DEBRA LEWIS Professor of Mathematics B.A., University of California, Berkeley Ph.D., University of California, Berkeley Course Page

Phone: 831-459-2969 Office: Baskin 359B Office Hours: By Appt. Email: lewis_at_ima_dot_umn_dot_edu

RESEARCH INTERESTS

Debra Lewis's research focuses on geometric mechanics, particularly Hamiltonian and Lagrangian systems with symmetry. Inviscid fluids, hyperelastic materials, and systems of coupled rigid bodies are a few important examples of Hamiltonian and Lagrangian systems. The properties of these systems, including conservation of total energy, angular or linear momentum, and related quantities, makes possible the rigorous, detailed analysis of key quantitative and qualitative features of the dynamics.

Much of Lewis's research is directed toward the efficient, explicit analysis of steady motions, that is, trajectories that coincide with the orbit of a one-parameter subgroup of the symmetry group of the system. Variational characterizations of steady motions date back to the nineteenth century, but the development of methods suitable for the analysis of the complex systems used to model current scientific applications is the subject of active research. Techniques from modern geometric mechanics and bifurcation theory can be used to extend classical techniques to very general classes of conservative systems.

Lewis is also interested in the design of algorithms for the numerical integration of conservative systems. Symmetries of mechanical systems and the associated conservation laws, such as conservation of linear and angular momentum, are typically not present in numerical simulations of these systems. This failure to respect the symmetries and conservation laws of the original system can lead to significant qualitative errors in simulations, particularly long-time simula-tions. Key features, such as equilibria, separatrices, and periodic orbits, may be ÔlostÕ or artificially introduced. Algorithms preserving the symmetries and one or more of the invariants of a conservative system appear to have some advantages over traditional numerical schemes. Lewis is currently developing several classes of conserving algorithms and investigating their performance.

SELECTED PUBLICATIONS

D. Lewis: Conservative and approximately conservative algorithms on manifolds. To appear.

D. Lewis: Stacked Lagrange tops. J. Nonlinear Sci. 8, 63-102 (1998).

D. Lewis and J. C. Simo: Conserving algorithms for the N-dimensional rigid body. Fields Inst. Comm. 10. 121-139 (1996).

D. Lewis and T. Ratiu: Rotating n-gon/kn-gon vortex configurations. J. Nonlinear Sci. 6, 385-414 (1996).

D. Lewis and J. C. Simo: Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups. J. Nonlinear Sci. 4. 253-299 (1994).

D. Lewis: Bifurcation of liquid drops. Nonlinearity 6 (4) (1993).

D. Lewis, J. Marsden, and J. C. Simo: Stability of relative equilibria I. The reduced energy-momentum method. Arch. Rational Mech. Anal. (1991), 115:15-59.

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