D'Alembert Principle and weak solutions of the Euler equations

Let M be a smooth manifold in an Euclidean space; consider the motion of a material point on M in absence of friction. The D'Alembert Principle says that the acceleration vector is orthogonal to the tangent space to M, and this fact defines the point trajectory provided the initial position and velocity are given. The flow of an ideal incompressible fluid can be regarded as a motion of a material point on the set D of volume-preserving diffeomorphisms of M embedded into an (infinite-dimensional) Euclidean space. However, the set D is not a smooth manifold, and it is hardly a manifold in any reasonable sense. We can define a substitute of a tangent space to D at every point, but these planes form a non-smooth and non-integrable distribution which makes the use of the D'Alembert Principle far from obvious. In this talk I propose a way to use this principle to define the trajectory, i.e. the fluid flow. The kinetic energy of the fluid should not be constant; in fact, it may decrease which is characteristic for turbulent motions. However, the accurate formulation of our method requires the tools of Nonstandard Analysis.