Impulse, Flow Force and Variational Principles

Abstract

This essay is a broad commentary on the pivotal notion of impulse conservation and its ramifications in theories of non-dissipative continuous systems. For such systems the dynamical equations typically admit a Hamiltonian formulation, which may be evident at once or may be found by various strategies; and then conservation laws for impulse are linked, through Noether's theorem, to spatial symmetries of the mathematical model. The primary example is that when a Hamiltonian continuous system is uniform (translation invariant) in the x-direction there is a conservation law It + Sx = 0, in which the net impulse density I may be an actual momentum density but is commonly not so. The flux S, called flow force, has many interesting properties, particularly concerning the variational characterization of steady motions; and the ideas in question can be focused more generally on the invariant S¯ = S−cl for steady translational wave motions with phase velocity c. The attributes of impulse and flow forçe presented by Hamiltonian systems of non-linear PDEs with a single space variable are reviewed in Section 2, supplementing which and highlighting connections with later topics Section 3 deals with an exemplary pseudo-differential equation. The general possibilities presented by the more difficult case of PDEs with several space variables are reviewed in Section 4. The remaining three Sections explore particular applications to fluid mechanics. In Section 5 several problems of vorticity dynamics are treated; Hamiltonian representations are found and their consequences examined for planetary waves, for waves in density-stratified fluids and for waves in axisymmetric swirling flows. In Section 6 various novel aspects of non-linear water-wave theory-are discussed; and in Section 7 Clebsch transformations of hydrodynamic problems are reappraised from the present standpoint.