of the Nonlinearity, for suggestions on the various versions of the manuscript.

A version of this paper was presented at the Workshop on Global Annalysis, Differential Geometry and Lie Algebras, Aristotle University of Thessaloniki, Dec. 16–18, 1993.

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University Politehnica of Bucharest Department of Mathematics I Splaiul Independentei 313 77206, Bucharest, Romania e–mail:udriste@mathem.pub.ro Fax:401/411.53.65