The topology of integrable systems with incomplete fields

Abstract

Liouville's theorem holds for Hamiltonian systems with complete Hamiltonian fields which possess a complete involutive system of first integrals; such systems are called Liouville-integrable. In this paper integrable systems with incomplete Hamiltonian fields are investigated. It is shown that Liouville's theorem remains valid in the case of a single incomplete field, while if the number of incomplete fields is greater, a certain analogue of the theorem holds. An integrable system on the algebra sl (3) is taken as an example.Bibliography: 11 titles.

Euler equations on the general linear group, cubic curves, and inscribed hexagons

“…Note that despite the fact that solutions located on R 3 components blow up, the topology of these components is still compatible with the Arnold-Liouville theorem. This phenomenon is explained in the first author's paper [5].…”

Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

“…Nonetheless, these questions can be answered for some systems: see [11]- [13]. We also mention several more general results concerning generalizations of Liouville's theorem to systems with incomplete flows: see [14] and [15].…”