Guy Casale

Suites de Godbillon–Vey et intégrales premières

Editions scientifiques et mElsevier SAS Godbillon-Vey sequences and first integrals Abstract. We prove that a 1-form on a zero characteristic differential field admits a fir st integral whith a special type of transcendance (Riccati type) if only if it admits a Godbillon-Vey sequence of length 3 on this field. c 2002 Acaddes sciences/ ´ Editions scientifiques et m´ edicales

Integrability of Hamiltonian systems with homogeneous potentials of degree zero

We derive necessary conditions for integrability in the Liouville sense of classical Hamiltonian systems with homogeneous potentials of degree zero. We obtain these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution View the MathML source of nonlinear equation gradV(d)=d. We prove that when the system is integrable the Hessian matrix V″(d) has only integer eigenvalues and is diagonalizable.

Dynamics of Rational Symplectic Mappings and Difference Galois Theory

In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.