The Dynamics of Relativistic Membranes I: Reduction to 2-dimensional Fluid Dynamics

We greatly simplify the light-cone gauge description of a relativistic membrane moving in Minkowski space by performing a field-dependent change of variables which allows the explicit solution of all constraints and a Hamiltonian reduction to a $SO(1,3)$ invariant $2+1$-dimensional theory of isentropic gas dynamics, where the pressure is inversely proportional to (minus) the mass-density. Simple expressions for the generators of the Poincar\'e group are given. We also find a generalized Lax pair which involves as a novel feature complex conjugation. The extension to the supersymmetric case, as well as to higher-dimensional minimal surfaces of codimension one is briefly mentioned.