A note on a variational formulation of electrodynamics

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1 Proeedings of the XV International Workshop on Geometry and Physis Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11 16, 006 Publ. de la RSME, Vol. 11 (007), A note on a variational formulation of eletrodynamis Antonio De Niola 1 and W lodzimierz M. Tulzyjew 1 Dipartimento di Matematia, Università di Bari Via Orabona, Bari, Italy adeniola@tin.it Valle San Benedetto, 6030 Monteavallo (MC), Italy Assoiated with INFN, Sezione di Napoli, Italy tulzy@libero.it Abstrat We present a variational formulation of eletrodynamis using de Rham even and odd differential forms. Relying on a variational priniple more omplete than the Hamilton priniple our formulation leads to field equations with external soures and permits the derivation of the onstitutive relations. Keywords: variational priniples, eletrodynamis. 000 Mathematis Subjet Classifiation: 49S05, 70S Introdution A general framework for variational formulations of physial theories was presented in [1]. Appliations to statis and dynamis of mehanial systems appear in [, 3]. This note presents an introdution to a more omplete variational formulation to be presented in a future publiation. Our formulation of eletrodynamis is speial relativisti. The use of de Rham even and odd differential forms ([5, 6]) permits a rigorous formulations of eletrodynamis and the desription of the transformation properties of eletromagneti fields relative to refletions (f. [4]). Relying on a variational priniple more omplete than the Hamilton priniple our formulation leads to field equations with external soures and permits the derivation of the onstitutive relations whih are usually postulated

2 A. De Niola and W.M. Tulzyjew 317 separately sine the variations normally onsidered are not general enough to derive them from the variational priniple. We interpret a domain in spae-time as an odd de Rham 4-urrent. This permits a treatment of different types of boundary problems in an unified way. In partiular we obtain a smooth transition to the infinitesimal version by using a urrent with a one point support.. Currents Let M be the affine Minkowski spae-time of speial relativity with the 4-dimensional model spae V and a metri tensor g : V V of signature (1,3). The vetor spae of even differential q-forms in M will be denoted by Φ q e(m) and spae of odd differential q-forms will be denoted by Φ q o(m). The symbol Φ q p(m) will be used to denote either of the two spaes when the distintion is not relevant. An even or odd de Rham urrent of dimension q on M is a linear funtion : Φ q p(m) R: A A. (1) Domains in spae-time will be treated as urrents. The boundary of a urrent is defined by assuming that Stokes theorem holds for all urrents as it holds for domains. In addition to domains in spae-time odd de Rham urrents most frequently used are the Dira urrents. A Dira urrent wδ(x) is an odd urrent of dimension 4 defined in terms of a point x M and an odd 4-vetor w. If A is an odd 4-form, then A = A(x),w. () wδ(x) 3. The spae of fields Let CM be the spae of odd 4-urrents with ompat supports in M. We onsider the set X(Φ 1 e(m);cm) of pairs (A,), where is an odd urrent of dimension 4 in M with a ompat support Sup() and A is an even 1-form A: U 1 ev (3) defined in an open set U M ontaining the support of. The symbols q ev and q ov denote respetively the vetor spaes of even and odd q-ovetors. The 1-form A will represent the eletromagneti potential. Its differential F = da is the eletromagneti field. A mapping κ: M 1 ev ev 4 ov (4) is said to be quadrati if for eah x M there exists a symmetri bilinear mapping δ κ x : ( 1 ev ev ) ( 1 ev ev ) 4 ov (5)

