Symmetry and Mechanics: Geometric Integration Techniques and Bifurcations of Relative Critical Points

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Mathematical Sciences: Symmetry and Mechanics: Geometric Integration Techniques and Bifurcations of Relative Critical Points DMS-0104292 Lewis The project consists of three main components: geometric integration using Lie group methods, numerical integration of conservative systems, and bifurcation theory of relative critical points. The first component involves the design of efficient, stable geometric integration schemes on Lie groups and homogeneous manifolds. The primary goals are minimization of computational costs with accurate capture of key features and facilitation of the adaptation of existing numerical codes to geometric schemes. The second component addresses the role of approximate or exact preservation of conserved quantities or structures in the numerical integration of conservative systems. Comparisons of the symplectic, energy, and momentum errors to overall algorithm performance, including accuracy of capture of such key features of the underlying system as (relative) equilibria and separatrices, will be carried out for a variety of algorithms, including variational algorithms. The third component addresses the variational characterization of relative equilibria, that is, steady motions, as critical points of appropriate functionals. The notion of a relative critical point unifies and generalizes the two most widely used characterizations; this generalization will facilitate the extension of existing bifurcation and stability results to a wide range of finite and infinite dimensional systems and will provide insights into the geometric structure of sets of relative equilibria. Specific applications of geometric integration methods include the Landau--Lifschitz--Ginzburg (LLG) equations for a micromagnetic field and various mathematical models of muscular hydrostats. The LLG equations model systems such as read/write heads on disk drives, nanocrystralline magnets, and magnetohydrodynamic materials; efficient, accurate numerical algorithms are needed to carry out simulations on the spatial and temporal scales relevant to industrial applications. Muscular hydrostats are structural components of many biological systems, including many kinds of worms, reptile and amphibian tongues, and sea anemones; plausible mathematical models of these systems are needed to test theories of the evolution of such systems. Conservative systems arise throughout physics and mechanics, playing a crucial role in both industrial and theoretical research. Numerical simulations that respect the conservation laws of such systems have been shown to be crucial in several fields, including astrophysics. The bifurcation analyses will be applied to problems in fluid mechanics, elasticity, and stellar dynamics.