Coadjoint orbits and a method of horizontal variations for incompressible fluids (Q1096477)

Two variational principles are formulated; one for a stationary system, and another one for a nonstationary system of Euler equations of motion of an incompressible fluid. Formulations consider the notion of horizontal variations and the notion of coadjoint orbit of Lie groups. The classical Euler equations are considered as a Hamiltonian system on a coadjoint orbit. The appropriate Hamilton principle is not obvious, since the system is not canonical. By using ``horizontal variations'' (which denote a calculus of conditional extrema on an orbit) we obtain a general, variational formulation of Euler-Arnold's equations. In hydrodynamics, the formulation is valid for classical and weak solutions of the Euler equations. On some special orbits (e.g. in the vortex theory) we can expect some simpler, canonical principles.