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Polynomial function models

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			In [[theoretical physics]], the '''Born–Infeld model''' is a particular example of what is usually known as a [[nonlinear electrodynamics]]. It was historically introduced in the 30s to remove the divergence of the electron's self-energy in classical electrodynamics by introducing an upper bound of the electric field at the origin. The Born-Infeld electrodynamics possesses a whole series of physically interesting properties:

			First of all the total energy of the electromagnetic field is finite and the electric field is regular everywhere.

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			Finally (and more technically) Born-Infeld theory can be seen as a covariant generalization of [[Mie scattering\|Mie's theory]], and very close to [[Einstein]]'s idea of introducing a nonsymmetric metric tensor with the symmetric part corresponding to the usual metric tensor and the antisymmetric to the electromagnetic field tensor.

			During the 1990s there was a revival of interest on Born-Infeld theory and its nonabelian extensions as they were found in some limits of [[string theory]].

			The model is named after physicists [[Max Born]] and [[Leopold Infeld]] who first proposed it.

			== Equations ==

			We will use the [[theory of relativity\|relativistic]] notation here as this theory is fully relativistic.

			The [[Lagrangian density]] is

			:<math>\mathcal{L}=-b^2\sqrt{-\det\left(\eta+{F\over b}\right)}+b^2</math>

			where η is the [[Minkowski metric]], ''F'' is the [[Faraday tensor]] and both are treated as square matrices so that we can take the [[determinant]] of their sum; ''b'' is a scale parameter. The maximal possible value of the electric field in this theory is ''b'', and the [[self-energy]] of point charges is finite. For electric and magnetic fields much smaller than ''b'', the theory reduces to [[Maxwell's equations\|Maxwell electrodynamics]].

			In 4-dimensional spacetime the Lagrangian can be written as

			:<math>\mathcal{L}=-b^2\sqrt{1-\frac{E^2-B^2}{b^2}-\frac{(\vec{E}\cdot\vec{B})^2}{b^4}}+b^2</math>

			where ''E'' is the electric field, and ''B'' is the magnetic field.

			In [[string theory]], gauge fields on a [[D-brane]] (that arise from attached open strings) are described by the same type of Lagrangian:

			:<math>\mathcal{L}=-T\sqrt{-\det\left(\eta+2\pi\alpha'F\right)}</math>

			where ''T'' is the tension of the D-brane.

			==Partial differential equation==
			<math>\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0</math>

			There are many travelling wave exact solutions of Born Infeld pde<ref>Yuanxi Xie and Jiashi Tang,New Explicit Exact Solutions of the Born Infeld Equation,2005</ref>

			<math>u[1] := 2atan(\sqrt((a0+b1)/(a0-b1))tan((1/2)\sqrt(a0^2-b1^2)\eta))+(1/2)*\pi</math>

			<math>u[2] := 2atan(\sqrt((a0+b1)/(a0-b1))cot((1/2)\sqrt(a0^2-b1^2)\eta))+(1/2)*\pi</math>

			<math>u[3] := 2atan(\sqrt(a1^2+b1^2)tan((1/2)\sqrt(a1^2+b1^2)\eta)/a1+b1/a1)</math>

			<math>u[4] := 2atan(\sqrt(a1^2+b1^2)cot((1/2)\sqrt(a1^2+b1^2)\eta)/a1+b1/a1)</math>

			<math>u[5] := 2atan(\sqrt(a1^2+b1^2)(coth(\sqrt(a1^2+b1^2)\eta)+csc(\sqrt(a1^2+b1^2)\eta))/a1+b1/a1)</math>

			<math>u[6] := 2atan(\sqrt(a1^2+b1^2)(cosh(\sqrt(a1^2+b1^2)\eta)+1)/(a1sinh(\sqrt(a1^2+b1^2)*\eta))+b1/a1)</math>

			<math>u[7] := 2atan(\sqrt(a1^2+b1^2)sinh(\sqrt(a1^2+b1^2)\eta)/(a1(cosh(\sqrt(a1^2+b1^2)*\eta)+1))+b1/a1)</math>

			<math>u[8] := 2atan(\sqrt((a0+b1)/(b1-a0))cot(\sqrt(b1^2-a0^2)\eta+sec(h\sqrt(b1^2-a0^2)\eta)))+(1/2)pi</math>

			<math>u[9] := 2atan(\sqrt((a0+b1)/(b1-a0))sinh(\sqrt(b1^2-a0^2)\eta)/(cosh(\sqrt(b1^2-a0^2)\eta)+1))</math>

			{{Gallery
			\|width=200
			\|height=172
			\|align=center
			\|File:Born Infeld equation animation3.gif\|Born Infeld equation animation3
			\|File:Born Infeld equation animation5.gif\|Born Infeld equation animation5
			\|File:Born Infeld equation animation6.gif\|Born Infeld equation animation6
			\|File:Born Infeld equation animation7.gif\|Born Infeld equation animation7
			\|File:Born Infeld equation animation9.gif\|Born Infeld equation animation9
			\|File:Born Infeld equation animation10.gif\|Born Infeld equation animation10
			}}

			==References==
			<references/>
			*[http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=2&index1=3228428 Born-Infeld theory on arxiv.org]

			{{DEFAULTSORT:Born-Infeld model}}
			[[Category:Quantum field theory]]
			[[Category:String theory]]


			{{physics-stub}}
			{{string-theory-stub}}

			[[ko:보른-인펠트 이론]]
			[[pl:Teoria Borna-Infelda]]

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Polynomial and rational function modeling - Revision history

66.57.167.91: /* Polynomial function models */

en>BattyBot: /* Historical */fixed CS1 errors: dates & General fixes using AWB (9816)

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