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Center for Nonlinear Studies |
The way we view hydrodynamics changed forever when Arnold made his revolutionary discovery [1] that the Euler equations for an ideal fluid represent geodesic motion on SDi_ (volume preserving diffeomorphisms) with respect to the L2 norm on the tangent space TSDi_' Xdiv _ SDi_, where Xdiv denotes the divergence-free vector fields. Arnold's famous paper has led to many further developments in continuum dynamics. These developments range, for example, from shallow-water solitons to shape analysis for computational anatomy. The developments of Arnold's discovery that we will discuss in this talk are based on a dual pair of momentum maps that emerge from the Euler-Poincare theory of Lagrangian reduction by symmetry when the symmetry is the Lie group of diffeomorphisms acting on a smooth manifold M, or on a space of smooth embeddings in M [2].
The examples we shall discuss as variations on the theme of dual momentum maps are: 1. Shallow-water solitons called peakons. 2. Jetlets: a new type of coherent particle-like fluid excitation that carries momentum and angular momentum, while preserving its circulation [3]. If time remains, we may also say a few words about the geometry of completely integrable continuum spin chains (strands) and stochastic extensions of these examples.
References
[1] Arnold, V. I., \Sur la geometrie differentielle des groupes de Lie de dimension
infinie et ses applications a l'hydrodynamique des fluides parfaits," Annales
de l'institut Fourier, 6, No. 1, 319{361 (1966).
[2] Holm, D. D., Marsden, J.E., \Momentum Maps and Measure-valued Solutions,"
in: The Breadth of Symplectic and Poisson Geometry, J.E. Marsden
and T.S. Ratiu, Editors, Birkhauser Boston, Boston, MA, 2004, Progr.
Math., 232, pp. 203{235.
[3] C.J. Cotter, D.D. Holm, H.O. Jacobs and D. M. Meier, A jetlet hierarchy
for ideal fluid dynamics. J Phys A To appear. Preprint at arXiv:1402.0086.
Host: Robert Ecke