Abstract
When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.
Abstract
The aim of this paper is to study a quasistatic contact problem between an
electro-elastic viscoplastic body with damage and an electrically conductive
foundation. The contact is modelled with an electrical condition, normal
compliance and the associated version of Coulomb’s law of dry friction in
which slip dependent friction is included. We derive a variational
formulation for the model and, under a smallness assumption, we prove the
existence and uniqueness of a weak solution.
The method to be described provides an alternative means of dealing with certain non-standard linear boundary-value problems. It is developed in several applications to the theory of gravity-capillary waves. The analysis is based on a variational formulation of the hydrodynamic problem, being motivated by and extending the original study by Benjamin and Scott [3].
A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.
Variational formulation of the problems of mechanics and its matrix analogy
