**Next:** Non-integrability
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Very broadly speaking, Ziglin's method [14] takes into consideration an analytic Hamiltonian for which a particular solution describing a curve non-homologous to zero on a certain Riemann surface is selected. Then, the non-integrability depends on the properties of the monodromy group associated to the variational equations along such a solution, and particularly on certain non-resonance conditions.

In the case of a Hamiltonian

where
is homogeneous of an integer degree ,
Yoshida [11] obtained very useful criteria,
based on Ziglin's results, by making use of the existence of a straight-line
solution of the form

where is a solution of and the constant vector is a solution to the algebraic equation .

In a standard way, the variational equations can be transformed into
a Gauss hypergeometric equation, with well known monodromy group and finally,
the non-integrability depends on the eigenvalues of ,
one of them always being equal to *k*-1.

In the planar case, *n*=2, the problem turns out to be non-integrable
if a coefficient referred to as the integrability coefficient, ,
lies in certain non-integrability regions *S _{k}* ([11];
p. 128, eq. (1.3)), which are derived from the fact that the trace of some
monodromy matrices is >2. For

are rationally independent, where the are the eigenvalues of .

The reformulation of these criteria into polar or spherical coordinates
presented in [6] has turned out to be useful
in dealing with orbital problems, since it allows us to simplify the calculations,
sometimes very drastically. In the planar case, the integrability coefficient
in polar coordinates is easily computed through

where
is the potential function, *k* being an integer but ,
and
is a solution to
such that .
This coefficient relates to Yoshida's integrability coefficient through
the equation .