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 the 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.