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Register a new userThe Euler-Poincare theory of Metamorphosis
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In the pattern matching approach to imaging science, the process of ``metamorphosis is template matching with dynamical templates. Here, we recast the metamorphosis equations of into the Euler-Poincare variational framework of and show that the metamorphosis equations contain the equations for a perfect complex fluid cite{Ho2002}. This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids. After developing the general theory, we reinterpret various examples, including point set, image and density metamorphosis. We finally discuss the issue of matching measures with metamorphosis, for which we provide existence theorems for the initial and boundary value problems.
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We study Euler-Poincare systems (i.e., the Lagrangian analogue of Lie-Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler-Poincare equations for a parameter dependent Lagrangian by using a variational principle of Lagrange dAlembert type. Then we derive an abstract Kelvin-Noether theorem for these equations. We also explore their relation with the theory of Lie-Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; so it does not produce a corresponding Euler-Poincare system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler-Poincare systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa-Holm equations, which have many potentially interesting analytical properties. These equations are Euler-Poincare equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather than L^2.
This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $alpha$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $alpha$. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnolds theorem), but with respect to a right invariant $H^1$ metric instead of the $L^2$ metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {it even in the presence of boundaries}.
The Green Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler Poincare theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.
A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by a recurrence relation that follows from the Cauchy invariants formulation of the Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more - and occasionally much more - efficient and less prone to instability than Eulerian Runge-Kutta methods, and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algorithm. We also develop tools of analysis adapted to the Cauchy-Lagrangian method, such as the monitoring of the radius of convergence of the time-Taylor series. Certain other fluid equations can be handled similarly.
It is shown that applying manifold learning techniques to Poincare sections of high-dimensional, chaotic dynamical systems can uncover their low-dimensional topological organization. Manifold learning provides a low-dimensional embedding and intrinsic coordinates for the parametrization of data on the Poincare section, facilitating the construction of return maps with well defined symbolic dynamics. The method is illustrated by numerical examples for the Rossler attractor and the Kuramoto-Sivashinsky equation. For the latter we present the reduction of the high-dimensional, continuous-time flow to dynamics on one- and two two-dimensional Poincare sections. We show that in the two-dimensional embedding case the attractor is organized by one-dimensional unstable manifolds of short periodic orbits. In that case, the dynamics can be approximated by a map on a tree which can in turn be reduced to a trimodal map of the unit interval. In order to test the limits of the one-dimensional map approximation we apply classical kneading theory in order to systematically detect all periodic orbits of the system up to any given topological length.
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