### Exotic galilean symmetry and non-commutative mechanics.

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We describe some recent results on a specific nonlinear hydrodynamical problem where the geometric approach gives insight into a variety of aspects.

We survey two problems illustrating geometric-topological and Hamiltonian methods in fluid mechanics: energy relaxation of a magnetic field and conservation laws for ideal fluid motion. More details and results, as well as a guide to the literature on these topics can be found in [3].

The Ito equation is shown to be a geodesic flow of ${L}^{2}$ metric on the semidirect product space $\widehat{{\mathrm{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}}^{s}\left({S}^{1}\right)\u2a00{C}^{\infty}\left({S}^{1}\right)}$, where ${\mathrm{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}}^{s}\left({S}^{1}\right)$ is the group of orientation preserving Sobolev ${H}^{s}$ diffeomorphisms of the circle. We also study a geodesic flow of a ${H}^{1}$ metric.