On variational formulations for steady water waves
The paper is concerned with the Hamiltonian formalism for the two-dimensional steady water-wave problem modelling the flows of inviscid, irrotational fluid in a domain of finite or infinite depth, bounded above by a free surface and driven by the effects of gravity and surface tension. The flows are stationary relative to a uniformly translated frame of reference. The boundary value problem is treated as a dynamical system and it is shown that the differential equations and the nonlinear free-boundary conditions can be obtained from a trajectory of a particular Hamiltonian system. The latter is a triple \((M, \Omega , H)\), where \(M\) is a manifold, \(H:M\to \mathbb{R} \) is a continuously-differentiable function called the Hamiltonian, and \(\Omega\) is a symplectic 2-form. The form \(\Omega\) is an assignment of a bounded alternating 2-linear map \(\Omega |_{m}:(TM|_{m})^2 \to \mathbb{R}\) to each point \(m\in M\), where \(TM|_{m}\) is the tangent space to \(M\) at \(m\). Each Hamiltonian system has an associated Hamiltonian vector field \(v_H \) defined by the formula \(dH|_m (v|_m )=\) \(\Omega |_m (v_H |_m ,v|_m )\) for all \(v|_m \in TM|_m \) where \(dH|_m \) is an element of \(T^{\ast}M|_m \), the cotangent space to \(M\) at \(m\), i.e. \(dH|_m \) is a 1-form defined on \(M\). Hamilton's equations for the Hamiltonian system \((M, \Omega , H)\) are the differential equations \(\gamma ^{'}=\) \(v_H |_{\gamma (x)}\) which determine the trajectories \(\gamma :(a,b)\to M\) of the Hamiltonian vector field. For the flows with surface tension no boundary conditions is imposed a priori to define \(M\). It is explained that the domain of \(v_H \) consists of points of \(M\) that have extra regularity and satisfy natural boundary conditions. The kinematic boundary conditions emerge as a consequence of the definition of \(v_H \), and the dynamic boundary condition follows from Hamiltonian equations. In the absence of surface tension, a Hamiltonian system is proposed in such a way that the dynamic free-boundary condition emerges as a criterion for membership of the domain of \(v_H \). The Lagrangian formalism is also developed and is shown to be equivalent to the Hamiltonian one by using the convex duality theory.
trajectories of Hamiltonian vector field
scientific article; zbMATH DE number 1045414