Nonclassical Electromagnetic Dynamics

Transcription

1 Nonclasscal Electromagnetc Dynamcs CONSTANTIN UDRISTE Unversty Poltehnca of Bucharest Department of Mathematcs Splaul Independente Bucharest ROMANIA DOREL ZUGRAVESCU Insttute of Geodynamcs Dr. Gerota Bucharest ROMANIA FLORIN MUNTEANU Insttute of Geodynamcs Dr. Gerota Bucharest ROMANIA Abstract: Our paper s concerned wth effects of specal forces on the moton of partcles. 1 studes the sngletme geometrc dynamcs nduced by electromagnetc vector felds 1-forms and by the Eucldean structure of the space. 3 defnes the second-order forms or vectors. 4 5 descrbes a nonclasscal electrc agnetc dynamcs produced by an electrc agnetc second-order Lagrangan, va the extremals of the energy functonal. 6 generalze ths dynamcs for a general second-order Lagrangan, havng n mnd possble applcatons for dynamcal systems comng from Bomathematcs, Economcal Mathematcs, Industral Mathematcs, etc. Key Words: geometrc dynamcs, second-order vectors, second-order Lagrangan, nonclasscal dynamcs. 1 Sngle-tme geometrc dynamcs nduced by electromagnetc vector felds Let U be a doman of lnear homogeneous sotropc meda n the Remannan manfold M = R 3, δ. Maxwell s equatons coupled PDEs of frst order dv D = ρ, rot H = J + t D, t = tme dervatve operator dv B = 0, rot E = t B wth the consttutve equatons B = µh, D = εe, on R U, reflect the relatons between the electromagnetc felds: E [V/m] electrc feld strength H [A/m] magnetc feld strength J [A/m ] electrc current densty ε [As/V m] permtvty µ [V s/am] permeablty B [T ] = [V s/m ] magnetc nducton agnetc flux densty D [C/m ] = [As/m ] electrc dsplacement electrc flux densty Snce dv B = 0, the vector feld B s source free, hence may be expressed as rot of some vector potental A,.e., B = rot A. Then the electrc feld strength s E = grad V t A. It s well-known that the moton of a charged partcle n the electromagnetc felds s descrbed by the ODE system Lorentz World-Force Law m d x = e E + dx B, x = x 1, x, x 3 U R 3, m s the mass, and e s the charge of the partcle. Of course, these are Euler-Lagrange equatons produced by the Lorentz Lagrangan L 1 = 1 mδ dx + eδ dx Aj ev. The assocated Lorentz Hamltonan s H 1 = m δ dx + ev. A smlar mechancal moton was dscovered by Popescu [7], acceptng the exstence of a gravtovortex feld represented by the force G + dx Ω, G s the gravtatonal feld and Ω s a vortex determnng the gyroscopc part of the force. The papers [5], [8], [9], [1], [14], [15] confrm the pont of vew of Popescu va geometrc dynamcs. 1.1 Sngle-tme geometrc dynamcs produced by vector potental A To conserve the tradtonal formulas, we shall refer to feld lnes of the vector potental A usng dmensonal homogeneous relatons. These curves are 31

2 solutons of the ODE system m dx = ea. Ths ODE system and the Eucldean metrc produce the least squares Lagrangan L = 1 δ dx + ea dxj + eaj = 1 mdx + ea The Euler-Lagrange equatons assocated to L are m d x A j = e x A dx j x j + e f A m x e ta, f A = 1 δ A A j s the energy densty produced by A. Equvalently, t appears a sngle-tme geometrc dynamcs m d x = edx B + e m f A e t A, B = rot A, a moton n a gyroscopc force [1], [14], [15] or a B-vortex dynamcs. The assocated Hamltonan s H = 1 δ dx ea dxj + eaj = m δ dx e f A. Remark. Generally, the sngle-tme geometrc dynamcs produced by the vector potental A s dfferent from the classcal Lorentz World-Force Law because L ml 1 = 1 e δ A A j + mev and the force e m f A t A s not the electrc feld E = V t A. In other words the Lagrangans L 1 and L are not n the same equvalence class of Lagrangans. 