Physis Notes Note 9 Marh 009 Lagrangian Formulation of the Combined-Field Form of the Maxwell Equations Carl E. Baum University of New Mexio Department of Eletrial and Computer Engineering Albuquerque New Mexio 873 Abstrat A lassial question in eletromagneti theory onerns the onstrution of a Lagrangian from whih the Maxwell equations an be derived. In this paper a Lagrangian is formulated for the (omplex) ombined Maxwell equation, inluding both eletri and magneti potentials. ` This work was sponsored in part by the Air Fore Offie of Sientifi Researh.
. Introdution One way to formulate the Maxwell equations is via a Lagrangion density, L, whih an be found in various tests, e.g., []. In this formalism the Lagrangian is a salar funtion of the various eletromagneti field omponents, urrent-density omponents, and potential (vetor and salar) omponents. A ertain ombination of these (forming L) is differentiated with respet to the oordinates (spae and time) and with respet to the spatial derivatives of the potentials, where the potential omponents and their derivatives (forming the fields) are regarded as separate variables for this purpose. (This is also referened as the Euler-Lagrange equation.) One an also write the Maxwell equations in ombined-field form, reduing these to a single url equation extended to the four-vetor form of the Maxwell equations, as well as the quaternion form []. In this paper we investigate the appliation of the Lagrangian to the four-vetor form of the ombined Maxwell equation.. Combined-Field Form of the Maxwell Equations. Three-vetor form The ombined Maxwell equation is [3] where jq Eq( r, t) jq Z J q( r, t) t Eq ( r, t) E( r, t) jq Z H( r, t) ombined field (.) Z jq J q( r, t) J ( r, t) J m( r, t) ombined urrent density Z / wave impedane of uniform, isotropi, frequeny-independent medium (e.g. free spae) / wave propagation speed (light speed in free spae) (.) q separation index Taking the divergene of (.) we have Eq( r, t) Dq( r, t) J q ( r, t) t t J q ( r, t) q ( r, t) t
jq q( r, t) ( r, t) m( r, t) ombined harge density (.3) Z This is extended to the potentials Aq ( r, t) A( r, t) jq Z Am ( r, t) q( r, t) ( r, t) jqq( r, t) (.) This gives the ombined field as Eq( r, t) q ( r, t) jq Aq( r, t) t (.5) with the integral representation r r J q ( r, t ) Aq ( r, t) dv r r V r r (, ) q r t q ( r, t) dv r r V (.6) These are related by the gauge ondition Aq ( r, t) (, ) 0 q r t (.7) t. Four-vetor form The spae-time oordinates take the form [3] 3
3 rp ( r, Tp) x, y, z, Tp r, r, r, r p x yx zx TpT p Tp jpt, p (.8) p additional separation index Here an overbar is used to indiate a -vetor. The del operator and laplaian beome p, Tp p p Tp t (.9) Then we have J pq, ( rp) J q ( r, t), jpq( r, t) (-urrent density) J pq, ( rp) 0 J q ( r, t) q( r, t) t (ontinuity) (.0) p Apq, ( rp) Aq ( r, t), j q( r, t) (-potential) p Apq, ( r, t) Aq ( r, t) (, ) 0 q r t (Lorentz gauge) t p Apq, ( rp) Jpq, ( rp) Jpq, ( rp ) Apq, ( rp) dv, rp r, jpt r r r r V (.) This leads to the -vetor form of the ombined Maxwell equation as p E pq, ( r p) jqzj pq, ( r p)
Epq, ( rp) 0 Ez E q y pqe q xq Ez 0 E q x pqe q y q Ey Ex 0 pqe q q z q pqex pqey pqez 0 q q q E T pq, ( r p ) (skew symmetri) (.) We also have Epq, ( rp) Epq, ( rp) 0 0 0 0 0 0 Epq, ( rp) Eq( r, t) Eq( r, t) 0 0 0 0 0 0 (.3) Where Eq( r,) t Eq( r,) t E( r,) t E( r,) t Z H( r,) t H( r,) t j qz E( r,) t H( r,) t (.) whih is preserved under Lorentz transformation. Assuming that (.) is nonzero (whih is not the ase for a plane wave), then the field four-dyadi has an inverse as E pq, ( rp) Eq( r, t) Eq( r, t) Epq, ( rp) (.5) i.e., exept for a salar multiplier it is its own inverse. It has determinant and trae with eigenvalues det Epq, rp Eq( r, t) Eq( r, t), trepq, ( rp) 0 (.6) / qep, qrp Eq( r, t) Eq( r, t) (.7) with eah sign taken twie..3 Dual of a four-dyadi Define dnmn,,, m / for any even permutation of (,, 3, ) -/ for any odd permutation of (,, 3, ) 0 otherwise (any two indies equal) dnmn,,, m tetradi or fourth-rank tensor (.8) 5
