Equal masses results for choreographies $n$-body problems

Abstract

We prove that equally spaced choreography solutions of a large class of $n$-body problems including the classical $n$-body problem and a subset of quasi-homogeneous $n$-body problems, have equal masses if the dimension of the space spanned by the point masses is $n-1$, $n-2$, or, if $n$ is odd, if the dimension is $n-3$. If $n$ is even and the dimension is $n-3$, then all masses with an odd label are equal and all masses with an even label are equal. Additionally, we prove that the same results hold true for any solution of an $n+1$-body problem for which $n$ of the point masses behave like an equally spaced choreography and the $n+1$st point mass is fixed at the origin. Furthermore, we deduce that if the curve along which the point masses of a choreography move has an axis of symmetry, the masses have to be equal if $n=3$ and that if $n=4$, if three of the point masses behave as stated and the fourth mass is fixed at a point, the masses of the first three point masses are all equal. Finally, we prove for the $n$-body problem in spaces of negative constant Gaussian curvature that if $n<6$, $n\neq 4$, equally spaced choreography solutions have to have equal masses, and for $n=4$ the even labeled masses are equal and the odd labeled masses are equal and that the same holds true for the $n$-body problem in spaces of positive constant Gaussian curvature, as long as the point masses do not move along a great circle. Additionally, we show that these last two results are also true for any solution to the $n+1$-body problem in spaces of negative constant Gaussian curvature and the $n+1$-body problem in spaces of positive constant Gaussian curvature respectively, for the case that $n$ of the point masses behave like an equally spaced choreography and the $n+1$st is fixed at a point.