Abstract. The averaging theory of first order is applied to study a generalized Yang-Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the non-integrable classical Yang-Mills Hamiltonian systems, in the sense of Liouville-Arnold, which have the isolated periodic orbits found with averaging theory, can not exist any second first integral of class C 1 . This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion.

In this paper we revisit the classification of the gauge transformations in the Euler top system using the generalized classical Hamiltonian dynamics of Nambu. In this framework the Euler equations of motion are bi-Hamiltonian and SL(2, R) linear combinations of the two Hamiltonians leave the equations of motion invariant, although belonging to inequivalent Lie-Poisson structures. Here we give the explicit form of the Hamiltonian vector fields associated to the components of the angular momentum for every single Lie-Poisson structure including both the asymmetric rigid bodies and its symmetric limits. We also give a detailed classification of the different Lie-Poisson structures recovering all the ones reported previously in the literature.

The super-separability of the three-body inverse-square Calogero systemWe study the motion of three masses in a plane interacting with a central potential proportional to 1/r 2 using the coordinates introduced recently by Piña. We show that this problem with four degrees of freedom ͑three angles and a distance related to the inertia moment of the system in these coordinates͒ is partially separable, and can be reduced to a problem with two degrees of freedom ͑two angles͒ with a new constant of motion. We find a symmetry of reflection ͑an involution͒ for this system and we use the symmetry lines to find periodic orbits in the angular coordinates. These orbits will not be periodic in general on the whole phase space because the coordinate of distance type grows as t when t→ϱ and it is unbounded. However, if the inertia moment of the system remains constant, they will be periodic on the whole phase space.

We apply the averaging theory of second order to study the periodic orbits for a generalized Hénon-Heiles system with two parameters, which contains the classical Hénon-Heiles system. Two main results are shown. The first result provides sufficient conditions on the two parameters of these generalized systems, which guarantee that at any positive energy level, the Hamiltonian system has periodic orbits. These periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy. The second result shows that for the non-integrable Hénon-Heiles systems in the sense of Liouville-Arnol'd, which have the periodic orbits analytically found with averaging theory, cannot exist any second first integral of class C 1 . In particular, for any second first integral of class C 1 , we prove that the classical Hénon-Heiles system and many generalizations of it are not integrable in the sense of Liouville-Arnol'd. Moreover, the tools we use for studying the periodic orbits and the non-Liouville-Arnol'd integrability can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.

Abstract. We apply the averaging theory of first order to study analytically families of periodic orbits for the cored and logarithmic Hamiltoniansandq 2 , which are relevant in the study of the galactic dynamic. We first show, after introducing a scale transformation in the coordinates and momenta with a parameter ε, that both systems give essentially the same set of equations of motion up to first order in ε. Then the conditions for finding families of periodic orbits, using the averaging theory up to firs order in ε, apply equally for both systems in every energy level H = h > 0 with H either H C or H L . We prove the existence of two periodic orbits if q is irrational, for ε small enough, and we give an analytic approximation for the initial conditions of these periodic orbits. Finally, the previous periodic orbits provide information about the non-integrability of the cored and the logarithmic Hamiltonian systems.