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Canonical variables for steep planar water waves over nonuniform bed

An explicit expression in terms of canonical variables is obtained for the Hamiltonian functional determining the fully nonlinear dynamics of two-dimensional potential flows of an ideal fluid with a free surface over an arbitrary nonuniform depth. The canonically conjugate variables are derived from the previously developed non-canonical conformal description of water waves over a strongly undulating bottom [V. P. Ruban, Phys. Rev. E {\bf 70}, 066302 (2004)]. Also an alternative approach to the problem is discussed, which gives weakly nonlinear Hamiltonian models of different orders.