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We consider Euler equations for potential flow of ideal incompressible fluidwith a free surface and infinite depth in two dimensional geometry. Bothgravity forces and surface tension are taken int account. A time-dependentconformal mapping is used which maps a lower complex half plane of theauxiliary complex variable $w$ into a fluid's area with the real line of $w$mapped into the free fluid's surface. We reformulate the exact Euleriandynamics through a non-canonical nonlocal Hamiltonian structure for a pair ofthe Hamiltonian variables. These two variables are the imaginary part of theconformal map and the fluid's velocity potential both evaluated of fluid's freesurface. The corresponding Poisson bracket is non-degenerate, i.e. it does nothave any Casimir invariant. Any two functionals of the conformal mappingcommute with respect to the Poisson bracket. New Hamiltonian structure is ageneralization of the canonical Hamiltonian structure of Ref. V.E. Zakharov, J.Appl. Mech. Tech. Phys. 9, 190 (1968) which is valid only for solutions forwhich the natural surface parametrization is single valued, i.e. each value ofthe horizontal coordinate corresponds only to a single point on the freesurface. In contrast, new non-canonical Hamiltonian equations are valid forarbitrary nonlinear solutions (including multiple-valued natural surfaceparametrization) and are equivalent to Euler equations. We also consider ageneralized hydrodynamics with the additional physical terms in the Hamiltonianbeyond the Euler equations. In that case we identified powerful reductionswhich allowed to find general classes of particular solutions.

Non-Canonical Hamiltonian Structure and Poisson Bracket for 2D Hydrodynamics with Free Surface