ABSTRACT
As above, let v = sgradH be an integrable Hamiltonian system restricted
to the compact isoenergy surface Q
, and let W
be its marked molecule.
Consider an arbitrary edge e of the molecule W
. Recall that it represents
a one-parameter family of tori. Suppose that, on some Liouville torus from this
family, we have chosen and xed an arbitrary basis in its fundamental group,
i.e., a pair of cycles (; ). According to the Liouville theorem, the trajectories
of the Hamiltonian system on the torus are windings (rational or irrational). This
means that there exists a coordinate system
('
mod 2; '
mod 2)
on the torus in which v is straightened and takes the form
v = a
@
@'
+ b
@
@'
:
Moreover, the coordinate lines of this coordinate system f'
= constg and
f'
= constg are homologous to the basis cycles and , respectively.
