Fractal Magnetic Dynamics around a Koch Fractal Electric Circuit

Transcription

1 8th WSEAS International Conference on SYSTEMS THEORY and SCIENTIFIC COMPUTATION ISTASC 08 Fractal Magnetic Dynamics around a Koch Fractal Electric Circuit CONSTANTIN UDRISTE University Politehnica of Bucharest Faculty of Applied Sciences Department of Mathematics Splaiul Independentei 1 ROMANIA udriste@mathem.pub.ro CRISTIAN GHIU University Politehnica of Bucharest Faculty of Applied Sciences Department of Mathematics Splaiul Independentei 1 ROMANIA 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; 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 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 []-[6], [7]-[9], [11]-[4]. The mathematical definition of these fields involves a smooth curve : [a, b] R, t 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 F x µ 0 J 4π dt x t x t, x R \, where µ 0 is the magnetic permeability constant. When 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 A of the magnetic field F is µ 0 J dt A x 4π x t, x R \. To simplify, we accept µ 0 J 4π. Also we remark that A and F are C vector fields. Proposition 1.1. Let ϕ : R R, ϕx T Rx T + v, x x 1, x, x, v R, R \ {0}, where R M R is an orthogonal matrix and ϕ be the differential of the map ϕ. Denote by F the magnetic field generated by the electric circuit, by A the vector potential of this field, and εϕ detr. The relations sgn εϕ F ϕ ϕx ϕ F x, or equivalently F ϕ x A ϕ ϕx 1 ϕ A x sgn εϕ ϕ F ϕ 1 x, 1 A ϕ x ϕ A ϕ 1 x hold true. Proof. The Biot-Savart-Laplace magnetic field on the exterior of the electric circuit : [a, b] R is defined by dt x t F x x t ISSN: ISBN:

2 8th WSEAS International Conference on SYSTEMS THEORY and SCIENTIFIC COMPUTATION ISTASC 08 b a t x t x t dt. The action of the function ϕ on the vector field F produces in the matrix language ϕ b a F ϕ ϕx dϕ t ϕx ϕt ϕx ϕt R t Rx t Rx t dt. On the other hand, if M col c 1 ; c ; c M R and u 1, u M,1 R, then Mu 1 Mu col c c ; c c 1 ; c 1 c u1 u. Particularly, if M is the orthogonal matrix R, we have either or detr 1 when c c c 1, c c 1 c, c 1 c c, detr 1 when c c c 1, c c 1 c, c 1 c c. It follows and col c c ; c c 1 ; c 1 c detrr Ru 1 Ru detrr u 1 u. Therefore F ϕ ϕx b a detr detr detr detr R t x t Rx t dt b t x t R a x t dt x t R R F x x t εϕ dt ϕ F x, where εϕ detr. To prove the second relation, we use the formula rot ϕ X ϕ 1 εϕ ϕ rot X ϕ 1, where X is an arbitrary vector field. We denote Let R row l 1 l l. Y x ϕ X ϕ 1 x Y 1 x, Y x, Y x, where x1 v 1 Y 1 x l 1 X x1 v 1 Y x l X x1 v 1 Y x l X l 1 + x v l 1 + x v l 1 +x v On the other hand, the vector field i j k rot Y x 1 x x Y 1 x Y x Y x is reprezented by the matrix rot Y l + x v l l + x v l l +x v l l J X ϕ 1 x l l J X ϕ 1 x l l 1 J X ϕ 1 x l l J X ϕ 1 x l 1 l J X ϕ 1 x l 1 l 1J X ϕ 1 x l l J X ϕ 1 x l l JX ϕ 1 x l l 1 J X ϕ 1 x l l JX ϕ 1 x l 1 l J X ϕ 1 x l 1 l 1 JX ϕ 1 x l l J X ϕ 1 x J X ϕ 1 x l l 1 J X ϕ 1 x J X ϕ 1 x l l J X ϕ 1 x J X ϕ 1 x l 1 l l, rot X ϕ 1 x l l 1, rot X ϕ 1 x l 1 l, rot X ϕ 1 x l l l l 1 rot X ϕ 1 x l 1 l l 1 det R l rot X ϕ 1 x l det R rot, R X ϕ x 1. ISSN: ISBN:

