  The classical energy minimization principles of Dirichlet and Thompson are
extended as minimization principles to acoustics, elastodynamics and
electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is done
by building upon ideas of Cherkaev and Gibiansky, who derived minimization
variational principles for quasistatics. In the absence of free current the
primary electromagnetic minimization variational principles have a minimum
which is the time-averaged electrical power dissipated in the body. The
variational principles provide constraints on the boundary values of the fields
when the moduli are known. Conversely, when the boundary values of the fields
have been measured, then they provide information about the values of the
moduli within the body. This should have application to electromagnetic
tomography. We also derive saddle point variational principles which correspond
to variational principles of Gurtin, Willis, and Borcea.
