  Many equations that arise in a physical context can be posed in the form of a
Hamiltonian system, meaning that there is a symplectic structure on an
appropriate phase space, and a Hamiltonian functional with respect to which
time evolution of their solutions can be expressed in terms of a Hamiltonian
vector field. It is known from the work of VE Zakharov that the equations for
water waves can be posed as a Hamiltonian dynamical system, and that the
equilibrium solution is an elliptic stationary point. In this article we
generalize the Hamiltonian formulation of water waves by Zakharov to a general
coordinatization of the dynamical free surface, which allows it to apply to
situations that include overturning wave profiles. This answers a question
posed to the author by T.~Nishida during the RIMS Symposium on Mathematical
Analysis in Fluid and Gas Dynamics that took place during July 6 - 8 2016.
