The interaction of charged particles with electromagnetic fields

The interaction of charged particles with electromagnetic fields 

We now turn to the construction of the appropriate perturbation Hamiltonian. To ensure the correct identification of conjugated variables the proper route is 

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 Lagrangian has the general form L = T — U, where T is the kinetic energy, or more generally the interaction-free part, and U is the generalized potential 

In a conservative system the generalized potential is independent of the velocity q. The extension to continuous systems is done with the introduction of a Lagrangian density 

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