On the Analyticity of Particle Trajectories in the Ideal Incompressible Fluid

A new proof is given of the fact that the particle trajectories of the ideal incompressible °uid are analytic curves, though the solutions of the Euler equations may have a ¯nite regularity. This is a consequence of a general fact that the geodesic exponential map on the group of volume preserving di®eomorphisms belonging to the Sobolev space is real-analytic. The proof is based on the general properties of holomorphic maps in complex Banach spaces.