Non-integrability of certain Hamiltonian systems. Applications of the Morales-Ramis differential Galois extension of Ziglin theory
Volume 58 / 2002
Banach Center Publications 58 (2002), 139-150
MSC: Primary 37J30; Secondary 70F20, 34Mxx
DOI: 10.4064/bc58-0-10
Abstract
The aim of this paper is to present two examples of non academic Hamiltonian systems for which the Morales-Ramis theory can be applied effectively. First, we investigate the Gross-Neveu system with $n$ degrees of freedom. Till now it has been proved that this system is not integrable for $n=3$. We give a simple proof that it is not completely integrable for an arbitrary $n\geq 3$. Our second example is a natural generalisation of the Jacobi problem of a material point moving on an ellipsoid. We formulate sufficient conditions for its non-integrability.
