## ABSTRACT

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 .