Abstract

Introduction

A popular approach in the modelling of nonlinear water waves is to make the approximations that the three-dimensional fluid velocity u is irrotational and divergent free, such that u = ∇ϕ and ∇ · u = ∇2ϕ = 0, and that the dynamics is inviscid, such that the dynamics is governed by variational and Hamiltonian dynamics [1, 2]. At least symbolically one can invert this Laplace equation for the interior potential ϕ and reduce the dynamics to the free surface, expressed in terms of the potential ϕs at the free surface and the position of this free surface. For non-overturning waves, this free surface dynamics can be expressed in terms of the water depth h = h(x, y, t) and ϕs(x, y, t) = ϕ(x, y, z = b + h, t) with horizontal coordinates x and y as well as time t. Here the fixed topography is denoted by b = b(x, y). The free surface thus lies at the vertical level z = b(x, y)+h(x, y, t), parametrised by x and y.

One then often considers the initial value problem governed by autonomous Hamiltonian dynamics for h and ϕs with initial conditions h(x, y,0) and ϕs(x, y,0) without any forcing or dissipation. In practical situations, however, waves are generated continuously by wave makers or temporarily by opening a sluice gate, both involving time dependent internal or boundary conditions. This implies that the dynamics is non-autonomous, including explicit dependence of the equations on time. Sometimes, these non-autonomous aspects can be included in the variational principles governing the wave dynamics.

We will therefore start to formulate finite-dimensional variational dynamics in which the variational principle indeed depends explicitly on time. The forced-dissipative nonlinear pendulum with the harmonic oscillator as linearisation is a first example of such a non-autonomous variational principle.