A New Topological Invariant of Hamiltonian Systems of Liouville-Integrable Differential Equations. An Invariant Portrait of Integrable Equations and Hamiltonians

Abstract

The author has discovered a topological invariant which characterizes Hamiltonian systems of Liouville-integrable differential equations. This chapter is devoted to topological characteristics of Liouville-integrable Hamiltonians. In particular, important ideas of symplectic topology are developed which were first suggested in papers by Novikov, Arnold, Gel’fand, Faddeev, Smale, Moser, and Kozlov. For simplicity we will consider only the case of equations on four-dimensional symplectic manifolds, although some of the results can be extended to a multidimensional case. The construction of the new invariant is based on the “Morse-type” theory for integrable systems of differential equations which was developed by the author in [282]–[284] (and in Chapter 2 of the present book). More precisely, we mean surgery on level surfaces for Bott integrals defined on isoenergy surfaces of integrable systems. Developing these results, we suggest a new topological invariant of integrable systems of differential equations: graph Γ, two-dimensional surface P² and embedding k: Γ → P²; in the non-resonance case all these geometrical objects do not depend on the choice of additional integral and describe, consequently, the integrable case (Hamiltonian) itself. It turns out that ths invariant can be effectively calculated. As an example, we calculate it for some classical cases of integrability of equations of motion of a heavy rigid body (the cases of Kovalevskaya, Goryachev-Chaplygin, a gyrostat).