Picard-Vessiot theory and Ziglin's theorem
In 1983 S. L. Ziglin has given a criterium for non-integrability of Hamiltonian systems in terms of a monodromy group of normal and reduced variational equations along an integral curve. The authors of the paper apply the Picard-Vessiot differential Galois theory to linear differential equations of Ziglin's theorem which are linear Hamiltonian equations of Fuchsian type with coefficients in the field \(K\) of meromorphic functions on a compact Riemann surface. They consider the Picard-Vessiot extension \(P = K\langle y_ 1,\dots,y_{2n}\rangle\) of the field \(K\) by means of a fundamental system of solutions \(y_ 1,\dots,y_{2n}\) and the Galois group \(G\) of \(P\) over \(K\) made of all the automorphisms of \(P\) leaving invariant the functions of \(K\). The authors prove that under the hypothesis of Ziglin's theorem the identity component \(G_ 0\) of the Galois group \(G\) is a complete diagonal subgroup of \(\text{Sp}(n,\mathbb{C})\) and the index \([G:G_ 0]\) is finite. Using these results the authors give a full investigation of non- integrability for 2-dimensional Hamiltonian systems, in particular, Lamé and Hénon-Heiles equations.
scientific article; zbMATH DE number 522671