Kinematical symmetries of three-dimensional incompressible flows

The motion of an incompressible fluid in Lagrangian coordinates involves infinitely many symmetries generated by the left Lie algebra of group of volume preserving diffeomorphisms of the three-dimensional domain occupied by the fluid. Utilizing a 1+3-dimensional Hamiltonian setting, an explicit realization of this symmetry algebra is constructed recursively. A dynamical connection is used to split the symmetries into reparametrization of trajectories and one-parameter family of volume preserving diffeomorphisms of the fluid domain. Algebraic structures of symmetries and Hamiltonian structures of their generators are inherited from the same construction. A comparison with the properties of two-dimensional flows is included.