Fractal Magnetic Dynamics around a Koch Fractal Electric ... - wseas

8th WSEAS International Conference on SYSTEMS THEORY and SCIENTIFIC COMPUTATION (ISTASC’08) Rhodes, Greece, August 20-22, 2008

Fractal Magnetic Dynamics around a Koch Fractal Electric Circuit CONSTANTIN UDRISTE CRISTIAN GHIU University Politehnica of Bucharest University Politehnica of Bucharest Faculty of Applied Sciences Faculty of Applied Sciences Department of Mathematics Department of Mathematics Splaiul Independentei 313 Splaiul Independentei 313 ROMANIA ROMANIA [email protected] Abstract: This is the first paper who presents original functional equations regarding the prefractal or fractal magnetic dynamics around a prefractal respectively fractal electric circuit of Koch type. These equations reflects a tensorial invariance with respect four geometric transformations. The main results refer to: (1) recurrence formulas for the prefractal magnetic fields and prefractal magnetic vector potentials; (2) the fractal magnetic field and the fractal magnetic vector potential obtained as fractal limits. Maple simulations regarding the field lines and constant level sets of magnetic energy are now in our attention. Due to the novelties, we develope a systematic language with words and sentences and grammar. Key–Words: electromagnetic theory, Koch fractal electric circuit, Koch fractal magnetic field, magnetic lines, magnetic energy, functional equations reflecting fractality.

1 Properties of Biot-Savart-Laplace magnetic field

Proposition 1.1. Let ϕ : R3 → R3 , (ϕ(x))T = αRxT + v, x = (x1 , x2 , x3 ), v ∈ R3 , α ∈ R \ {0},

We will focus on the properties of vector fields which are particularly important in physics and multidisciplinary applications, namely, the magnetic fields around electrical circuits [2]-[6], [7]-[9], [11]-[34]. The mathematical definition of these fields involves a smooth curve γ : [a, b] → R3 , kγ(t)k ˙ = 1, which represents the electric wire, and a constant J which stands for the curent intensity. More precisely, the Biot-Savart-Laplace magnetic field is defined by a path dependent curvilinear integral Z ¡ ¢ µ0 J dγ(t) × (x − γ(t)) , x ∈ R3 \ γ, Fγ x = 4π γ kx − γ(t)k3

where R ∈ M3 (R) is an orthogonal matrix and ϕ∗ be the differential of the map ϕ. Denote by Fγ the magnetic field generated by the electric circuit γ, by Aγ the vector potential of this field, and ε(ϕ) = det(R) . The relations ´ ¡ ¢ (sgn α)ε(ϕ) ³ ϕ F (x) , Fϕ◦γ ϕ(x) = ∗ γ α2 ³ ´ ¡ ¢ 1 ϕ∗ Aγ (x) Aϕ◦γ ϕ(x) = | α| or equivalently ´ (sgn α)ε(ϕ) ³ −1 ϕ F (ϕ (x)) , Fϕ◦γ (x) = ∗ γ α2

where µ0 is the magnetic permeability constant. When γ is a closed curve or a = −∞, b = ∞ we have a true magnetic vector field. Otherwise we have a fictive magnetic vector field. The vector potential Aγ of the magnetic field Fγ is Z ¡ ¢ µ0 J dγ(t) , x ∈ R3 \ γ. Aγ x = 4π γ kx − γ(t)k

³ ´ ¡ ¢ 1 ϕ∗ Aγ (ϕ−1 (x)) Aϕ◦γ x = | α| hold true. Proof. The Biot-Savart-Laplace magnetic field on the exterior of the electric circuit γ : [a, b] → R3 is defined by Z ¡ ¢ dγ(t) × (x − γ(t)) Fγ x = kx − γ(t)k3 γ

To simplify, we accept µ0 J = 4π. Also we remark that Aγ and Fγ are C ∞ vector fields.

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Z

b

= a

γ(t) ˙ × (x − γ(t)) dt. kx − γ(t)k3

where X is an arbitrary vector field. We denote   l1 R = row  l2  . l3

The action of the function ϕ on the vector field Fγ produces (in the matrix language) ³ ¡ ¢´| Fϕ◦γ ϕ(x)

