SQUARE AND RHOMBUS CENTRAL CONFIGURATIONS

1.IntroductionTheproblemofdeterminingthecelltralconfigurationsofnbodymotionsisclassical.Infact,asearlyasinthe18thcelltury,LagrangeandEuleralreadysolvedtheproblemofcentralconfigurationsofthreebodies,namely,threecollinearsolutionsandoneequilateraltrian...

SQUARE AND RHOMBUS CENTRAL CONFIGURATIONS

In this paper, we consider the flat central configurations of bodies using the characteristic set method. We completely solve two special cases of four planets, namely, the square and the rhombus. For the square case,we obtain that a square is a central configuration only in the case where the masses are equal and there exactly are two different square central configurations determined by the mass and the angular velocity; for the rhombus case, we obtain that if a rhombus is a central configuration, then the masses of the diagonal venices must be equal. Furthermore, there are two or three or four different rhombus central configurations determined by the masses and the angular velocity.