  We consider models for elastic liquids, such as solutions of flexible
polymers. They introduce a relaxation time $\lambda$ into the system, over
which stresses relax. We study the kinematics of the problem, and clarify the
relationship between Lagrangian and Eulerian descriptions, thereby showing
which polymer models correspond to a nonlinear elastic deformation in the limit
$\lambda\rightarrow\infty$. This allows us to split the change in elastic
energy into reversible and dissipative parts, and thus to write an equation for
the total energy, the sum of kinetic and elastic energies. As an illustration,
we show how the presence or absence of an elastic limit determines the fate of
an elastic thread during capillary instability, using novel numerical schemes
based on our insights into the flow kinematics.
