Open problems in the theory of finite-dimensional integrable systems and related fields
Two conferences in 2013 addressed the topic of ``Geometry and Dynamics of Integrable Systems''. This paper collects, describes, and puts into context open problems in six different areas. First the authors note that the term ``integrability'' has come to mean many different but related things in mathematical physics and geometry. In a broad sense, integrability for dynamical systems means the existence of a foliation of phase space by leaves invariant under the flow. The most frequently used sense, however, is that of Liouville integrability with explicit integration by quadrature. The areas where open problems are provided are the following: (1) Liouville integrable systems coming from two compatible Poisson structures (bi-Hamiltonian systems); (2) extension of the idea of integrability to non-Hamiltonian systems; (3) induced action derived from integrable systems; (4) integrability for ordinary differential equations and its relation to classic Liouville integrability; (5) dynamical complexity; and (6) integrability for nonholonomic systems. The conference participants provided more than two dozen conjectures and open problems.
scientific article; zbMATH DE number 6376737
scientific article