A Note on the Non-Integrability of some Hamiltonian Systems with a Homogeneous Potential Juan J. Morales-Ruiz Departament de Matem`atica Aplicada II Universitat Polit` ecnica de Catalunya Pau Gargallo 5, E-08028 Barcelona, Spain E-mail: morales@ma2.upc.es Jean Pierre Ramis Laboratoire Emile Picard Universit´ e Paul Sabatier 118, route de Narbonne Toulouse, Cedex, France E-mail: ramis@picard.ups-tlse.fr March 2, 2001 Abstract We obtain a non-integrability result on Hamiltonian Systems with a homogeneous potential with an arbitrary number of degrees of freedom which generalizes a Yoshida’s Theorem [7]. Except for the cases when the degree of homogeneity of the potential is equal to two or minus two, only a discrete set of families of these type of potentials are compatible with the complete integrability condition. We illustrate this result with two examples: the collinear problem of three bodies and a highly symmetrical family introduced by Umeno ([6]). 1 Introduction The purpose of this note is to give a simple non-integrability criterion for complex Hamil- tonian Systems with homogeneous potentials, i.e., of the type H (x, y)= T + V = 1 2 (y 2 1 + ... + y 2 n )+ V (x 1 , ..., x n ), where V is a homogeneous function of integer degree k. We consider this as a first (non completely academic) application in order to test the results of our previous paper [4]. In this way, we improve some Yoshida’s results even for two degrees of freedom (see [7]) and we avoid the arithmetical problems related with the non-resonance assumptions in Ziglin’s Theorem or its generalizations, that are exclusively based on an analysis of the monodromy group of the variational equations (see [6]). As two concrete examples we study the collinear homogeneous problem of three parti- cles, studied by Yoshida in [7], and the n-degrees of freedom system with potential V = 1 s  x s i 1 x s i 2 ··· x s i r , 1