LAGRANGIAN AND HAMILTONIAN-FORMALISM FOR DISCONTINUOUS FLUID AND GRAVITATIONAL-FIELD

The barotropic ideal fluid with step and delta-function discontinuitie s coupled to Einstein's gravity is studied. The discontinuities repres ent star surfaces and thin shells; only nonintersecting discontinuity hypersurfaces are considered. No symmetry (such as, e.g., the spherica l symmetry) is assumed. The symplectic structure as well as the Lagran gian and the Hamiltonian variational principles for the system are wri tten down. The dynamics is described completely by the fluid variables and the metric on the fixed background manifold. The Lagrangian and t he Hamiltonian are given in two forms: the volume form, which is ident ical to that corresponding to the smooth system, but employs distribut ions, and the surface form, which is a sum of volume and surface integ rals and employs only smooth variables. The surface form is completely four or three covariant (unlike the volume form). The spacelike surfa ces of time foliations can have a cusp at the surface of discontinuity . Geometrical meaning of the surface terms in the Hamiltonian is given . Some of the constraint functions that result from the shell Hamilton ian cannot be smeared so as to become differentiable functions on the (unconstrained) phase space. Generalization of the formulas to more ge neral fluid is straightforward.