Integrable Systems Associated to Curves in Flat Galilean and Lorentzian Manifolds

This article examines the relationship between geometric Poisson brackets and integrable systems in flat Galilean, and Lorentz manifolds. First, moving frames are used to calculate differential invariants of curves and to write invariant evolution equations. The moving frames are created to ensure that the Galilean moving frame is the limit of the Lorentz one as c → ∞. Then, associated integrable evolutions and their bi-Hamiltonian structures are found, using the parallelism of Euclidean and Lorentzian cases. The Galilean case is particularly significant because the Galilean group is not semisimple, yet it can be considered as a limit of the (semisimple) Lorentzian case. The Galilean integrable systems and Hamiltonian structures are compared to the c→∞ limit of the Lorentzian ones.