Central Configurations and Total Collisions for Quasihomogeneous n-Body Problems

TL;DR

This paper studies the dynamics of n-body problems with specific potential functions, analyzing central configurations, collisions, and their relationships to special solutions, including new theoretical insights and generalizations.

Contribution

It generalizes Moulton's theorem for central configurations and explores properties of collision orbits in Manev-type potentials, linking configurations to special solutions.

Findings

Generalized Moulton's theorem for central configurations

Qualitative properties of collision orbits in Manev-type case

New relationships between configurations, equilibria, and homothetic solutions

Abstract

We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}$ with $a,b,\alpha,\beta$ constants, $0\le a<b$. To analyze the dynamics of the problem, we first prove some properties related to central configurations, including a generalization of Moulton's theorem. Then we obtain several qualitative properties for collision and near-collision orbits in the Manev-type case $a=1$. At the end we point out some new relationships between central configurations, relative equilibria, and homothetic solutions.