1. Sngle-tme geometrc dynamcs produced by magnetc nducton B Snce we want to analyze the geometrc dynamcs usng unts of measure, the magnetc flow must be descrbed by m dx = λb,. 3 the unt measure for the constant λ s [kgm 3 /V s ]. The assocated least squares Lagrangan s L 3 = 1 δ dx λb dxj λbj = 1 mdx λb Ths gves the Euler-Lagrange equatons sngle-tme magnetc geometrc dynamcs m d x = λdx λ rot B + m f B + λ t B, f B = 1 δ B B j = 1 B s the magnetc energy densty. Ths s n fact a dynamcs under a gyroscopc force or n J-vortex. The assocated Hamltonan s H 3 = 1 δ dx λb dxj + λbj = m δ dx λ f B. 1.3 Sngle-tme geometrc dynamcs produced by electrc feld E The electrc flow s descrbed by m dx = λe, the unt measure of the constant λ s [kgm /V s]. It appears the least squares Lagrangan L 4 = 1 δ dx λe dxj λej = 1 mdx λe wth Euler-Lagrange equatons sngle-tme electrc geometrc dynamcs m d x = λdx λ rot E + m f E + λ t E, f E = 1 δ E E j = 1 E.,

3 s the electrc energy densty; here we have n fact a dynamcs n t B-vortex. The assocated Hamltonan s H 4 = 1 δ dx λe dxj + λej = m δ dx λ f E. Open problem. As s well-known, charged partcles n the magnetc feld of the earth spral from pole to pole. Smlar motons are also observed n laboratory plasmas and nferred for electrons n metal subjected to an external magnetc feld. Tll now, these motons justfed by the classcal Lorentz World-Force Law. Can we justfy such motons usng the geometrc dynamcs produced by sutable vector felds n the sense of present paper and the papers [5], [7]-[1], [14], [15]? Potentals assocated to electromagnetc forms Let U R 3 = M be a doman of lnear homogeneous sotropc meda. Whch mathematcal object shall we select to model the electromagnetc felds; vector felds or forms? It was shown [1] that, the magnetc nducton B, the electrc dsplacement D, and the electrc current densty J are all -forms; the magnetc feld H, and the electrc feld E are 1-forms; the electrc charge densty ρ s a 3-form. The operator d s the exteror dervatves and the operator t s the tme dervatve. In terms of dfferental forms, the Maxwell equatons on U R can be expressed as dd = ρ, dh = J + t D db = 0, de = t B. The consttutve relatons are D = ε E, B = µ H, the star operator s the Hodge operator, ε s the permtvty, and µ s the scalar permeablty. The local components E, = 1,, 3, of the 1- form E are called electrc potentals, and the local components H, = 1,, 3, of the 1-form H are called magnetc potentals. Snce the electrc feld E, and the magnetc feld H are 1-forms [1], n the Sectons 4-5 we combne our deas [8]-[16] wth the deas of Emery [3] and Foster [4], creatng a nonclasscal electrc or magnetc dynamcs. Fnally, we generalze the results to nonclasscal dynamcs nduced by a second-order Lagrangan Secton Potental assocated to electrc 1-form E Let us consder the functon V : R U R, t, x V t, x and the Pfaff equaton dv = E or the equvalent PDE system V x = E, = 1,, 3. Of course, the complete ntegrablty condtons requre de = 0 electrostatc feld, whch s not always satsfed. In any stuaton we can ntroduce the least squares Lagrangan L 9 = 1 V V δ x + E x j + E j = 1 dv + E. These produce the Euler-Lagrange equaton Posson equaton V = dv E, and consequently V must be the electrc potental. For a lnear sotropc materal, we have D = εe, wth ρ = dv D = εdv E. We get the potental