3 8th WSEAS International Conference on SYSTEMS THEORY and SCIENTIFIC COMPUTATION ISTASC 08 where J X is the Jacobian matrix of X. Therefore rot ϕ X ϕ 1 εϕ ϕ rot X ϕ 1. The vector field X A satisfies the relations rot ϕ A ϕ 1 εϕ ϕ rot A ϕ 1 εϕ ϕ F ϕ 1 F ϕ, whence it follows that potential for F ϕ. Koch curve in R 1 ϕ A ϕ 1 is a vector Let us start with a simple construction of the Koch curve [1], [7], [10]. This is a planar curve in R, but we describe it as a curve in R. We begin with a segment 0 which joins the points 1 1, 0, 0 and, 0, 0, parametrized [ by 0 t t, 0, 0, t 1, 1 ]. The initial object 0 is also called initiator. The prefractal n+1 is the union of four curves obtained by the following geometric transformations upon the prefractal n. The first curve: we translate n with the vector 1, 0, 0 and the resulted curve is scaled down by the factor 1/. Since these operations are realized by the geometric transformation ϕ 1 : R R, ϕ 1 x 1 x + 1, 0, 0, x x 1, x, x, the first curve is ϕ 1 n. The second curve: we rotate n with π, then we translate with the vector 1 4, 4, 0 and the resulted curve is scaled down by the factor 1/. Mathematically, we use the geometric transformation ϕ : R R, ϕ x T x x 1, x, x and we build the curve ϕ n. x 1 x + x , The third curve: we rotate n with π, then we 1 translate with the vector 4, 4, 0, and the resulted curve is scaled down by the factor 1/. If we concentrate these actions into the geometric transformation ϕ : R R, ϕ x T x x +, x 0 x x 1, x, x, the third curve is ϕ n. The fourth curve: we translate n with the vector 1, 0, 0, and the resulted curve is scaled down by the factor 1/. These operations are realized by the geometric transformation ϕ 4 : R R, ϕ 4 x 1 x + 1, 0, 0, x x 1, x, x. The fourth curve is ϕ 4 n. We obtain a sequence of prefractal curves 0, 1,..., n,... of Koch type and the self-similarity is built into the construction process. The fractal character is represented by the geometric transformations ϕ 1, ϕ, ϕ, ϕ 4 and their successive compositions. It is well-known that the Koch sequence of prefractals n is convergent to the Koch fractal curve which admits a continuous parametrization. The Koch curve lim n is the object which one obtains if one repeats the construction steps infinitely often [7]. Fractal magnetic field around a Koch fractal electric circuit We consider the Koch prefractal n as an electric circuit and we denote by F n the associated prefractal Biot-Savart-Laplace magnetic field. This magnetic field is related to the geometric transformations ϕ 1, ϕ, ϕ, ϕ 4 by the recurrence formula from the next Theorem.1. 1 The prefractal Biot-Savart- Laplace magnetic field F n, arround the electrical circuit n of Koch type, satisfies the recurrence relation F n+1 x 9ϕ 1 F n ϕ 1 1 x +9ϕ F n ϕ 1 x + 9ϕ F n ϕ 1 x ISSN: ISBN:

4 8th WSEAS International Conference on SYSTEMS THEORY and SCIENTIFIC COMPUTATION ISTASC 08 with where +9ϕ 4 F n ϕ 1 4 x F 0 x 1, x, x ρx 1, x, x 0, x, x x +, x x ρx 1, x, x x x + x x x x + x The prefractal vector potential A n of the prefractal magnetic field F n satisfies the recurrence relation with A n+1 x ϕ 1 A n ϕ 1 1 x + ϕ A n ϕ 1 x +ϕ A n ϕ 1 x + ϕ 4 A n ϕ 1 4 x A 0 x 1, x, x x 1 1 ln + x + x x , 0, 0. x x + x x 1 1 Proof. 1 Since n+1 ϕ 1 n ϕ n ϕ n ϕ 4 n, we can write d n+1 t x n+1 t F n+1 x n+1 x n+1 t ϕ 1 n ϕ n ϕ n ϕ 4 n It follows dϕ 1 n t x ϕ 1 n t x ϕ 1 n t dϕ n t x ϕ n t x ϕ n t dϕ n t x ϕ n t x ϕ n t dϕ 4 n t x ϕ 4 n t x ϕ 4 n t. F n+1 x F ϕ1 n x + F ϕ n x +F ϕ n x + F ϕ4 n x. We use the Proposition 1.1 1, εϕ 1. We obtain the recurrence formula F n+1 x 9ϕ 1 F n ϕ 1 1 x +9ϕ F n ϕ 1 x + 9ϕ F n ϕ 1 x +9ϕ 4 F n ϕ 1 4 x, which reflects an invariance with respect to the geometric transformations ϕ 1, ϕ, ϕ, ϕ 4. We proceed analogously for the statement, regarding the prefractal vector potential A n. The initial terms F 0 and A 0 are the magnetic field respectively its potential vector field for an initiator segment. These fields are determined in the paper [18]. The energy of the magnetic field F n is f n 1 F n. For the moment, we accept that the associated Koch prefractal magnetic field sequence F n is convergent to the Koch fractal magnetic field F. The fractality of the magnetic field F is reflected by an invariance formula with respect to the geometric transformations by ϕ 1, ϕ, ϕ, ϕ 4, in the sense of following Theorem.. The Koch fractal magnetic field F is a solution of the vector functional equation F x 9ϕ 1 F ϕ 1 1 x + 9ϕ F ϕ 1 x +9ϕ F ϕ 1 x + 9ϕ 4 F ϕ 1 4 x. The fractal vector potential A of the Koch fractal magnetic field is a solution of the vector functional equation A x ϕ 1 A ϕ 1 1 x + ϕ A ϕ 1 x +ϕ A ϕ 1 x + ϕ 4 A ϕ 1 4 x. 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. ISSN: ISBN:

