48 research outputs found
Birkhoffian formulation of the dynamics of LC circuits
We present a formulation of general nonlinear LC circuits within the
framework of Birkhoffian dynamical systems on manifolds. We develop a
systematic procedure which allows, under rather mild non-degeneracy conditions,
to write the governing equations for the mathematical description of the
dynamics of an LC circuit as a Birkhoffian differential system. In order to
illustrate the advantages of this approach compared to known Lagrangian or
Hamiltonian approaches we discuss a number of specific examples. In particular,
the Birkhoffian approach includes networks which contain closed loops formed by
capacitors, as well as inductor cutsets. We also extend our approach to the
case of networks which contain independent voltage sources as well as
independent current sources. Also, we derive a general balance law for an
associated "energy function".Comment: 26 pages, 2 figures. Z. Angew. Math. Phys. (ZAMP), accepted for
publicatio
Nonlinear Two-Dimensional Water Waves with Arbitrary Vorticity
We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for the dynamics of the point vortex and the time evolution of the free surface variables
Travelling wave solutions for some two-component shallow water models
21 pages, 13 figures, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceIn the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyse the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models.In particular, we show the existence of new type of solutions
Variational derivation of two-component Camassa-Holm shallow water system
By a variational approach in the Lagrangian formalism, we derive the
nonlinear integrable two-component Camassa-Holm system (1). We show that the
two-component Camassa-Holm system (1) with the plus sign arises as an
approximation to the Euler equations of hydrodynamics for propagation of
irrotational shallow water waves over a flat bed. The Lagrangian used in the
variational derivation is not a metric.Comment: to appear in Appl. Ana
A geometric Birkhoffian formalism for nonlinear RLC networks
The aim of this paper is to give a formulation of the dynamics of nonlinear
RLC circuits as a geometric Birkhoffian system and to discuss in this context
the concepts of regularity, conservativeness, dissipativeness. An RLC circuit,
with no assumptions placed on its topology, will be described by a family of
Birkhoffian systems, parameterized by a finite number of real constants which
correspond to initial values of certain state variables of the circuit. The
configuration space and a special Pfaffian form, called Birkhoffian, are
obtained from the constitutive relations of the resistors, inductors and
capacitors involved and from Kirchhoff's laws. Under certain assumptions on the
voltage-current characteristic for resistors, it is shown that a Birkhoffian
system associated to an RLC circuit is dissipative. For RLC networks which
contain a number of pure capacitor loops or pure resistor loops the Birkhoffian
associated is never regular. A procedure to reduce the original configuration
space to a lower dimensional one, thereby regularizing the Birkhoffian, it is
also presented. In order to illustrate the results, specific examples are
discussed in detail.Comment: 30 pages, 2 figure