Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid

Abstract

In this chapter we study Lagrangian hydrodynamical systems (LHSs) of an ideal incompressible fluid. These systems turn out to be systems of diffuse matter with a holonomic constraint in the sense of Sect. 5. (To compare, we recall that an LHS of an ideal barotropic fluid considered in Sect. 24.B is an LHS of diffuse matter with an external force.) In Sect. 25 we define LHSs of an ideal incompressible fluid and study some of their general properties. In particular, we show that such an LHS can be described by a C^∞-smooth equation on the integral manifolds of the constraint, even though the Euler equations lose smoothness. This equation is given by the spray of the metric. Sect. 26 contains a construction of a special infinite-dimensional constraint on the group of volume-preserving diffeomorphisms. The LHS of an ideal incompressible fluid on a manifold without boundary subject to this constraint describes the motion of a fluid on a manifold with boundary given beforehand. We continue studying these LHSs in Sect. 27. In particular, applying the results of Sect. 26, we prove a regularity theorem for a flow of an ideal incompressible fluid on a manifold with boundary.