# Symmetry in n-body problem via group representations

@inproceedings{Zhou2021SymmetryIN, title={Symmetry in n-body problem via group representations}, author={Tingjie Zhou and Zhihong Xia}, year={2021} }

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. We… Expand

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SHOWING 1-10 OF 16 REFERENCES

Convex central configurations for the n-body problem☆

- Mathematics
- 2004

Abstract We give a simple proof of a classical result of MacMillan and Bartky (Trans. Amer. Math. Soc. 34 (1932) 838) which states that, for any four positive masses and any assigned order, there is… Expand

Periodic Solutions of an N-Body Problem

- Mathematics
- 2004

This thesis develops methods to identify periodic solutions to the n-body problem by representing gravitational orbits with Fourier series. To find periodic orbits, a minimization function was… Expand

Introduction to Hamiltonian Dynamical Systems and the N-Body Problem

- Mathematics, Physics
- 1991

This book gives a systematic grounding in the theory of Hamiltonian differential equations from a dynamical systems point of view. It develops a solid foundation for students to read some of the… Expand

Measure of degenerate relative equilibria. I

- Mathematics
- 1976

Relative equilibria of three positive masses always correspond to the well known central configurations found by Euler and Lagrange-the collinear and equilateral triangle solutions of the three body… Expand

Four-body co-circular central configurations

- Mathematics
- 2012

We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central… Expand

Finiteness of relative equilibria of the four-body problem

- Mathematics
- 2006

We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on… Expand

On the role and the properties ofn body central configurations

- Mathematics
- 1980

The role central configurations play in the analysis ofn body systems is outlined. Emphasis is placed on collision orbits, expanding gravitational systems, andn body ‘zero radial velocity’ surfaces.… Expand

Central configurations with many small masses

- Mathematics
- 1991

Abstract By using the method of analytical continuation, we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n. It turns out that the numbers… Expand

Finiteness of central configurations of five bodies in the plane

- Mathematics
- 2012

We prove there are finitely many isometry classes of planar central configurations (also called relative equilibria) in the Newtonian 5-body problem, except perhaps if the 5-tuple of positive masses… Expand

REPRESENTATION THEORY FOR FINITE GROUPS

- 2014

We cover some of the foundational results of representation theory including Maschke’s Theorem, Schur’s Lemma, and the Schur Orthogonality Relations. We consider character theory, constructions of… Expand