Extremal Properties and Hamiltonian Structure of the Euler Equations

  • T. G. Shepherd
Part of the NATO ASI Series book series (NSSE, volume 218)


The Euler equations describing perfect-fluid motion represent a Hamiltonian dynamical system. The Hamiltonian framework implies, via Noether’s theorem, that symmetries are connected with extremal properties. For example, steady solutions are conditional extrema of the energy under smooth perturbations that preserve vortex topology, as pointed out originally by Arnol’d (1966a). In the three-dimensional context such isovortical perturbations consist of those that maintain the strength and topology of vortex tubes; in the two-dimensional context they consist of those that maintain the vorticity of each fluid element, which is a considerably stronger constraint. Just as with temporal symmetry, solutions that are independent of a spatial coordinate are conditional extrema of the associated momentum invariant under isovortical perturbations; and steadilytranslating solutions are conditional extrema of a linear combination of the energy and the momentum. This connection between symmetries and extremal properties is exploited to particular effect in Arnol’d’s (1966c) nonlinear stability theorems and their recent extensions and applications. Finally, a class of algorithms is described that can find true energy and energy-momenta extrema (under isovortical perturbations) of the two- and three-dimensional Euler equations, when they exist.


Euler Equation Poisson Bracket Nonlinear Stability Steady Solution Hamiltonian Structure 
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© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • T. G. Shepherd
    • 1
  1. 1.Department of PhysicsUniversity of TorontoTorontoCanada

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