arXiv subjects

D. D. Holm

Publications and source records attributed to D. D. Holm.

41 records · Page 3Linked to original sources

arXiv1999

Lagrangian Reduction, the Euler--Poincaré Equations, and Semidirect Products

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given.

chao-dyn↗

arXiv1999

A connection between the Camassa-Holm equations and turbulent flows in channels and pipes

In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant $α$ version of the Camassa-Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order $α$ distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale $α$ is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant $α$ region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that $α$ decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply $α$ is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the Camassa-Holm equations provide a promising theoretical framework from which to understand some turbulent flows.

chao-dyn↗

arXiv1999

Integrable vs Nonintegrable Geodesic Soliton Behavior

We study confined solutions of certain evolutionary partial differential equations (pde) in 1+1 space-time. The pde we study are Lie-Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler-Poincare equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from the L2 norm of the velocity. These pde possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call ``pulsons.'' We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. In contrast, head-on antisymmetric collisons of pulsons tend to form singularities.

solv-int↗

arXiv1998

The Euler-Poincare Equations and Semidirect Products with Applications to Continuum Theories

We study Euler-Poincare 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-Poincare 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; so it does not produce a corresponding Euler-Poincare system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler-Poincare 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-Poincare equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather than L^2.

chao-dyn↗

arXiv1998

The Maxwell-Vlasov equations in Euler-Poincare 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-Poincare form for right invariant motion on the diffeomorphism group of position-velocity phase space, 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-Poincare 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-Poincare equations and its meaning in the plasma context.

chao-dyn↗