Let ¡ ¢ Y (x) = ϕ∗ ◦ X ◦ ϕ−1 (x) = Y1 (x), Y2 (x), Y3 (x) ,

¡ ¢| ¡ ¢| d(ϕ ◦ γ)(t) × ϕ(x) − ϕ(γ(t)) ¡ ¢| = k ϕ(x) − ϕ(γ(t)) k3 ϕ◦γ ³ ´ ³ ´ | × αR(x − γ(t))| Z b αR(γ(t)) ˙ = dt. kαR(x − γ(t))| k3 a ¡ ¢ On the other hand, if M = col c1 ; c2 ; c3 ∈ M3 (R) and u1 , u2 ∈ M3,1 (R), then ¡ ¢¡ M u1 ×M u2 = col c2 ×c3 ; c3 ×c1 ; c1 ×c2 · u1 ×u2 ). Z

where ´ ³x − v 1 1 | x2 − v2 | x3 − v3 | l1 + l2 + l3 α α α ´ ³x − v 1 1 | x2 − v2 | x3 − v3 | l1 + l2 + l3 Y2 (x) = αl2 X α α α ´ ³x − v 1 1 | x2 − v2 | x3 − v3 | l1 + l2 + l3 . Y3 (x) = αl3 X α α α On the other hand, the vector field ¯ ¯ ¯ i j k ¯¯ ¯ ¯ ∂ ∂ ∂ ¯¯ rot Y = ¯¯ ∂x2 ∂x3 ¯¯ ¯ ∂x1 ¯Y1 (x) Y2 (x) Y3 (x)¯ Y1 (x) = αl1 X

Particularly, if M is the orthogonal matrix R, we have either det(R) = 1 when c2 × c3 = c1 , c3 × c1 = c2 , c1 × c2 = c3 , or

is reprezented by the matrix ¡ ¢ ¡ ¢   l3 JX ¡ϕ−1 (x)¢l2| − l2 JX ¡ϕ−1 (x)¢l3| (rot Y )| =  l1 JX ¡ϕ−1 (x)¢l3| − l3 JX ¡ϕ−1 (x)¢l1|  l2 JX ϕ−1 (x) l1| − l1 JX ϕ−1 (x) l2|

det(R) = −1 when c2 × c3 = −c1 , c3 × c1 = −c2 , c1 × c2 = −c3 . It follows ¢ ¡ col c2 × c3 ; c3 × c1 ; c1 × c2 = det(R)R and

¡ ¡ ¢¢| ¡ ¢ l3 JX ¡ϕ−1 (x)¢l2| − l3 ¡JX ¡ϕ−1 (x)¢¢ l2| | =  l1 JX ¡ϕ−1 (x)¢l3| − l3 ¡JX ¡ϕ−1 (x)¢¢ l1| | l2 JX ϕ−1 (x) l1| − l1 JX ϕ−1 (x) l2|  ³ ¡ ¢ ¡ ¡ ¢¢| ´ | l3 JX ϕ−1 (x) − JX ϕ−1 (x) l  ³ ¡ ´2 ¢ ¡ ¡ ¢¢ | |  =  l1 JX ϕ−1 (x) − JX ϕ−1 (x) l  ³ ¡ ´3 ¢ ¡ ¡ ¢¢ | | l2 JX ϕ−1 (x) − JX ϕ−1 (x) l1 ¡ ¢   hl2 × l3 , ¡rot X ¢ ◦ ϕ−1 (x)i =  hl3 × l1 , ¡rot X ¢ ◦ ϕ−1 (x)i  hl1 × l2 , rot X ◦ ϕ−1 (x)i   l2 × l3 ³¡ ´| ¢ =  l3 × l1  rot X ◦ ϕ−1 (x) l1 × l2   l1 ³¡ ´| ¢ = (det R)  l2  rot X ◦ ϕ−1 (x) l3 ³¡ ´| ¢ (det R) αR rot X ◦ ϕ−1 (x) , = α 

¡ Ru1 × Ru2 = det(R)R u1 × u2 ).

Therefore

³ ¡ ¢´| Fϕ◦γ ϕ(x) ¡ ¢ Z b | × (x − γ(t))| det(R) · R (γ(t)) ˙ = dt |α| · kR(x − γ(t))| k3 a ³ Z b (γ(t)) ´ | × (x − γ(t))| ˙ det(R) ·R dt = |α| k(x − γ(t))| k3 a ¡ ³ Z dγ(t)¢| × ¡x − γ(t)¢| ´ det(R) ¡ ¢| ·R = |α| k x − γ(t) k3 γ ´´| ³ ´| ε(ϕ) ³ ³ det(R) · αR Fγ (x) = ϕ∗ Fγ (x) , = α|α| | α|

where ε(ϕ) = det(R). To prove the second relation, we use the formula ¡ ¢ ε(ϕ) ϕ∗ ◦ (rot X) ◦ ϕ−1 , rot ϕ∗ ◦ X ◦ ϕ−1 = α ISSN: 1790-2769

