( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.

Transcription

1 9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can be extended discete set of coodinates i to a continuous fomulation. Each point in a egion of space eithe finite o infinite will be associated with a continuous vaiable, φ( xt, ). (9.) The set of vaiables constitute a system with an infinite numbe of degees of feedom a field. In this fomulation, the Lagangian of the field vaiable is functional of the field. A functional is a mapping fom a space of functions, the Lagangians, to a set of eal numbes. This mapping is given by, Lt = L φ( xt,, ) φ ( xt, ). (9.) The functional Lt, depends on the value of φ and φ at all points in space at simultaneous time. Hamilton s pincipal can now be applied to this functional by defining a vaiation of the functional as, [ ] [ ] [ ] δf [ φ] δφ δφ( x) δf φ = F φ+δφ F φ, xdx. (9.) The tem in Eq. (9.) δf[ φ] δφ ( x) is called the functional deivative of the functional F [ φ ] with espect to the functional φ at a spatial point x. This deivative decibes how the functional is changing when the values of the function φ is vaied at the point x. The functional deivative is fomally defines as, +εδ( ) δf Fgx x y Fgx = lim δgy ε 0 ε (9.4) whee δ( x y) is the Diac delta function. The deivative in Eq. (9.4) displays all the popeties of a standad deivative. [] If F and G ae two functionals, than thei poduct is a functional deivative defined as, [ FG] g( x) G( F g ( x) ) F( G ( x) ) δ δ = δ δ + δ δ, which is the Leibniz popety. If F g is a functional that is well behaved in the inteval in the function space aound g = 0, than Copyight 000, 00 9

2 The functional deivative can now be applied to the Lagangian in Eq. (9.) to give, δl δl δl φφ, = δφ ( x) + δφ ( x) dx δφ( x) δφ ( x) (9.5) The Lagangian can now be integated to poduce the action function, usually denoted by W φφ,, which is a functional of φ and φ. By integating ove the time inteval t = to t =+ to give, t t t δ W =δ L φφ, dt, t t δl δl = δφ ( xt) + δφ δφ( xt, ) δφ ( xt, ) t δl δl = δφ δφ( xt, ) δφ ( xt, ) Since Hamilton s stationay function is,, xt, dtdx,, t t xtdtdx,. (9.6) δw φφ, =δ L φφ, dt = 0, (9.7) the Eule Lagangian can be genealized to Classical Field theoy as, L δl = 0. (9.8) φ δφ Applying the pincipals developed in 8 to Maxwell s equations began as ealy as the 870 s. Maxwell made use of Hamilton s pincipal in his Teatise [Buch85]. By fomulating the electomagnetic field as a continuous dynamic medium, Maxwell built on the ideas put foth by William Thomson and Pete Tait in [Thom6]. Thomson and Tait consideed Hamilton s pincipal as a fundamental fomulation of physics the functional F can be expanded in a Taylo seies n [ ] =!,,, δ n [ ] ( δ δ n ). F g n dx dx dx F g g x g x g = 0 9 Copyight 000, 00

3 (the) pincipal of Least Action is a useful guide to kinetic investigations. We ae stongly impessed with the conviction that a much moe pofound impotance will be attached to it in the theoy of seveal banches of physical science now beginning to eceive dynamical explanation his method of Vaying Action which undoubtedly become a most valuable aid in the futhe genealization [Thom6] Vol. 6. Restating Hamilton s pincipal equies that the path taken by a physical system between two states at specified times and with fixed values of the vaiables at these times such that the value of the function TVdt, whee T is the kinetic enegy V is the potential enegy, must be an extemimum such that, t t0 δ TVdt = 0. In this fom Hamilton s pincipal is capable of geneating both the equations of motion of the electodynamic system and the bounday conditions fo any continuous field with localized foms of enegy. By adopting Hamilton s pincipal as the fundamental fomulation of electomagnetics, then evey poblem in field theoy educes to finding an appopiate expession fo the field s potential and kinetic enegies. [] The essence of the Lagangian o Hamiltonian appoach consists of two pats: (i) the specification of the genealized coodinates which fix the state of the field and (ii) a choice of expessions, in tems of these coodinates and thei spatial and tempoal deivatives. To make use of the esults developed in 8, the Lagangian, L, fo the electomagnetic field in vacuum will be constucted. The f 's ae given by the vecto and scala potentials and the field equations take the fom, A = 0 φ A t φ.(9.9) = 0 The potentials have to be subjected to the futhe constaint of the Loentz condition, As is always the case this simplification of the situation is not univesally tue. Independent conditions on the bounday values o genealized coodinates must be imposed whee the fields enegies ae functions. Copyight 000, 00 9