equaton for a homogeneous materal ε = constant V = ρ ε. For the charge free space, we have ρ = 0, and then the potental V satsfes the Laplace equaton V = 0 harmonc functon. The Lagrangan L 9 produces the Hamltonan H 9 = 1 dv E, dv + E, and the momentum-energy tensor feld T j = V x j = V V x j x + E L L 9 δ V j x L 9 δj.. Potental assocated to magnetc 1-form H Now, we consder the Pfaff equaton dϕ = H or the PDE system ϕ x = H, = 1,, 3, ϕ : R U R, t, x ϕt, x. The complete ntegrablty condtons dh = 0 are satsfed only for partcular cases. If we buld the least squares Lagrangan L 10 = 1 δ ϕ x H ϕ x j H j

4 = 1 dϕ H, then we obtan the Euler-Lagrange equaton Laplace equaton and consequently ϕ must be the magnetc potental. The Lagrangan L 10 produces the Hamltonan H 10 = 1 dϕ H, dϕ + H..3 Potental assocated to 1-form potental A Snce db = 0, there exsts an 1-form potental A satsfyng B = da. Now let us consder the Pfaff equaton dψ = A or the equvalent PDEs system ψ x = A, = 1,, 3, ψ : R U R, t, x ψt, x. The complete ntegrablty condtons da = 0 are satsfed only for B = 0. If we buld the least squares Lagrangan L 11 = 1 ψ ψ δ x A x j A j = 1 dψ A, then we obtan the Euler-Lagrange equaton Laplace equaton ψ = 0. The Lagrangan L 11 gves the Hamltonan H 11 = 1 dψ A, dψ + A and the momentum-energy tensor feld T j = ψ ψ x j x A L 11 δj. Open problem. Fnd nterpretatons for the extremals of least squares Lagrangans L 1 = 1 da B = 1 δk δ jl A x j A j x B Ak x l A l x l B kl L 13 = 1 de + tb + 1 dh J td + 1 dd ρ + 1 db, whch are not solutons of Maxwell equatons. Remarks. 1 There are a lot of applcatons of prevous type n appled scences. One of the most mportant s n the materal strength. In problems assocated wth the torson of a cylnder or prsm, one has to nvestgate the functonal z z Jz = x y + y + x dxdy D for an extremum. The Euler-Lagrange equaton, z x + z y = 0, shows that the extremals are harmonc functons. Of course, J has no global mnmum pont snce the PDE system z z = y, x y = x s not completely ntegrable. All prevous examples n ths secton are sngle-component potentals. Smlarly we can ntroduce the mult-component potentals. 3 Second-Order Forms and Vectors Now we wll concentrate on certan geometrc deas that are very mportant n the physcal and stochastc applcatons. To avod too much repetton, M wll denote a dfferentable manfold of dmenson n, and all the functons are of class C. Let x = x x,, = 1,..., n be a changng of coordnates on M. Then we ntroduce the symbols D = x x, D j = x x x j. For a dfferentable functon f : M R we use the smplfed coordnate expresson fx = fx x, the frst order dervatves f,, f, and the second order dervatves f, j, f,. These are connected by the rule f,, f, j = f,, f, D D j 0 D D j j. 1 The par frst-order partal dervatves, secondorder partal dervatves possesses a tensoral change law that the second dervatve, by tself, lacks. Ths par was used n the classcal works lke contact element or lke jet. If a curve s gven by the parametrc equatons x = x t, t I, then the precedng dffeomorphsm modfes the par ẍ, ẋ ẋ T as follows ẍ ẋ = ẋ D, ẍ = ẍ D + ẋ ẋ j D, ẋ ẋ j = D D 0 D Dj j ẍ ẋ ẋ j. The par acceleraton, square of velocty s suggested by the equatons of geodescs ẍ k t + Γ k xtẋ tẋ j t = 0. Consequently, when frst and second dervatves come nto play together, then matrces of blocks such 34

