  In this paper, we consider a free boundary problem of the incompressible
elatodynamics, a coupling system of the Euler equations for the fluid motion
with a transport equation for the deformation tensor. Under a natural force
balance law on the free boundary with the surface tension, we establish its
well-posedness theory on a short time interval. Our method is the vanishing
viscosity limit by establishing a uniform a priori estimates with respect to
the viscosity. As a by-product, the inviscid limit of the incompressible
viscoelasticity (the system coupling with the Navier-Stokes equations) is also
justified. We point out that based on a crucial new observation of the inherent
structure of the elastic term on the free boundary, the framework here is
established solely in standard Sobolev spaces, but not the co-normal ones used
in \cite{MasRou}.