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      


π The third curve: we rotate γn with − , then we 3 ´ ³ 1 √3 , , 0 , and the retranslate with the vector 4 4 sulted curve is scaled down by the factor 1/3. If we concentrate these actions into the geometric transformation ϕ3 : R3 → R3 ,   √   1  3 1 0 x1 12 √ 2 2 √ 1  (ϕ3 (x))T =  − 3 1 0   x2 + 123 , 2 2 3 x3 0 0 0 1

where JX is the Jacobian matrix of X. Therefore ¡ ¢ ¡ ¢ ε(ϕ) ϕ∗ ◦ rot X ◦ ϕ−1 . rot ϕ∗ ◦ X ◦ ϕ−1 = α The vector field X = Aγ satisfies the relations ¡ ¢ ε(ϕ) ϕ∗ ◦ (rot Aγ ) ◦ ϕ−1 rot ϕ∗ ◦ Aγ ◦ ϕ−1 = α ε(ϕ) ϕ∗ ◦ Fγ ◦ ϕ−1 = |α|Fϕ◦γ , α ¢ 1 ¡ ϕ∗ ◦ Aγ ◦ ϕ−1 is a vector whence it follows that |α| potential for Fϕ◦γ . =

2

x = (x1 , x2 , x3 ), the third curve is ϕ3 (γn ). The fourth curve: we translate γn with the vector (1, 0, 0), and the resulted curve is scaled down by the factor 1/3. These operations are realized by the geometric transformation

Koch curve in R3

Let us start with a simple construction of the Koch curve [1], [7], [10]. This is a planar curve ( in R2 ), but we describe it as a curve in R3 . We begin with a segment γ0 which joins the ´ ³1 ´ ³ 1 , 0, 0 , parametrized points − , 0, 0 and 2 h2 1 1 i . The initial by γ0 (t) = (t, 0, 0), t ∈ − , 2 2 object γ0 is also called initiator. The prefractal γn+1 is the union of four curves obtained by the following geometric transformations upon the prefractal γn . The first curve: we translate γn with the vector (−1, 0, 0) and the resulted curve is scaled down by the factor 1/3. Since these operations are realized by the geometric transformation

ϕ4 : R3 → R3 , ³1 ´ 1 , 0, 0 , x = (x1 , x2 , x3 ). ϕ4 (x) = x + 3 3 The fourth curve is ϕ4 (γn ). We obtain a sequence of prefractal curves γ0 , γ1 , ..., γn , ... of Koch type and the self-similarity is built into the construction process. The fractal character is represented by the geometric transformations ϕ1 , ϕ2 , ϕ3 , ϕ4 and their successive compositions. It is well-known that the Koch sequence of prefractals γn is convergent to the Koch fractal (curve γ) which admits a continuous parametrization. The Koch curve γ = lim γn is the object which one obtains if one repeats the construction steps infinitely often [7].

ϕ1 : R3 → R3 , ϕ1 (x) =

³ 1 ´ 1 x + − , 0, 0 , x = (x1 , x2 , x3 ), 3 3

3

the first curve is ϕ1 (γn ). π The second curve: we rotate γn with , then we 3 translate√with the vector ´ ³ 1 3 , 0 and the resulted curve is scaled down − , 4 4 by the factor 1/3. Mathematically, we use the geometric transformation ϕ2 : R3 → R3 ,   √   1  3 1 −√12 − 0 x 1 2 2 1  √3  T 3   1 (ϕ2 (x)) =  0  x2 + 12 , 2 2 3 x3 0 0 0 1 x = (x1 , x2 , x3 ) and we build the curve ϕ2 (γn ). ISSN: 1790-2769

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Fractal magnetic field around a Koch fractal electric circuit

We consider the Koch prefractal γn as an electric circuit and we denote by Fγn the associated prefractal Biot-Savart-Laplace magnetic field. This magnetic field is related to the geometric transformations ϕ1 , ϕ2 , ϕ3 , ϕ4 by the recurrence formula from the next Theorem 2.1. 1) The prefractal Biot-SavartLaplace magnetic field Fγn , arround the electrical circuit γn of Koch type, satisfies the recurrence relation ³ ´ Fγn+1 (x) = 9ϕ1∗ Fγn (ϕ−1 (x)) 1 ³ ´ ³ ´ −1 +9ϕ2∗ Fγn (ϕ−1 (x)) + 9ϕ F (ϕ (x)) 3∗ γ n 2 3 ISBN: 978-960-6766-96-1


³ ´ +9ϕ4∗ Fγn (ϕ−1 (x)) 4

³ ´ 1 We use the Proposition 1.1 α = , ε(ϕ) = 1 . We 3 obtain the recurrence formula ³ ´ Fγn+1 (x) = 9ϕ1∗ Fγn (ϕ−1 (x)) 1

with Fγ0 (x1 , x2 , x3 ) = ρ(x1 , x2 , x3 )