For an antisymmetri dyadi X 0 X, X,3 X, X, 0 X,3 X, X,3 X,3 0 X 3, X, X, X3, 0 X T (.9) The dual dyadi is also antisymmetri as with dual nm nm 0 X3, X, X,3 X3, 0 X, X,3 X, X, 0 X, X,3 X,3 X, 0 X dnmnm,,, X, (.0) dual dual X X (.) Applying this to the ombined field dyadi gives [3] E, r pqe, r dual p q p p q p (.) whih, exept for a possible sign hange, makes the ombined-field four-dyadi self dual.. Four-dyadi field tensor related to potentials From [3] we have T T Ep, q( rp) jq pap, q( rp) pap, q( rp) p dual pap, q( rp) pap, q( rp) (.3) 3. Classial Lagrangian Formulation Following [] we have (in his units) 6
L F, J A, L F, r A r rv L J A (3.) giving the Maxwell equations F, r F, J 0 Bz By jex Bz 0 Bx jey By Bx 0 jez jex jey jez 0 (3.). Combined-Field Lagrangian So now let us try something of the form (in MKS units) L E ; pq, ( rp) Jpq, ( rp) Apq, ( rp) (.) onstant to be determined. First term ( An / rm terms) The first term involves the derivatives of the potentials with respet to the four oordinates. So we write L E ; pq, rp Eq( r, t) Eq( r, t) Eq ( r, t) (.) from (.) (or (.3)). This also takes the form in (.) in terms of E and H. 7
In terms of the potentials we have dual () () Epq, rp jqx jpx T () X papq, ( r p) papq, ( r p) () () X dual X () () Epq, rp jqx jqx pqepq, rp (.3) Where () X and () X are expliitely exhibited in [3]. For L we then have (suppressing the p, q subsripts on the vetor-potential omponents, and spae-time oordinates) L E ; pq, ( rp) () () jqxnm, jpx, nm () () () () X X pqx Xnm, () () (3) L L L () () () () L L Xnm, Xnm, A n An (zero for n = m, 6 nonzero terms) rm rn mn (3) A AA3 A L pq r r r r3 A A3 A A r3 r r r A A3 A A r3 r r r (.) Now we form the x dyadi 8
() () L L An / rm An / rm A A A 0 A3 A A r r r3 r r r A A A 0 A3 A A r r r3 r r r A 3 A A3 A A 0 3 A r r3 r r3 r r3 A A A A A A 3 0 r r r r r3 r () () T X X (.5) Similarly we form (3) L An / rm A3 A A A A A3 0 r r3 r r r3 r A A3 A 0 A A3 A r3 r r r r r3 pq A A A A A 0 A r r r r r r A3 A A A3 A A 0 r r3 r3 r r r () () T pqx pqx (.6) Summing these we have L () () X pqx An / r m jqe pq, ( rp) T q papq, ( rp) papq, ( rp) T pdual papq, ( rp) papq, ( rp) (.7) with the last result from (.3). 9
Now we have (from (.)) p L An / rm jq p Epq, ( rp) ZJ pq, ( rp) Jpq, ( rp) (.8). Seond term ( An terms) From (.) we have the seond term L Jpq, ( rp) Apq, ( rp) (.9) Taking the derivatives with respet to the A pq, four omponents gives a gradient as L An J pq, ( rp) (.0).3 Combining both terms Combining the An / rm and An derivatives of the Lagrangian to give the required zero variational form, we have p L L An / rm An Jpq, ( rp) Jpq, ( rp) 0 (.) This requires / (.) To give our Lagrangian 0
L Enmpq, ;, ( rp) Jpq, ( rp) Apq, ( rp) Eq( r,) t Eq( r,) t J q ( r,) t Aq( r,) t q( r,) t q( r,) t (.3) By our previous derivation this Lagrangian is differentiable in the usual Lagrangian way to give the Maxwell equation (.) in -vetor/tensor form. 5. Conluding Remarks It is thus possible to onstrut a Lagrangian for the ombined-field Maxwell equations, expressed in fourvetor/dyadi form. This works for all hoies of q (field-separation index) and p (time-separation index). Referenes. C. E. Baum, The Combined Field in Quaternian Form, Physis Note 3, Otober 003.. J. J. Sakurai, Advaned Quantum Mehanis, Addison-Wesley, 967. 3. C. E. Baum and H. N. Kritikos, Symmetry in Eletromagnetis, h., pp. -90, in C. E. Baum and H. N. Kritikos (eds.), Eletromagneti Symmetry, Taylor & Franis, 995.