3 318 A note on a variational formulation of eletrodynamis suh that the mappings κ x = κ(x,, ) and δ κ x are in the relation κ x (a,f) = 1 δ κ x ((a,f)(a,f)), (6) for eah (a,f) 1 ev ev. We will use the set of all quadrati mappings (4) to introdue an equivalene relation in the set X(Φ 1 e(m);cm). Pairs (A,) and (A, ) are equivalent if κ (x,a,da ) = κ (x,a,da) (7) for eah quadrati mapping (4). Equivalene lasses will be alled fields. Our fields are similar to those used by Freed in [8]. The spae of fields will be denoted by Q(Φ 1 e(m);cm) or simply Q. The equivalene lass of (A,) will be denoted by q(a,). The symbol q will denote a generi element of Q. There is a natural projetion ε: Q CM : q(a,) from the spae of fields to the spae CM of urrents in M whih is similar to a vetor fibration. Eah fibre ε 1 () of the projetion ε is a vetor spae denoted by Q(Φ 1 e(m);) or Q. 4. Funtions, vertial vetors and ovetors in the spae of fields With eah quadrati mapping (4) we assoiate the funtion k: Q(Φ 1 e(m);cm) R: q(a,) κ (x,a,da). (8) Funtions onstruted in this way will be onsidered differentiable. The spae of suh funtions will be denoted by K(Φ 1 e(m);cm). It is easy to verify that these differentiable funtions separate points of Q, i.e. if k(q ) = k(q) for eah k K(Φ 1 e(m);cm), then q = q. The tangent spae to the vetor spae Q = ε 1 () is the spae Q itself. It follows that the vertial tangent bundle of the vetor fibration ε is the spae VQ = Q Q = {(q,δq) Q Q; ε(q) = ε(δq)}. (9) (ε,ε) There is no obvious hoie of the bundle dual to VQ. Using the fibre derivatives of funtions k K(Φ 1 e(m);cm) as models of ovetors we obtain the following result. A ovetor p is an equivalene lass of triples (G,J,) of an odd -form G: U ov, an odd 3-form J : U 3 ov, and a urrent with support ontained in U. The objets G and J are interpreted as the eletromagneti indution and the urrent respetively. Elements (G,J,) and (G,J, ) are equivalent if = and ( 1 J δa 1 4π d (G δa) ) = ( 1 J δa 1 d (G δa) 4π ), (10) for eah δa: U 1 ev. The equivalene lass of (G,J,) is denoted by p(g,j,).

4 A. De Niola and W.M. Tulzyjew 319 The vetor spae Π of ovetors assoiated with the urrent is the dual of the spae Q with the pairing ( 1 p(g,j,),q(δa,) = J δa 1 ) d (G δa). (11) 4π The spae of all ovetors is the union Π = CR Π. There is a natural projetion ε : Π CM : p(g,j,). The phase spae is the spae Ph = Q Π = {(q,p) Q Π; ε(q) = ε (p)}. (1) (ε,ε ) The symbol Ph will denote the set Q Π Ph. 5. A virtual ation priniple for eletrodynamis In this setion a variational priniple for eletrodynamis similar to the virtual ation priniple of analytial mehanis (see [3]) will be formulated. The ation is the differentiable funtion W : Q(Φ 1 e(m);cm) R: q(a,) L (A,dA) (13) derived from the quadrati Lagrangian density L: 1 e V ev 4 ov : (a,f) 1 8π f, eg 1 (f) g. (14) We are using the symbol g to denote the odd 4-ovetor derived from the metri tensor g (see [4] or [7]), while the symbol eg 1 denotes the inverse mapping of eg: ev ev haraterized by the equality eg(v 1 v ) = g(v 1 ) g(v ) for even simple -vetors. A phase ph = (q(a,),p(g,j,)) satisfies the virtual ation priniple if the equality DW(q,δq) p,δq = 0 (15) holds for eah virtual displaement δq = q(δa,) Q. For eah urrent the dynamis assoiated with the urrent is the set D Ph of phases whih satisfy the virtual ation priniple. The dynamis is the subset D = CR D of the phase spae P h defined above. A phase spae trajetory is a triple of differential forms (A,G,J): U 1 ev ov 3 ov. (16) The dynamis of a system an also be represented as a set D of phase spae trajetories (A,G,J) suh that for eah urrent with support inluded in U the phase ph = (q(a,),p(g,j,)) is in D. The equation (15) is too abstrat to be used diretly. A more onrete expression of the virtual ation priniple will be given in the Proposition 1.

5 30 A note on a variational formulation of eletrodynamis The left interior multipliations are the operations : q p V q p V q q pp V, (17) defined for q q by w a,w = a,w w. The parity pp whih appears in this definition is onstruted by assigning the numerial values +1 and 1 to e and p respetively. The parity of the multivetor w must math the parity of the multiovetor w a. Proposition 1 A phase ph = (q(a, ), p(g, J, )) satisfies the virtual ation priniple if and only if the equality 1 ( ( d( 4π e g 1 da ) ) ( g δa d(( e g 1 da ) ) )) g δa ( 1 = J δa 1 ) d (G δa), (18) 4π is satisfied for eah virtual displaement δq = q(δa,). A phase spae trajetory belongs to the dynamis D, if and only if it satisfies the virtual ation priniple for eah urrent with support inluded in its domain of definition. There is a haraterization of the dynamis of phase spae trajetories in terms of differential equations. This is shown in the following propositions. Theorem A phase spae trajetory (A,G,J) belongs to the dynamis D if and only if it satisfies the Euler-Lagrange equation and the onstitutive relation ( d( e g 1 da ) ) g = 4π J (19) G = ( eg 1 da ) g. (0) The onstitutive relation (0) produed by our variational priniple orresponds to the momentum-veloity relation of analytial mehanis. Proposition 3 A phase spae trajetory (A, G, J) satisfies the Euler-Lagrange equation and the onstitutive relation if and only if it satisfies the Maxwell s equations and the onstitutive relation with F = da. dg = 4π J (1) G = ( eg 1 F ) g, ()

