Topologically Distinct Collision-Free Periodic Solutions for the N -Center Problem

Roberto Castelli*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


This work concerns the planar N-center problem with homogeneous potential of degree - α (α∈ (1 , 2)). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the N-center problem with α∈ (1 , 2) is proven. In case α= 1 a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.

Original languageEnglish
Pages (from-to)941-975
Number of pages35
JournalArchive for Rational Mechanics and Analysis
Issue number2
Publication statusPublished - 1 Feb 2017


Dive into the research topics of 'Topologically Distinct Collision-Free Periodic Solutions for the N -Center Problem'. Together they form a unique fingerprint.

Cite this