Models for Small-Scale Structure on Cosmic Strings: I. Mathematical Formalism

We describe the formalism of a quantitative analytic model for the evolution of realistic wiggly (as opposed to Goto-Nambu) cosmic strings. The model is particularly suited for describing the evolution of small-scale structure on string networks. We discuss model solutions in the extreme limit where the wiggles make up a high fraction of the total energy of the string network (which physically corresponds to the tensionless limit) and also provide a brief discussion of the opposite (linear) limit where wiggles are a small fraction of the total energy. A companion paper will discuss the detailed modelling and scaling behavior of the small-scale wiggles in the general model, together with a basic comparison with numerical simulations.