4 φ A + = 0. (9.0) The Lagangian which povides the equations in Eq. (8.X) is given by, L = ( + ) 8π E B. [] (9.) The B and E fields can now be expessed in tems of the potentials, A and φ as, A L = φ ( A). (9.) 8π Expanding Eq. (9.) in Catesian coodinates, φ A φ A φ A L = + + 8π q q q A A A A A A. q q q q q q (9.) Using the Lagange elationship, L L L + t A, (9.4) q A A gives fo the A component of the vecto potential, The facto 8π is completely abitay and is used so in the integation of the electomagnetic field fo chaged bodies. Thee ae othe changes in the notation used in this section, including the absence of the absolute value signs aound the E and B field vaiables it will be assumed that the eal natue of these vaiables is undestood by the eade. In all instances the field vaiables ae assumed to be functions of time and space although no explicit paametes ae shown in the equations. Whee thee is a explicit dependence on a position vaiable a subscipt will be used to indicate the dependent vaiable. In geneal the vaiable q will be used fo a genealized coodinate value in place of o x. 9 4 Copyight 000, 00

5 L φ A A A A A = + + 4π q q q q q q q A φ = A + +, A + 4π q A = 4π A., (9.5) with simila equations fo A anda components. A simila expansion can be developed fo the scala potential, such that, φ L = φ. (9.6) 4π A Lagangian which includes the Loentz condition and epesents the electomagnetic field independent of any constaints is given by, A φ L = φ ( A) A+. (9.7) 8π Consideing the Lagangian fo a electomagnetic field, with chage souces, is the next step in the development of the complete Hamiltonian. The potential equations as oiginally developed in Eq. (4.) ae, A = j φ A t φ. (9.8) = ρ The Lagangian that will esult in these equations is Eq. (9.7) plus ρ A v φ, whee ρ is the chage density and v is the an additional tem velocity of the chage, so that the adiation and inteaction Lagangian is given by, A φ L = φ ( A) A+ +ρ( A v φ). (9.9) 8π Eq. (9.9) now defines the Lagangian fo a chaged paticle moving in a electomagnetic field poduced by the chage ρ and cuent j. The Lagangian fo the entie field, including the paticles themselves, is Copyight 000,

6 given by integating each individual chage with the volume using L = L dv to give the field Lagangian as, T A φ L = + φ ( A) A+ +ρ( A v φ), (9.0) 8 π which esults in a desciption of both the equations of motion fo chaged paticles embedded in the field and the equations fo the electomagnetic field itself. Eq. (9.0) epesents the total Lagangian of a set of paticles inteacting with the electomagnetic field. The dynamical vaiables of the paticles fom a discete set involving the components of the position and the velocity d dt. Fo the electomagnetic field, it is the field potentials and the fields which epesent the genealized coodinates of the Lagangian. The total Lagangian has thee tems, the Lagangian fo the paticles, L, the Lagangian fo the adiated field, L and the obj Lagangian fo the inteaction between the electomagnetic field and the paticles, L int. The total Lagangian is then given by, L=Lobj+ Lad+ L int (9.) ad whee, L mq, (9.) obj = and E B L ad = + dv, (9.) and L = [ ja ρφ]. (9.4) int Regouping the adiation and inteaction Lagangian tems allows the e intoduction of the Lagangian Density. = L + + ρφ, (9.5) E B [ ja ] and the following fom fo the Standad Lagangian, 9 6 Copyight 000, 00

7 L = mq + L dv. (9.6) i It should be noted that the inteaction tem is local with the cuent density at the point multiplied by the vecto potential. In the adiated field Lagangian the spatial deivatives of the potentials come about fom E and B, which descibes the coupling between the fields at each point. This coupling is the oigin of the popagation of the fee field. 9.. FIELD ENERGY DENSITY Using the Hamiltonian expessions given in 8, Eq. (9.6) seves as the stating point fo computing the Hamiltonian of the electomagnetic field in the absence of chaged paticles and field souces, that is the Hamiltonian of a feely popagating electomagnetic wave. The contibution of the fist and last tems in Eq. (9.6) is detemined by Eq. (8.44). The emaining contibution will be fom the tems in the field quantities alone, whee, H = H dv, (9.7) H = π φ φ φ E A E B A, φ φ E E A A E B, = + φ π 8π φ. = E + B + E φ+ A The total Hamiltonian given by Eq. (8.46) is now, v s H = HS q, p ρ AsdV + s q 8π + ρφ E + B + E φ+ A φ dv. (9.8) (9.9) Whee H S is the Hamiltonian function of the dynamical system. When the field vaiables ae taken as discete, the odinay Hamiltonian canonical equations can be used fo the dynamical as well as fo the electomagnetic vaiables. This fist tem of this Hamiltonian is the Copyight 000,

8 kinetic enegy of the dynamically system. The sum of the last two tems unde the integal sign is zeo, due the Loentz condition. The fist and fouth tems unde the integal sign cancel since, ρφ+ E φ dv= φ + φ dv, 4π 4π 4π E E { } = φe dv, 4π (9.0) = 0. If the field potential vanish sufficiently apidly at infinity what emains is the adiation field enegy density, ( E + B ) dv. (9.) 8π 9 8 Copyight 000, 00