5 8th WSEAS International Conference on SYSTEMS THEORY and SCIENTIFIC COMPUTATION ISTASC 08 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. Acknowledgements: Partially supported by Grant CNCSIS 86/008 and by 15-th Italian- Romanian Executive Programme of S&T Cooperation for , University Politehnica of Bucharest. References: [1] J. N. Cederberg, A Course in Modern Geometries, Springer-Verlag New York, 001. [] V. Ciancio, C. Udrişte, Ioffe-Stefănescu Magnetic Trap, Rev. Roum. Sci.Techn.-Electrotechn. et Energ., 49, 004, [] C. Dumitrescu, Approximations des Lignes Magnétiques Utilisant la Transformation Lie, in. Gr. Tsagas Ed., Proc. of the Workshop on Global Analysis, Differential Geometry and Lie Algebras, 1996, BSG Proceedings, 1-, Geometry Balkan Press, [4] J. D. Jackson, Electrodinamică Clasică, Editura Tehnică, Bucureşti, [5] A. Lupaşcu, C. Udrişte, C. Ghiu, New Results on Ioffe-Ştefănescu Magnetic Trap, Proceedings of the -rd International Colloquim of Mathematics in Engineering and Numerical Physics MENP-, BSG Proceedings 1, , Geometry Balkan Press, Bucharest, 005. [6] D. Opriş, C. Udrişte, Pole Shifts Explained by Dirac Delay in a Stefănescu Magnetic flow, Analele Universităţii Bucureşti, LIII, 1 004, [7] H-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, New Frontiers of Science, Springer- Verlag New York, 199. [8] E. Petrişor, A Study of the Rimmer Bifurcation of Symmetric Fixed Points of Reversible Diffeomorphisms, Balkan Journal of Geometry and Its Applications,, 1998, [9] E. Petrişor, Heteroclinic Connections in the Dynamics of a Reversible Magnetic-Type Vector Field, Physica D, , [10] H. Sagan, Space-Filling Curves, Springer- Verlag New York, [11] S. S. Ştefănescu, M. Nabighian, Magnetic Field Lines of Two Equal Rectilinear Currents, St. Cerc. Fiz., 1960, 56-58; Rev. Roum. Geol. Geophys. et. Geogr.-Geophys., 1, 1987, 95-11; Asupra liniilor de câmp magnetic ale emiţătorului AB, Probleme de Geofizică, 1, 1961, [1] S. S. Ştefănescu, C. Udrişte, Magnetic Field Lines Around Filiform Electrical Circuits of Right Angle Type, Sci. Bull., U.P.B., Series A: Appl. Math. Phys. 55, 1-199, -18. [1] A. Udrişte, C. Udrişte, Properties of the Magnetic Lines and Surfaces, Proceedings of the rd Conference on Geometry and Topology, pp [14] A. Udrişte, C. Udrişte, Magnetic Dynamics Around Electrical Circuits, in. Gr. Tsagas Ed., Global Analysis, Differential Geometry and Lie Algebras, 1995, BSG Proceedings 1, 109-1, Geometry Balkan Press, [15] A. Udrişte, C. Udrişte, Dynamics Induced by a Magnetic Field, Ed. J. Szenthe, New Developments in Differential Geometry, Budapest 1996, pp , Kluwer Academic Publishers. [16] C. Udrişte, Geometric Dynamics, Southeast Asian Bulletin of Mathematics, Springer-Verlag, 4 000, 1-. [17] C. Udrişte, Geometric Dynamics, Kluwer Academic Publishers, Dordrecht/Boston/London, Mathematics and Its Applications, 51, 000. [18] C. Udrişte, M. Postolache Editors, Magnetic Fields Generated by Piecewice Rectilinear Circuits, Geometry Balkan Press, Bucharest, [19] C. Udrişte, M. Postolache, Atlas of Magnetic Geometric Dynamics, Geometry Balkan Press, Bucharest, 001. [0] C. Udrişte, A. Udrişte, Electromagnetic Dynamical Systems, Balkan Journal of Geometry and Its Applications,, 1998, [1] C. Udrişte, S. Udrişte, Biot-Savart-Laplace Dynamical Systems, Balkan Journal of Geometry and Its Applications,, 1998, [] C. Udrişte, V. Bălan, A. Udrişte, Magnetic Fields Generated by Piecewice Rectilinear Configurations, in. Gr. Tsagas Ed., Proc. of the Workshop on Global Analysis, Differential Geometry and Lie Algebras, 1996, BSG Proceedings, , Geometry Balkan Press, ISSN: ISBN:

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