Continuous System of Fields

In practical applications the independent variables of the system are often not discrete but described by a set of continuous functions, so-called fields. For simplicity we may assume here that our system is specified by / scalar and real-valued fields (p = p (r, t) which each depend on space and time. Set-ups of this kind can be found in electrodynamics and the mechanics of liquids and continuous media, for example. In field theory the role of space and time has been modified as compared to the mechanics of discrete particles discussed above. The time coordinate is no longer singled out as simply parametrizing the spatial motion of the particles but is now a coordinate completely equivalent to the spatial ones. This may be seen as a first glance toward the inevitable necessity of relativistic formulations, and, indeed, the paramoxmtcy of field theories for the development of modem physics is inseparably connected to their covariant relativistic stmcture. 

The Lagrangian L is given as the spatial integral over the Lagrangian density C over all available space V, 

Varying the generalized variables in exactly the same way as before for discrete systems, the Euler-Lagrange equations of field theory follow from the Hamiltonian principle and read 

Obviously, the scalar product between the nabla operator and the partial derivative of has to be interpreted as the divergence of the corresponding vector field. These equations of motion determine the actual form of the fields. Their solution is, however, much more involved than for discrete systems. The canonical momentum field conjugate to the field (pi is analogously defined by 

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