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

$$H=\displaystyle\frac{1}{2}{\bf p}^2+V({\bf q})\, ,$$

where $V({\bf q})$ is homogeneous of an integer degree $k\neq 0,\pm 2$, 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

$${\bf q}={\bf c}\phi(t)\, , \qquad {\bf p}={\bf c}\dot{\phi}(t)\, ,$$

where $\phi(t)$ is a solution of $\,\ddot{\phi}(t)+\phi ^{k-1}=0\,$and the constant vector ${\bf c}$ is a solution to the algebraic equation ${\bf c}=V_{{\bf q}}({\bf c})$.

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 $V_{{\bf qq}}({\bf c})$, 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, $\lambda = \mbox{\rm Tr} V_{{\bf qq}}({\bf c})-(k-1)$, 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 n>2, as shown in [13], non-integrability follows if the parameters

$$\Delta \rho _i =\sqrt{1+\displaystyle\frac{8 k \lambda _i}{(k-2)^2}}$$

are rationally independent, where the $\lambda _i$ are the eigenvalues of $V_{{\bf qq}}({\bf c})$.

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

$$\lambda _p=\displaystyle\frac{W_{\theta\theta }(\theta _{0})}{kW(\theta _{0})}\, ,$$

where $V(r,\theta )=r^kW(\theta )$ is the potential function, k being an integer but $k\neq 0,\pm 2$, and $\theta _0$ is a solution to $W_\theta (\theta )=0$ such that $W(\theta _0)\neq 0$. This coefficient relates to Yoshida's integrability coefficient through the equation $\lambda _p= \lambda -1$.