6 A. De Niola and W.M. Tulzyjew The Dynamis in a ompat domain Let the urrent onsist in integrating an odd 4-form on a ompat domain K M with smooth boundary K. Field onfigurations, tangent vetors and ovetors are equivalene lasses of equivalene relations based on the equalities (7) and (10). It follows that a field q = q(a,k) is represented by the restrition A K : K 1 ev (3) of the potential A to the domain K. A tangent vetor δq = q(δa,k) is represented by the restrition (δa) K : K 1 ev (4) of the variation δa to the domain K. A ovetor p = p(g,j,k) is represented by the pair of the restritions G K : K ov, J K : K 3 ov (5) of the eletromagneti indution G to the boundary K of the domain K and of the urrent J to the interior K of the domain K. The dynamis in the domain K is the set D K Ph of phases satisfying the virtual ation priniple. It is haraterized by the following proposition. Proposition 4 A phase ph = (q(a,k),p(g,j,k)), defined in a ompat domain K, belongs to the dynamis D K if and only if the Euler-Lagrange equation ( d( e g 1 da ) ) g K = 4π J K (6) and the onstitutive relation ( ( G K = e g 1 da ) ) g K (7) are satisfied. 7. The Lagrangian formulation The Lagrangian formulation of dynamis is the infinitesimal limit of the formulation in a ompat domain with the domain shrinking to a point. A formal method whih greatly simplifies the passage to the infinitesimal limit is to replae the ompat domain whih is used exlusively as domain of integration with the urrent = δ(x)w, where δ(x) is the Dira delta funtion in x M and w 4 ov is an odd 4-vetor, with w 0. The onstrution of infinitesimal fields, tangent vetors and ovetors is based on the equalities (7) and (10) whih in this ase redue to pairings of odd 4-ovetors with the odd 4-vetor w 0. It follows that an infinitesimal field q = q(a,) is represented by the pair (A(x),F(x)) 1 ev ev, (8)

7 3 A note on a variational formulation of eletrodynamis a tangent vetor δq = q(δa,) is represented by the pair (δa(x),δf(x)) 1 ev ev, (9) and a ovetor p = p(g,j,) is represented by the pair (G(x),dG(x) 4π J(x) ) ov 3 ov. (30) The pairing p,δq defined by the equality (11) assumes the form p,δq L = 1 (dg(x) 4π ) 4π J(x) δa(x) + G(x) δf(x),w. (31) We have onstruted the spae of infinitesimal fields Q δ = 1 ev ev and the spae of infinitesimal ovetors Π δ = ov 3 ov. Hene, the infinitesimal phase spae is Ph δ = Q δ Π δ = 1 ev ev ov 3 ov. The infinitesimal ation is W(q(A, δ(x)w)) = L(A(x), F(x)), w and the infinitesimal dynamis is the set { D δ = (a,f,g,h) Ph δ ; (δa,δf) 1 e V e V DL(a,f,δa,δf) = 1 (h δa + g δf) 4π }. (3) Applied to a phase ph = (q(a,δ(x)w),p(g,j,δ(x)w)), with w 0, the ation priniple results in the equations G(x) = ( eg 1 (F(x)) ) g, dg(x) = 4π J(x). (33) Referenes [1] W. M. Tulzyjew, The origin of variational priniples, in the volume Classial and quantum integrability (Warsaw, 001), Banah Center Publ., 59, Polish Aad. Si., Warsaw (003) [] G. Marmo, W. M. Tulzyjew and P. Urbański, Dynamis of autonomous systems with external fores, Ata Physia Polonia B, 33 (00), [3] A. De Niola and W. M. Tulzyjew, A variational formulation of analytial mehanis in an affine spae, Rep. Math. Phys., 58 (006), [4] G. Marmo, E. Paraseoli and W. M. Tulzyjew, Spae-time orientations and Maxwell s equations, Rep. Math. Phys. 56 (005), [5] J. A. Shouten, Tensor Analysis for Physiists, Oxford University Press, London, [6] G. de Rham, Variétés Differentiables, Hermann, Paris, 1955.

8 A. De Niola and W.M. Tulzyjew 33 [7] A. De Niola, Geometri Foundations of Classial Field Theory, PhD thesis, Bari, 006. [8] D. Freed, Classial field theory and Supersimmetry, IAS Park City Mathematis Series Vol. 11, IAS, Prineton, 001.