Helicity invariants in 3D: kinematical aspects

The Euler equation for an incompressible fluid is analysed to obtain a symplectic set-up for the helicity conservation law. This analysis is modified to more general dynamical equations of incompressible fluids including the equations of barotropic fluids, superconductivity and magnetohydrodynamics. Dynamics of magnetic and cross helicities are studied. Starting from the description of motion, the underlying symmetry principle for generalized helicity conservations is shown to be the kinematical one. The connection between Lagrangian and Eulerian conservation laws for helicity densities turns out to be the same as the conformal equivalence of a Poisson bracket algebra to infinitely many local Lie algebras of functions.