Origin and infinity manifolds for mechanical systems with homogeneous potentials
As a particular case of the application of blow up procedures to vector fields, they have led to the understanding of singularities due to either total collapse or escape of particles for the differential equations of celestial mechanics. The authors generalize this situation to mechanical systems where the potential is any homogeneous function, instead of the gravitational potential (which is homogeneous of degree -1). One defines essentially two general blow up transformations, but for some negative energy cases the blow up is defined for each particular problem. These transformations define on any energy level an origin and an infinity manifold, which describe the behavior of solutions approaching the origin or escaping to infinity in the configuration space. There is an inversion between the behaviors at the origin and at infinity when the sign of the degree of homogeneity of the potential is changed. Local behavior is determined at most by the sign of the energy. Of the two general blow ups, one is a direct generalization of that one first defined by \textit{R. McGehee} [Invent. Math. 27, 191-227 (1974; Zbl 0297.70011)] in celestial mechanics. It is shown here that these blow up equations can be written in canonical form, by first reducing to a contact structure. In the first sections of the paper, motivation of the blow up concept from algebraic geometry is given, and detailed examples of mechanical systems are considered. The global flow in one of them was analyzed lately by \textit{P. Atlea} [The charged isosceles 3-body problem, preprint (rev.)]. A related problem was studied by \textit{E. Lacomba} and \textit{F. Peredo} [Escape-equilibrium solutions in the repulsive coulombian isosceles 3-body problem, preprint (rev.)].
scientific article; zbMATH DE number 4074005
scientific article