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

Phone: 831-459-2718 Office: Baskin 359B Office Hours: By Appt. Email: lewis@ucsc.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. Fundamental properties of these systems, e.g. conservation of total energy and momentum, or a variational formulation, facilitate the analysis of crucial features of teh dynamics.

Lewis is 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 respected by conventional numerical schemes.  Key features of the dynamics, such as equilibria, separatrices, and periodic orbits, may be lost or artificially introduced unless methods designed to preserve the underlying structures are used. 

Lewis is currently working on the extension of key constructs and results in geometric mechanics to biological systems, particularly biomechanical control systems and population dynamics. The guiding dogma of geometric mechanics - that nature "optimizes" and "balances" - is as relevant to biological processes as it is to physical ones, but the development and analysis of biologically meaningful cost functions requires new insights and techniques.

SELECTED PUBLICATIONS

D. Lewis: Optimal control with moderation incentives, preprint.

D. Lewis, N. Nigam, and P. Olver: Connections for general group actions, Communications in Contemporary Mathematics  7 No. 7 (2005), 341-374.

D. Lewis and P. Olver: Geometric Integration Algorithms on Homogeneous Manifolds, Foundations of Computational Mechanics  2 No. 4 (2002), 363-392.

F. Fasso and D. Lewis: Stability Properties of the Riemann Ellipsoids, Archive for Rational Mechanics and Analysis 158 (2001), 259-292.

D. Lewis and M. Shub: The Distribution of the maximum condition number on great circles through a fixed 2 X; 2 real matrix, Linear Algebra and its Applications 297 (1999), 193-202.

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

D. Lewis: Lagrangian block diagonalization, The Journal of Dynamics and Differential Equations No. 1 (1992), 1-42.

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

D. Lewis and J. C. Simo: Nonlinear stability of rotating pseudo-rigid bodies, The Proceedings of the Royal Society of London 427 (1990), 281-319.

D. Lewis: Nonlinear stability of a circular rotating liquid drop, The Archive for Rational Mechanics and Analysis 106 No. 4 (1989), 287-333.

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