Geometrical approach to Hamiltonian fluids

Differential geometry based on the Cartan calculus of differential forms is applied to investigate invariant properties of equations that describe the motion of continuous media. The advantage of geometrical methods is that they provide an universal approach to the problem of invariance in hydrodynamics and magnetohydrodynamics. The main feature of the approach is that the hydrodynamic quantities are considered as geometrical objects and that the notion of invariance is formulated in terms of Lie derivatives. The solutions of the equations for invariant fields can be written in terms of Lagrange variables. This gives a generalized Cauchy formulation. A procedure for the construction of invariant fields using operations of differential geometry is presented. The formalism for finite-dimensional, canonical systems is extended to the case of continuous media. Similar to finite-dimensional systems, Hamiltonian fluids axe defined as those that conserve some exact two-form. It is shown that the equations of motion of charged barotropic fluids and of ideal plasmas belong to this class of equations. New type of scalar invariants for Hamiltonian fluids are constructed. It is the opinion of the authors that the application of these methods will be very fruitfull for the future development of the theory of fluid and plasma dynamics