The equations of motion for an isentropic elastic material give rise to additional “conservation laws” that are not linearly related to the original system of equations. At the same time, the natural “entropy function” for this system is the mechanical energy, which is not convex. In 1985 Dafermos presented his theory of “involutions” which used the extra conservation laws to compensate for the lack of convexity of the mechanical energy. This resulted in a new method for proving the short time existence of classical solutions.

Recently we developed an alternative theory. After representing the continuity equations as a set of three exact differential forms dx₁, dx₂, and dx₃, we find that the extra conservation laws can be derived using higher-order differential forms dx_i∧dx_j and dx₁∧dx₂∧dx₃. When the stored energy function is polyconvex, adding the extra conservation laws to the original equations of motion results in a symmetric hyperbolic system---in which case the existence theory for short-time, classical solutions fits into a standard theory.

We present both theories, and compare them.