A new formulation of the Lagrangian equations for the evolution of the Rayleigh-Taylor instability in inviscid incompressible fluids is presented. A set of exact coupled-mode equations which govern the evolution of the velocity field and the nonlinear motion of the surface is derived. Unlike the traditional Eulerian mode expansion, which requires an infinitely growing number of modes at the nonlinear stage, the present expansion coverages very rapidly. The use of the formalism for analyzing the nonlinear stage of the instability is demonstrated by analytical and numerical solutions. {copyright} {ital 1996 The American Physical Society.}