Integrable systems associated to curves in flat Galilean and Minkowski spaces

Abstract

This article examines the relationship between geometric Poisson brackets and integrable systems in flat Galilean and Minkowski spaces. First, moving frames are used to calculate differential invariants of curves and to write invariant evolution equations. The Galilean moving frame is the limit of the Minkowski one as c ! 1. Then, associated integrable evolutions and their bi-Hamiltonian structures are found, using the parallelism of Euclidean and Minkowski cases. The Galilean case is significant because its group is not semisimple, yet it can be considered as a limit of the (semisimple) Minkowski case. The Galilean integrable systems and Hamiltonian structures are compared to the c ! 1 limit of the Minkowski ones.

Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

“…On the other hand, it is known that a lot of completely integrable systems are described as bi-Hamiltonian systems, from which the existence of many first integrals can be deduced (Magri's theorem [22,27]). In this context, many of motions of curves as above have been studied from the viewpoint of bi-Hamiltonian systems recently [1,2,3,4,5,6,7,8,21,23,24,31]. The purpose of this paper is to construct a multi-Hamiltonian structure associated to the higher KdV flows on each level set of Hamiltonian functions in a geometric way (Theorem 7).…”

Moving frames, Geometric Poisson brackets and the KdV-Schwarzian evolution of pure spinors

“…The recent literature in this subject is extensive, with many authors finding many different geometric realizations and linking curve evolutions, Hamiltonian structures and background geometry to integrable systems, using many different points of view. See for example [2], [1], [8], [9], [16], [20], [21], [26], [25], [28], [39], [27], [37], [40], [41], a list that it is not meant to be, by any means, exhaustive.…”