  We consider Euler equations for potential flow of ideal incompressible fluid
with a free surface and infinite depth in two dimensional geometry. Both
gravity forces and surface tension are taken int account. A time-dependent
conformal mapping is used which maps a lower complex half plane of the
auxiliary 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 Eulerian
dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of
the Hamiltonian variables. These two variables are the imaginary part of the
conformal map and the fluid's velocity potential both evaluated of fluid's free
surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not
have any Casimir invariant. Any two functionals of the conformal mapping
commute with respect to the Poisson bracket. New Hamiltonian structure is a
generalization of the canonical Hamiltonian structure of Ref. V.E. Zakharov, J.
Appl. Mech. Tech. Phys. 9, 190 (1968) which is valid only for solutions for
which the natural surface parametrization is single valued, i.e. each value of
the horizontal coordinate corresponds only to a single point on the free
surface. In contrast, new non-canonical Hamiltonian equations are valid for
arbitrary nonlinear solutions (including multiple-valued natural surface
parametrization) and are equivalent to Euler equations. We also consider a
generalized hydrodynamics with the additional physical terms in the Hamiltonian
beyond the Euler equations. In that case we identified powerful reductions
which allowed to find general classes of particular solutions.
