Lagrangian equation 0 electrodynamics

The Lagrangian (850) shows that 0(3) electrodynamics is consistent with the Proca equation. The inhomogeneous field equation (32) of 0(3) electrodynamics is a form of the Proca equation where the photon mass is identified with a vacuum charge-current density. To see this, rewrite the Lagrangian (850) in vector form as follows ... 

There is rub to this construction. This Proca equation is really only applicable on a scale that approaches high-energy physics where the A 3 boson has appreciable influence. This will be only at a range of 10 17 cm. On the scale of atomic physics 10 3 cm, where quantum optics is applicable, this influence will be insignificant. In effect on a scale where the Al 3 does not exist, as it has decayed into pion pairs, the duality is established and there is no Lagrangian for the B 3 field. This puts us back to square one, where we must consider non-Abelian electrodynamics as effectively U(l) electrodynamics plus additional nonLagrangian and nonHamiltonian symmetries. 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

Not only mechanics but all of physics can be derived from the principle of least action. There are appropriate Lagrangian functions for electrodynamics, quantum mechanics, hydrodynamics, etc., which all allow us to derive the basic equations of the respective discipline from the principle of least action. In this sense, the principle of least action is the most powerful economy principle known in physics since it is sufficient to know the principle of least action, and the rest can be derived. Nature as a whole seems to be organized according to this principle. The principle of least action can be found under various names in nearly every branch of science. For instance the principle of least cost in economy or Fermat s principle of least time in optics. 

In 1892 Helmholtz inquired whether we...can cast the empirically known laws of electrodynamics, as they are formulated in Maxwell s equations, in the form of a minimal principle [44]. Indeed such a minimal principle exists in the form of the principle of least action For a system of n degrees of freedom there exists a Lagrangian L qi,qi,t) such that the action integral... 

The above considerations leads to the somewhat troubling question of whether (128) represents the true non-relativistic limit of the Dirac equation in the presence of external fields. Referring back to (110) we have certainly obtained the non-relativistic limit of the free-particle part Lm, but we have in fact retained the interaction term as well as the Lagrangian of the free field. In order to obtain the proper non-relativistic limit, we must consider what is the non-relativistic limit of classical electrodynamics. This task is not facilitated by the fact that, contrary to purely mechanical systems, the laws of electrodynamics appear in different unit systems in which the speed of light appears differently. In the Gaussian system Maxwell s laws are given as... 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

This interaction Lagrangian density may depend explicitly on the space-time coordinates x and the 4-velocity u via the charge-current density T. However, as far as only the equation of motion for the electrodynamic field is concerned they do not represent dynamical variables. Lorentz invariance of this interaction term is obvious, and gauge invariance of the corresponding action is a direct consequence of the continuity equation for the charge-current density f, cf. Eq. (3.162),... 

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