14,802 research outputs found
Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications
We develop the necessary tools, including a notion of logarithmic derivative
for curves in homogeneous spaces, for deriving a general class of equations
including Euler-Poincar\'e equations on Lie groups and homogeneous spaces.
Orbit invariants play an important role in this context and we use these
invariants to prove global existence and uniqueness results for a class of PDE.
This class includes Euler-Poincar\'e equations that have not yet been
considered in the literature as well as integrable equations like Camassa-Holm,
Degasperis-Procesi, CH and DP equations, and the geodesic equations
with respect to right invariant Sobolev metrics on the group of diffeomorphisms
of the circle
Discrete Euler-Poincar\'{e} and Lie-Poisson Equations
In this paper, discrete analogues of Euler-Poincar\'{e} and Lie-Poisson
reduction theory are developed for systems on finite dimensional Lie groups
with Lagrangians that are -invariant. These discrete
equations provide ``reduced'' numerical algorithms which manifestly preserve
the symplectic structure. The manifold is used as an approximation
of , and a discrete Langragian is
construced in such a way that the -invariance property is preserved.
Reduction by results in new ``variational'' principle for the reduced
Lagrangian , and provides the discrete
Euler-Poincar\'{e} (DEP) equations. Reconstruction of these equations recovers
the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are
naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP
algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is
shown that when , the DEP and DLP algorithms for a particular
choice of the discrete Lagrangian are equivalent to the
Moser-Veselov scheme for the generalized rigid body. %As an application, a
reduced symplectic integrator for two dimensional %hydrodynamics is constructed
using the SU approximation to the volume %preserving diffeomorphism group
of
Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations
We deal with Lagrangian systems that are invariant under the action of a
symmetry group. The mechanical connection is a principal connection that is
associated to Lagrangians which have a kinetic energy function that is defined
by a Riemannian metric. In this paper we extend this notion to arbitrary
Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new
fashion and we show how solutions of the Euler-Lagrange equations can be
reconstructed with the help of the mechanical connection. Illustrative examples
confirm the theory.Comment: 22 pages, to appear in J. Phys. A: Math. Theor., D2HFest special
issu
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
The Maxwell–Vlasov equations in Euler–Poincaré form
Low's well-known action principle for the Maxwell–Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamilton's principle for the Eulerian description of Low's action principle then casts the Maxwell–Vlasov equations into Euler–Poincaré form for right invariant motion on the diffeomorphism group of position-velocity phase space, [openface R]6. Legendre transforming the Eulerian form of Low's action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler–Poincaré equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie–Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell–Vlasov Poisson structure is known, whose ingredients are the Lie–Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born–Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin–Noether theorem for Euler–Poincaré equations and its meaning in the plasma context
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