(0, −x3 , x2 ) , x22 + x23

³ ´ ³ ´ −1 (x)) +9ϕ2∗ Fγn (ϕ−1 (x)) + 9ϕ F (ϕ 3∗ γn 2 3 ³ ´ +9ϕ4∗ Fγn (ϕ−1 4 (x)) ,

where −x1 + 21 ρ(x1 , x2 , x3 ) = q (x1 − 21 )2 + x22 + x23

which reflects an invariance with respect to the geometric transformations ϕ1 , ϕ2 , ϕ3 , ϕ4 . We proceed analogously for the statement 2), regarding the prefractal vector potential Aγn . The initial terms Fγ0 and Aγ0 are the magnetic field respectively its potential vector field for an initiator (segment). These fields are determined in the paper [18].

x1 + 12 . +q (x1 + 21 )2 + x22 + x23 2) The prefractal vector potential Aγn of the prefractal magnetic field Fγn satisfies the recurrence relation Aγn+1 (x) ´ ³ ´ −1 = 3ϕ1∗ Aγn (ϕ−1 1 (x)) + 3ϕ2∗ Aγn (ϕ2 (x)) ³ ´ ³ ´ −1 +3ϕ3∗ Aγn (ϕ−1 (x)) + 3ϕ A (ϕ (x)) 4∗ γn 3 4 ³

fγn

For the moment, we accept that the associated Koch prefractal magnetic field sequence Fγn is convergent to the Koch fractal magnetic field Fγ . The fractality of the magnetic field Fγ is reflected by an invariance formula with respect to the geometric transformations by ϕ1 , ϕ2 , ϕ3 , ϕ4 , in the sense of following Theorem 2.2. The Koch fractal magnetic field Fγ is a solution of the vector functional equation ³ ´ ³ ´ −1 −1 Fγ (x) = 9ϕ1∗ Fγ (ϕ1 (x)) + 9ϕ2∗ Fγ (ϕ2 (x))

with Aγ0 (x1 , x2 , x3 ) q ! Ã (x1 − 21 )2 + x22 + x23 − x1 + 21 ·(1, 0, 0) . = ln q (x1 + 21 )2 + x22 + x23 − x1 − 21 Proof. 1) Since γn+1 = ϕ1 (γn ) ∪ ϕ2 (γn ) ∪ ϕ3 (γn ) ∪ ϕ4 (γn ), we can write Z dγn+1 (t) × (x − γn+1 (t)) Fγn+1 (x) = kx − γn+1 (t)k3 γn+1 ¡ ¢ Z d(ϕ1 ◦ γn )(t) × x − (ϕ1 ◦ γn )(t) = kx − (ϕ1 ◦ γn )(t)k3 ϕ1 (γn ) ¡ ¢ Z d(ϕ2 ◦ γn )(t) × x − (ϕ2 ◦ γn )(t) + kx − (ϕ2 ◦ γn )(t)k3 ϕ2 (γn ) ¡ ¢ Z d(ϕ3 ◦ γn )(t) × x − (ϕ3 ◦ γn )(t) + kx − (ϕ3 ◦ γn )(t)k3 ϕ3 (γn ) ¡ ¢ Z d(ϕ4 ◦ γn )(t) × x − (ϕ4 ◦ γn )(t) . + kx − (ϕ4 ◦ γn )(t)k3 ϕ4 (γn )

³ ´ ³ ´ −1 +9ϕ3∗ Fγ (ϕ−1 3 (x)) + 9ϕ4∗ Fγ (ϕ4 (x)) . The fractal vector potential Aγ of the Koch fractal magnetic field is a solution of the vector functional equation ³ ´ ³ ´ −1 Aγ (x) = 3ϕ1∗ Aγ (ϕ−1 (x)) + 3ϕ A (ϕ (x)) 2∗ γ 1 2 ³ ´ ³ ´ −1 +3ϕ3∗ Aγ (ϕ−1 (x)) + 3ϕ A (ϕ (x)) . 4∗ γ 3 4

4

Applications in science and technology

There are many branches of Science and Technology in which the fractal theory plays a central role and faces fascinating challenges, but here is the first time when a fractal vector field is described by functional equations.

It follows Fγn+1 (x) = Fϕ1 ◦γn (x) + Fϕ2 ◦γn (x) +Fϕ3 ◦γn (x) + Fϕ4 ◦γn (x). ISSN: 1790-2769

The energy of the magnetic field Fγn is 1 = kFγn k2 . 2

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The fractal magnetic field described in this paper is useful in multidisciplinary reasearch (fractal theory, differential geometry, electro-magnetic engineering, biomedical engineering, magnetobiology, etc). Particularly, we are interested to build fractal antenna, fractal magnetic traps, etc or to positively manipulate plant germination and growth via a fractal magnetic field.

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Acknowledgements: Partially supported by Grant CNCSIS 86/2008 and by 15-th ItalianRomanian Executive Programme of S&T Cooperation for 2006-2008, University Politehnica of Bucharest.

[13] A. Udris¸te, C. Udris¸te, Properties of the Magnetic Lines and Surfaces, Proceedings of the 23rd Conference on Geometry and Topology, pp. 203-208.

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