Abstract:

The topic of this talk are non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. After recalling the basic theoretical results a general methodology is presented to deal with these problems with algebraic/analytic tools. Several examples will be presented to illustrate the methods, like a family of Hamiltonian systems which require the use of order $k$ variational equations, for arbitrary values of $k$, to prove non-integrability. Using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter for which the lack of integrability cannot be decided by using first order variational equations. In a similar way the degenerate cases of the Swinging Atwood's Machine are studied. For quite general problems the required analytical estimates can be unfeasible. Then a numerical method is introduced to give strong evidence of non-integrability, by using jet transport procedures along arbitrary complex paths to arbitrary order. The talk will be concluded by several comments on the meaning of the non-integrability from a dynamical points of view, depending on the problem.

This is a summary of joint works with J.J. Morales, J-P. Ramis and R. Martínez.