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 .
