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# Brief Summary of Yoshida's Theorem

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 Sk ([11]; p. 128, eq. (1.3)), which are derived from the fact that the trace of some monodromy matrices is >2. For n>2, as shown in [13], non-integrability follows if the parameters

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 .

Next: Non-integrability of the Truncated Up: NON-INTEGRABILITY OF THE MOTION Previous: Introduction

1998-10-24