Applying the symmetry groups to study the n body problem

Zhihong Xia*, Tingjie Zhou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We introduce an algebraic method to study local stability in the Newtonian n-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. Using representation theory of finite groups, we explicitly found the eigenvalues of certain 8×8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.

Original languageEnglish (US)
Pages (from-to)302-326
Number of pages25
JournalJournal of Differential Equations
StatePublished - Feb 15 2022


  • Eigenvalues
  • Representation theory
  • Symmetry
  • The Newtonian n-body problem

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


Dive into the research topics of 'Applying the symmetry groups to study the n body problem'. Together they form a unique fingerprint.

Cite this