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Applying the symmetry groups to study the n body problem

Abstract: We introduce an algebra method to study the Newtonian n-body problem. The main idea is considering the symmetry of the central configurations and using the representation theory of groups to simplify the calculation of the eigenvalues. For the 4 body problem, eigenvalues can be solved by simple calculation with trace in the representation. For the 3+1 body problem, we also use this technique to discover the relationship between eigenvalues and the parameter m.