On the equilateral triangle solution tothe three-body problem

Abstract

The N-body problem has no general solution for morethan two particles. Various restricted solutions such as the equilateral trianglesolutions of the three-body problem can,however, be found. Here, we discuss howto find and understand such restricted solutions. We dothis using generalized coordinates and the Lagrangianmethod. We note that the restricted solutions can beviewed as solutions arising from the imposition ofconstraints that do not require any constraint forces.We also note that an N-body problem leads to a centralforce problem provided the distances between theparticles are equal. In three dimensions this allows theparticles to be at the vertices of an equilateraltriangle or a regular tetrahedron. We presentcorresponding restricted solutions to the three- andthe four-body problems, respectively.