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 $t_5$ 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

The SPS as lead-ion accelerator

In 1995 the CERN SPS was used during two months to accelerate fully stripped ions of the Pb208 isotope from the equivalent proton momentum of 13 GeV/c to 400 GeV/c. The radio frequency swing which is needed in order to keep the synchronism during acceleration is too big to have the SPS cavities deliver enough voltage for all frequencies. In a first stage, the beam is accelerated from 13 GeV/c to 26 GeV/c using the fixed frequency mode. During this stage the beam is grouped in four 2msec batches, separated by 3msec holes during which the frequency is changed in order to keep synchronism. At 26 GeV the beams are de-bunched and recaptured in order to fill the 3msec holes. From there on the lead ions are then accelerated up to 400 GeV/c with the normal frequency program. The de-bunching and recapture at 26 GeV improved the effective spill at extraction by a factor of three. Intensities up to 3.9 1010 charges could be obtained at 400 GeV/c. The total efficiency of the two RF captures was 64%