5 as K = D D j 0 D D j j, K 1 = D D 0 D Dj j are useful for changng the components of pars of objects. Defnton. Any par ω, ω admttng the changng law 1, ω s an 1-form and ω s symmetrc, wll be called a second-order form on M. A second-order vector feld on M s a second-order dfferental operator, wth no constant term. A second-order form can be wrtten as θ = θ d x + θ dx, the components θ and θ = θ j are smooth functons. A second-order vector feld wrtes Xf = l D f + l D f, the components l = l j and l are smooth functons. The theory of second-order vectors or forms appears n Emery [3], wth applcatons n stochastc problems, and n Foster [4], suggestng new pont of vew about the felds theory. We use these deas to defne meanngful second-order Lagrangans knetc potentals and to study ther extremals. 4 Dynamcs nduced by secondorder electrc form Let E be the electrc potentals. The usual dervatve E,j may be decomposed nto skew-symmetrc and symmetrc parts, E,j = 1 E,j E j, + 1 E,j + E j,. The skew-symmetrc part vortex m = 1 E,j E j, s called Maxwell tensor feld gvng the opposte of the tme dervatve of magnetc nducton. The symmetrc part 1 E,j + E j, s not an ordnary tensor, but the par E, 1 E,j + E j, s a second-order object [3]-[4]. Let ω be a general object such that E, ω s a second-order object. The dfference 0, ω 1 E,j + E j, s a second-order object determned by g = ω 1 E,j + E j,. If ω s symmetrc, then g s a symmetrc tensor feld; f we add detg 0, then g can be used as a sem-remann metrc. In ths context the equalty E, ω = E, 1 E,j + E j, + 0, g shows that the valuable objects 1 E,j + E j, come from electrcty and mate wth gravtatonal potentals g. Consequently they have to be electrogravtatonal potentals. The precedng potentals determne an electrc energy second-order Lagrangan, L el = E xt, t d x t + 1 E,j + E j, xt, t dx tdxj t. If the most general energy second-order Lagrangan s gven by L ge = E xt, t d x t+ω xt, t dx tdxj t, and the gravtatonal energy frst-order Lagrangan s L g = g xt, t dx tdxj t, then L ge = L el + L g. To smplfy, let us take E = Ex. Snce the dstrbuton generated by the electrc 1-form E s gven by the Pfaff equaton E xdx = 0, the electrc energy Lagrangan s zero along ntegral curves of ths dstrbuton. Also the general energy second-order Lagrangan for E = Ex, ω = ω x determnes the energy functonal b a E xt d x We denote + ω xt dx tdxj t. ω k = 1 ω kj, + ω k,j ω,k the Chrstoffel symbols of ω, E k = 1 E k, + E j,k E,jk. 35

6 Theorem. The extremals of the energy functonal are descrbed by the Euler-Lagrange ODEs d x + ω k E k dx xa = x a, xb = x b. = 0, geodescs wth respect to an Otsuk connecton [6]. Proof. Usng the second-order Lagrangan L = E d x the Euler-Lagrange equatons L x k d L dx k + ω dx, + d L d x k = 0 transcrbe lke the equatons n the theorem. To a Lagrangan there may corresponds a feld theory. Consequently we obtan a feld theory havng as bass the general electrogravtatonal potentals. The pure gravtatonal potentals are gven by g = ω 1 E,j + E j,. We defne Γ k = ω k E k. It s verfed that Γ k = g k + m k, g k are the Chrstoffel symbols of g, and m k = m,k + m k,j s the symmetrzed dervatve of the Maxwell tensor m = 1 E,j E j,. Corollary. The extremals of the energy functonal 1 are descrbed by the Euler-Lagrange ODEs d x + g kj + m kj dx xa = x a, xb = x b. = 0, geodescs wth respect to an Otsuk connecton [6]. 