Differential Galois Theory and Non-integrability of Hamiltonian Systems

abelian action algebraic algebraic group analytic appear apply assume belong bundle canonical Chapter coefficients complex compute condition connection consider contained coordinates Corollary corresponding defined definition denote derivative differential equation differential Galois group element equilibrium point equivalent Example existence fact field finite Furthermore Galois group germ give given global group G Hamiltonian system Hence holomorphic identity component independent infinity integral curve invariant involutive isomorphism Lame Lemma Lie algebra linear manifold matrix meromorphic monodromy group natural necessary neighborhood non-integrability normal obtain orbit origin parameters particular periodic Poisson polynomial possible potential proof Proposition proved rational recall reduction relation remark resp respectively result Riemann surface Section singular points situation solutions structure studied subgroup subset symplectic Theorem theory trivial values variables variational equation vector vector bundle vector field write zero Ziglin