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

Abstract

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.