5 Dynamcs nduced by secondorder magnetc form Let H be the magnetc potentals. The usual dervatve H,j may be decomposed nto skew-symmetrc and symmetrc parts, H,j = 1 H,j H j, + 1 H,j + H j,, M = 1 H,j H j, 36 s the Maxwell tensor feld vortex gvng the sum between the electrc current densty and the tme dervatve of the electrc dsplacement. The par H, 1 H,j + H j, s a second-order object. If H, ω s a general second-order object, then the dfference g = ω 1 H,j + H j, represents the gravtatonal potentals a metrc provded that ω s symmetrc, g s a 0, tensor feld and detg 0. Consequently the valuable objects 1 H,j +H j,, whch come from magnetsm and mate, have to be magnetogravtatonal potentals. The precedng potentals produce the followng energy Lagrangans: 1 magnetc energy second-order Lagrangan, L ma = H xt, t d x t + 1 H,j + H j, xt, t dx tdxj t; gravtatonal energy frst-order Lagrangan, L g = g xt, t dx tdxj t; 3 general energy second-order Lagrangan, L ge = H xt, t d x + ω xt, t dx tdxj t. These satsfy the relaton L ge = L ma + L g. To smplfy, let us take H = Hx. Snce the dstrbuton generated by the magnetc 1-form H s gven by the Pfaff equaton H xdx = 0, the magnetc energy Lagrangan s zero along ntegral curves of ths dstrbuton. Also the general energy second-order Lagrangan for H = Hx, ω = ω x determnes the energy functonal b H xt d x a t + ω xt dx tdxj t 3 determned by a second-order Lagrangan whch s lnear n acceleraton. We denote ω k = 1 ω kj, + ω k,j ω,k

7 the Chrstoffel symbols of ω, H k = 1 H k, + H j,k H,jk. Theorem. The extremals of the energy functonal 3 are solutons of the Euler-Lagrange ODEs d x + ω k H k dx xa = x a, xb = x b. = 0, geodescs wth respect to an Otsuk connecton [6]. To an energy Lagrangan there may corresponds a feld theory. Consequently we obtan a feld theory havng as bass the general magnetogravtatonal potentals. The pure gravtatonal potentals are components of a Remann or sem-remann metrc g = ω 1 H,j + H j,. We ntroduce Γ k = ω k H k. It s verfed the relaton Γ k = g k +M k, g k are the Chrstoffel symbols of g, and M k = M,k + M k,j s the symmetrzed dervatve of the Maxwell tensor M = 1 H,j H j,. Corollary. The extremals of the energy functonal 3 are solutons of the Euler-Lagrange ODEs d x + g kj + M kj dx xa = x a, xb = x b. = 0, geodescs wth respect to an Otsuk connecton [6]. 6 Dynamcs nduced by secondorder general form Now we want to extend the precedng explanatons snce they can be appled to dynamcal systems comng from Bomathematcs, Economcal Mathematcs, Industral Mathematcs etc. Let ω be gven potentals gven form. The usual partal dervatve ω,j may be decomposed as ω,j = 1 ω,j ω j, + 1 ω,j + ω j,, M = 1 ω,j ω j, s a Maxwell tensor feld vortex. The par ω, 1 ω,j + ω j, s a second-order form. If ω, ω s a general second-order form, then we suppose that the dfference g = ω 1 ω,j + ω j, represents the components of a metrc,.e., ω s symmetrc, g s a 0, tensor feld and detg 0. The precedng potentals produce the followng energy Lagrangans: 1 potental-produced energy Lagrangan, L pp = ω xt, t d x t + 1 ω,j + ω j, xt, t dx tdxj t; gravtatonal energy Lagrangan, L g = g xt, t dx tdxj t; 3 general energy Lagrangan, L ge = ω xt, t d x whch verfy L ge = L pp + L g. + ω,jxt, t dx ; The Pfaff equaton ω xdx = 0, = 1,..., n defnes a dstrbuton on M. The valuable objects 1 ω,j + ω j, are the components of the second fundamental form of that dstrbuton [6]. The potental-produced energy Lagrangan s zero along ntegral curves of the dstrbuton generated by the gven 1-form ω = ω x. In the autonomous case, the general energy Lagrangan produces the energy functonal b ω xt d x a t + ω xt dx tdxj t. 4 Ths energy functonal s assocated to a partcular Lagrangan L of order two. We denote ω k = 1 ω kj, + ω k,j ω,k 37

