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

Published online by Cambridge University Press:  26 April 2006

Theodore G. Shepherd
Theodore G. Shepherd
Affiliation:
Department of Physics, University of Toronto, Toronto M5S 1A7, Canada
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Abstract

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 et al. 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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 213 , April 1990 , pp. 573 - 587
Copyright
© 1990 Cambridge University Press

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