14,971 research outputs found
Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation
We study the dynamics of measure-valued solutions of what we call the EPDiff
equations, standing for the {\it Euler-Poincar\'e equations associated with the
diffeomorphism group (of or an -dimensional manifold )}.
Our main focus will be on the case of quadratic Lagrangians; that is, on
geodesic motion on the diffeomorphism group with respect to the right invariant
Sobolev metric. The corresponding Euler-Poincar\'e (EP) equations are the
EPDiff equations, which coincide with the averaged template matching equations
(ATME) from computer vision and agree with the Camassa-Holm (CH) equations in
one dimension. The corresponding equations for the volume preserving
diffeomorphism group are the well-known LAE (Lagrangian averaged Euler)
equations for incompressible fluids. We first show that the EPDiff equations
are generated by a smooth vector field on the diffeomorphism group for
sufficiently smooth solutions. This is analogous to known results for
incompressible fluids--both the Euler equations and the LAE equations--and it
shows that for sufficiently smooth solutions, the equations are well-posed for
short time. In fact, numerical evidence suggests that, as time progresses,
these smooth solutions break up into singular solutions which, at least in one
dimension, exhibit soliton behavior. With regard to these non-smooth solutions,
we study measure-valued solutions that generalize to higher dimensions the
peakon solutions of the (CH) equation in one dimension. One of the main
purposes of this paper is to show that many of the properties of these
measure-valued solutions may be understood through the fact that their solution
ansatz is a momentum map. Some additional geometry is also pointed out, for
example, that this momentum map is one leg of a natural dual pair.Comment: 27 pages, 2 figures, To Alan Weinstein on the occasion of his 60th
Birthda
Variational Principles for Lagrangian Averaged Fluid Dynamics
The Lagrangian average (LA) of the ideal fluid equations preserves their
transport structure. This transport structure is responsible for the Kelvin
circulation theorem of the LA flow and, hence, for its convection of potential
vorticity and its conservation of helicity.
Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational
framework that implies the LA fluid equations. This is expressed in the
Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated
for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure
The Euler-Poincaré Equations in Geophysical Fluid Dynamics
Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold:
1. Euler-Poincaré equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained;
2. an abstract Kelvin-Noether theorem is derived for such systems.
By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincaré form and discuss their corresponding Kelvin-Noether theorems and potential
vorticity conservation laws. The various levels of GFD approximation are related by substituting a sequence of velocity decompositions and asymptotic expansions into Hamilton's principle for the Euler equations of a rotating
stratified ideal incompressible fluid. We emphasize that the shared properties of this sequence of approximate ideal GFD models follow directly from their Euler-Poincaré formulations. New modifications of the Euler-Boussinesq
equations and primitive equations are also proposed in which nonlinear dispersion adaptively filters high wavenumbers and thereby enhances stability and regularity without compromising either low wavenumber behavior or geophysical balances
An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow
In this paper we consider the Hamiltonian formulation of the equations of
incompressible ideal fluid flow from the point of view of optimal control
theory. The equations are compared to the finite symmetric rigid body equations
analyzed earlier by the authors. We discuss various aspects of the Hamiltonian
structure of the Euler equations and show in particular that the optimal
control approach leads to a standard formulation of the Euler equations -- the
so-called impulse equations in their Lagrangian form. We discuss various other
aspects of the Euler equations from a pedagogical point of view. We show that
the Hamiltonian in the maximum principle is given by the pairing of the
Eulerian impulse density with the velocity. We provide a comparative discussion
of the flow equations in their Eulerian and Lagrangian form and describe how
these forms occur naturally in the context of optimal control. We demonstrate
that the extremal equations corresponding to the optimal control problem for
the flow have a natural canonical symplectic structure.Comment: 6 pages, no figures. To appear in Proceedings of the 39th IEEEE
Conference on Decision and Contro
Forward velocity effects on fan noise and the influence of inlet aeroacoustic design as measured in the NASA Ames 40 x 80 foot wind tunnel
The inlet radiated noise of a turbofan engine was studied. The principal research objectives were to characterize or suppress such noise with particular regard to its tonal characteristics. The major portion of this research was conducted by using ground-based static testing without simulation of aircraft forward speed or aircraft installation-related aeroacoustic effects
The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories
We study Euler–Poincaré systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract Kelvin–Noether theorem for these equations. We also explore their relation with the theory of Lie–Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa–Holm equations, which have many potentially interesting analytical properties. These equations are Euler–Poincaré equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather thanL^2
The fish fauna of the Iwokrama Forest
Fishes were collected from the rivers in and around the Iwokrama Forest during January-February and November-December 1997. Four hundred species of fish were recorded from forty families in ten orders. Many of these fishes are newly recorded from Guyana and several are thought to be endemic. The number of species recorded for the area is surprising given the low level of effort and suggests that this area may be particularly important from a fish diversity perspective. This paper focuses on species of particular interest from a management perspective including those considered economically important, rare or endangered. The paper is also the basis for developing fisheries management systems in the Iwokrama Forest and Rupununi Wetlands
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