Accessibility navigation


A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

Shepherd, T. G. ORCID: https://orcid.org/0000-0002-6631-9968 (1990) A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids. Journal of Fluid Mechanics, 213 (1). pp. 573-587. ISSN 0022-1120

[img]
Preview
Text - Published Version
· Please see our End User Agreement before downloading.

1MB

It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing.

To link to this item DOI: 10.1017/S0022112090002452

Abstract/Summary

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

Item Type:Article
Refereed:Yes
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
ID Code:32986
Publisher:Cambridge University Press

Downloads

Downloads per month over past year

University Staff: Request a correction | Centaur Editors: Update this record

Page navigation