Nonintegrability of the Dragt–Finn model of magnetic confinement: a Galoisian-group approach

We prove rigorously that the Hamiltonian H of a model of magnetic confinement of Dragt and Finn (DF-model) is nonintegrable. More precisely, we show that for certain nonzero values M of the component M3 of the angular momentum vector along the rotational symmetry axis there does not exist a real analytic function in involution with and functionally independent of H and M3 which has a meromorphic extension to a suitable region in the complexified phase space of H. For M≠0, let KM be the Hamiltonian of two degrees of freedom obtained by reducing H with respect to M3 and K̂M its holomorphic extension to the complexified phase space of KM. For each such M, we show that the differential Galois group (DGG) of the normal variational equation (NVE) of a suitable complexified nonequilibrium curve to K̂M is not of Ziglin type. By a result implicit in recent work of Morales-Ruiz and Ramis which follows more directly from results explicitly stated by Churchill, this entails that K̂M is nonintegrable in a suitable sense, from which the nonintegrability of H follows. We point out the advantage of using this approach to prove nonintegrability in cases such as the DF-model, in which the NVE considered is embedded into a non-Fuchsian equation.