On variational formulations for steady water waves
type: Item
instance of: scholarly article
title: On variational formulations for steady water waves
zbMATH Open document ID: 0888.76008
author: Mark D. Groves
author: John F. Toland
published in: Archive for Rational Mechanics and Analysis
publication date: 1998-05-13T00:00:00Z
review text: 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 …
reviewed by: V. V. Shelukhin
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76M30
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 1045414
zbMATH Keywords: gravity
zbMATH Keywords: surface tension
zbMATH Keywords: boundary value problem
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: symplectic 2-form
zbMATH Keywords: trajectories of Hamiltonian vector field
zbMATH Keywords: kinematic boundary conditions
zbMATH Keywords: dynamic boundary condition
zbMATH Keywords: Lagrangian formalism
zbMATH Keywords: convex duality theory
MaRDI profile type: Publication
full work available at URL: s002050050028
OpenAlex ID: W2021702070
DOI: 10.1007/S002050050028
description: scientific article; zbMATH DE number 1045414