  The supporting worldsheet of a string, membrane, or other higher dimensional
brane, is analysed in terms of its first, second, and third fundamental
tensors, and its inner and outer curvature tensors. The dynamical equations
governing the models appropriate for phenomena such as (superconducting) cosmic
strings and cosmic domain walls are developed in a general framework (allowing
for both electromagnetic and Kalb Ramond background coupling). It is shown how
the surface stress momentum energy density tensor determines the propagation
characteristics of small ``wiggle" perturbations of the worldsheet. Attention
is then focussed on special features of strings (using the transonic model with
tension T inversely proportional to the energy density U as a particularly
important example). A quadratic Hamilton-Jacobi formulation is shown to govern
equilibium states and other conservative string configurations sharing a
symmetry of the (gravitational, electromagnetic, and Kalb-Ramond) background,
including stable ring states that may be cosmologically important.
