60 research outputs found
Damping of quasi-2D internal wave attractors by rigid-wall friction
The reflection of internal gravity waves at sloping boundaries leads to
focusing or defocusing. In closed domains, focusing typically dominates and
projects the wave energy onto 'wave attractors'. For small-amplitude internal
waves, the projection of energy onto higher wave numbers by geometric focusing
can be balanced by viscous dissipation at high wave numbers. Contrary to what
was previously suggested, viscous dissipation in interior shear layers may not
be sufficient to explain the experiments on wave attractors in the classical
quasi-2D trapezoidal laboratory set-ups. Applying standard boundary layer
theory, we provide an elaborate description of the viscous dissipation in the
interior shear layer, as well as at the rigid boundaries. Our analysis shows
that even if the thin lateral Stokes boundary layers consist of no more than 1%
of the wall-to-wall distance, dissipation by lateral walls dominates at
intermediate wave numbers. Our extended model for the spectrum of 3D wave
attractors in equilibrium closes the gap between observations and theory by
Hazewinkel et al. (2008)
On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian
It is shown that a class of dynamical systems (encompassing the one recently
considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both
quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the
separability of these systems; the second one is obtained trough a non
canonical map whose form is directly suggested by the associated Nijenhuis
tensor.Comment: 11 pages, AMS-LaTex 1.
Optimum control strategies for maximum thrust production in underwater undulatory swimming
Fish, cetaceans and many other aquatic vertebrates undulate their bodies to
propel themselves through water. Numerous studies on natural, artificial or
analogous swimmers are dedicated to revealing the links between the kinematics
of body oscillation and the production of thrust for swimming. One of the most
open and difficult questions concerns the best kinematics to maximize this
later quantity for given constraints and how a system strategizes and adjusts
its internal parameters to reach this maximum. To address this challenge, we
exploit a biomimetic robotic swimmer to determine the control signal that
produces the highest thrust. Using machine learning techniques and intuitive
models, we find that this optimal control consists of a square wave function,
whose frequency is fixed by the interplay between the internal dynamics of the
swimmer and the fluid-structure interaction with the surrounding fluid. We then
propose a simple implementation for autonomous robotic swimmers that requires
no prior knowledge of systems or equations. This application to aquatic
locomotion is validated by 2D numerical simulations
Non-symplectic symmetries and bi-Hamiltonian structures of the rational Harmonic Oscillator
The existence of bi-Hamiltonian structures for the rational Harmonic
Oscillator (non-central harmonic oscillator with rational ratio of frequencies)
is analyzed by making use of the geometric theory of symmetries. We prove that
these additional structures are a consequence of the existence of dynamical
symmetries of non-symplectic (non-canonical) type. The associated recursion
operators are also obtained.Comment: 10 pages, submitted to J. Phys. A:Math. Ge
Quasi-BiHamiltonian Systems and Separability
Two quasi--biHamiltonian systems with three and four degrees of freedom are
presented. These systems are shown to be separable in terms of Nijenhuis
coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with
an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis
coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May
1997
The quasi-bi-Hamiltonian formulation of the Lagrange top
Starting from the tri-Hamiltonian formulation of the Lagrange top in a
six-dimensional phase space, we discuss the possible reductions of the Poisson
tensors, the vector field and its Hamiltonian functions on a four-dimensional
space. We show that the vector field of the Lagrange top possesses, on the
reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set
of separation variables for the corresponding Hamilton-Jacobi equation.Comment: 12 pages, no figures, LaTeX, to appear in J. Phys. A: Math. Gen.
(March 2002
Quantization of Nonstandard Hamiltonian Systems
The quantization of classical theories that admit more than one Hamiltonian
description is considered. This is done from a geometrical viewpoint, both at
the quantization level (geometric quantization) and at the level of the
dynamics of the quantum theory. A spin-1/2 system is taken as an example in
which all the steps can be completed. It is shown that the geometry of the
quantum theory imposes restrictions on the physically allowed nonstandard
quantum theories.Comment: Revtex file, 23 pages, no figure
Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
We discuss the Boussinesq system with stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.Comment: 20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published
in Theor. Math. Phy
Quantized W-algebra of sl(2,1) and quantum parafermions of U_q(sl(2))
In this paper, we establish the connection between the quantized W-algebra of
and quantum parafermions of that a
shifted product of the two quantum parafermions of
generates the quantized W-algebra of
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