Section 10.1. General Scheme for Topological Analysis of the Liouville Foliation
A.V. Bolsinov, A.T. Fomenko · 2004
Chapter 10 Methods of Calculation of Topological Invariants of Integrable Hamiltonian Systems 10.1. GENERAL SCHEME FOR TOPOLOGICAL ANALYSIS OF THE LIOUVILLE FOLIATION 10.1.1. Momentum Mapping Let v = sgradH be an integrable Hamiltonian system on a four-dimensional symplectic manifold M . We assume that the Hamiltonian H is given in an explicit way, as well as the additional integral f . It should be noted that the integral f is not uniquely dened and can be replaced by an arbitrary function of f and H . We recall that, in the non-resonant and non-degenerate case, the topology of the Liouville foliation does not depend on the specic choice of the integral f . In this case, the molecules corresponding to two dierent Bott integrals f and f will coincide. We may use this fact by choosing possibly simplest integral f . The recommendation is that, among dierent possible integrals, we choose a function f which has the least number of critical points. For example, sometimes, to make f better, it is useful to extract the root: p f .