  In Cygnus A and other classical FR II double radio sources, powerful opposing
jets from the cores of halo-centered galaxies drive out into the surrounding
cluster gas, forming hotspots of shocked and compressed cluster gas at the jet
extremities. The moving hotspots are sandwiched between two shocks. An
inner-facing shock receives momentum and cosmic rays from the jet and creates
additional cosmic rays that form a radio lobe elongated along the jet axis. An
outer-facing bow shock moves directly into the undisturbed group or cluster
gas, creating a cocoon of shocked gas enclosing the radio lobe. We describe
computations that follow the self-consistent dynamical evolution of the shocked
cluster gas and the relativistic synchrotron-emitting gas inside the lobes.
Relativistic and non-relativistic components exchange momentum by interacting
with small magnetic fields having dynamically negligible energy densities. The
evolution of Cygnus A is governed almost entirely by cosmic ray energy flowing
from the hotspots. Mass flowing into hotspots from the jets is assumed to be
small, greatly reducing the mass of gas flowing back along the jet, common in
previous calculations, that would disrupt the spatial segregation of
synchrotron-loss ages observed inside FR II radio lobes. We compute the
evolution of the cocoon when the velocity and cosmic ray luminosity of the
hotspots are constant and when they vary with time. If cosmic rays mix with
cluster gas in hotspots before flowing into the radio lobe, the thermal gas is
heated to mildly relativistic temperatures, producing an unobserved pressure
inside the lobe.
