On the singularities of generalized solutions to $n$--body type problems

Abstract

The validity of Sundman-type asymptotic estimates for collision solutions is established for a wide class of dynamical systems with singular forces, including the classical $N$--body problems with Newtonian, quasi--homogeneous and logarithmic potentials. The solutions are meant in the generalized sense of Morse (locally --in space and time-- minimal trajectories with respect to compactly supported variations) and their uniform limits. The analysis includes the extension of the Von Zeipel's Theorem and the proof of isolatedness of collisions. Furthermore, such asymptotic analysis is applied to prove the absence of collisions for locally minimal trajectories.