Acta Applicandae Mathematicae 11 (1988), 259-284. 259 © 1988 by Kiuwer Academic Publishers. Origin and Infinity Manifolds for Mechanical Systems with Homogeneous Potentials* ERNESTO A. LACOMBA** Departamento de Matemdticas, Universidad Aut6noma Metropolitana Unidad Iztapalapa. Apdo. Postal 55-534, CP. 09340 Mdxico, D.F. and LUIS A. IBORT Departamento de Ffsica Te6rica, Universidad Complutense, 28040 Madrid, Spain (Received: 16 July 1987; revised: 12 January 1988) Abstract. We study manifolds describing the behavior of motions close to the origin and at infinity of configuration space, for mechanical systems with homogeneous potentials. We find an inversion between these behaviors when the sign of the degree of homogeneity is changed. In some cases, the blow up equations can be written in canonical form, by first reducing to a contact structure. A motivation for the use of blow-up techniques is given, and some examples are studied in detail. AMS subject clanificatiom (1980). 58F05, 70H05, 70H15, 70F35. Key words. Blow up, conservative mechanical system, homogeneous function, Hamiltonian system, canonical transformation, symplectic manifold, contact manifold. 1. Introduction We generalize a McGehee [13] blow up at the origin and a Lacomba and Sim6 [10] blow up at infinity in celestial mechanics for the case of any homogeneous potential. The behavior at infinity for any negative degree of homogeneity has been sketched in the joint paper [10], while Devaney [5] has previously studied the behavior at the origin. For a positive degree of homogeneity, there have been studies of blow up at the origin for zero energy and 2 degrees of freedom by Losco et al. [12]. By looking at apparent difficulties for blowing up the origin for nonzero energy in those examples, one realizes that there should be some inversion exchanging the behaviors of blow ups at the origin and at infinity when the sign of the degree of * Research partially supported by CONACyT (Mexico), under grants PCCBNAL 790178 and PCCBBNA 022553. ** Member of CIFMA (Mexico). On sabbatical leave at the University of Barcelona during the year 1987-88.