8 the Chrstoffel symbols of ω, Ω k = 1 ω k, + ω j,k ω,jk. Theorem. The extremals of the energy functonal 4 are solutons of the Euler-Lagrange ODEs d x + ω k Ω k dx xa = x a, xb = x b = 0, geodescs wth respect to an Otsuk connecton [6]. To an energy Lagrangan there may correspond a feld theory. We ntroduce Γ k = ω k Ω k. After some computatons we fnd Γ k = g k +m k, g k are the Chrstoffel symbols of g, and m k = M,k + M k,j s the symmetrzed dervatve of the Maxwell tensor feld M. Corollary. The extremals of the energy functonal 4 are solutons of the Euler-Lagrange ODEs d x + g kj + m kj dx xa = x a, xb = x b = 0, geodescs wth respect to an Otsuk connecton [6]. Open problems: 1 Fnd the lnear connectons n the sense of Crampn [] assocated to the precedng second-order ODEs. Analyze the second varatons of the precedng energy functonals. Analyze the symmetres of the precedng second order dfferental systems see also [16]. 3 Fnd practcal nterpretatons for motons known as geometrc dynamcs, gravtovortex moton and second-order force moton see also [5]. Acknowledgements: Partally supported by Grant CNCSIS 86/ 007 and by 15-th Italan- Romanan Executve Programme of S&T Cooperaton for , Unversty Poltehnca of Bucharest. References: [1] A. Bossavt, Dfferental forms and the computaton of felds and forces n electromagnetsm, Eur. J. Mech., B, Fluds, 10, no 51991, [] C. Crampn, A lnear connecton assocated wth any second order dfferental equaton feld, L.Tamassy and J.Szenthe eds, New Developments n Dfferental Geometry, 77-85, Kluwer Academc Publshers, [3] M. Emery, Stochastc Calculus n Manfolds, Sprnger-Verlag, [4] B. L. Foster, Hgher dervatves n geometry and physcs, Proc. R. Soc. Lond. A , [5] D. Isvoranu, C. Udrste, Flud flow versus Geometrc Dynamcs, 5-th Conference on Dfferental Geometry, August 8-September, 005, Mangala, Romana; BSG Proceedngs 13, pp. 70-8, Geometry Balkan Press, 006. [6] T. Otsuk, General connectons, Math. J. Okayama Unversty , 7-4. [7] I. N. Popescu, Gravtaton, Edtrce Nagard, Roma, Italy, [8] C. Udrşte, Geometrc dynamcs, Southeast Asan Bulletn of Mathematcs, Sprnger-Verlag, 4, 1 000, [9] C. Udrşte, Geometrc Dynamcs, Mathematcs and Its Applcatons, 513, Kluwer Academc Publshers, Dordrecht, Boston, London, 000. [10] C. Udrşte, Dynamcs nduced by second-order objects, Global Analyss, Dfferental Geometry and Le Algebras, BSG Proceedngs 4, Ed. Grgoros Tsagas, Geometry Balkan Press 000, [11] C. Udrşte, Mult-tme dynamcs nduced by 1- forms and metrcs, Global Analyss, Dfferental Geometry and Le Algebras, BSG Proceedngs 5, Ed. Grgoros Tsagas, Geometry Balkan Press 001, [1] C. Udrşte, Geodesc moton n a gyroscopc feld of forces, Tensor, N. S., 66, 3 005, [13] C. Udrşte, O. Dogaru, Convex nonholonomc hypersurfaces, The Mathematcal Hertage of C.F. Gauss, , Edtor G. M. Rassas, World Scentfc Publ. Co. Sngapore, [14] C. Udrşte, M. Ferrara, D. Oprş, Economc Geometrc Dynamcs, Monographs and Textbooks 6, Geometry Balkan Press, Bucharest, 004. [15] C. Udrşte, Tools of geometrc dynamcs, Buletnul Insttutulu de Geodnamcă, Academa Română, 14, 4 003, 1-6; Proceedngs of the XVIII Workshop on Hadronc Mechancs, honorng the 70-th brthday of Prof. R. M. Santll, the orgnator of hadronc mechancs, Unversty of Karlstad, Sweden, June 0-, 005; Eds.

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