论文标题
编舞的相等质量结果$ n $ - 身体问题
Equal masses results for choreographies $n$-body problems
论文作者
论文摘要
我们证明，在包括经典$ n $的问题（包括经典$ n $ body问题）的大型$ n $机体问题的舞蹈编排解决方案中，如果点质量为$ n-1 $，$ n-2 $，或者，如果$ n $ n $ n n off nigemension $ n-3 $ n-3 $ n-3 $ n-3 $ n-3 $ n-3 $ n-3 $ n-3 $ n $ n-3 $。如果$ n $均匀，并且尺寸为$ n-3 $，那么所有具有奇数标签的质量都相等，并且所有标签的质量都相等。此外，我们证明，对于$ n+1 $ body问题的任何解决方案，点质量的$ n $的表现就像均等编排，而$ n+1 $ st点质量的质量固定在原点上。此外，我们推断出，如果沿着编舞移动的点质量的曲线具有对称性的轴，则如果$ n = 3 $，质量必须相等，如果$ n = 4 $，则三个点质量的行为表现为所述，并且第四个质量的质量是固定的。 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群众不会沿着大圆圈移动。此外，我们表明，对于分别为正恒定高斯曲率和$ n+1 $ 1 $的$ n+1 $ body问题的任何解决方案的最后两个结果也是正确的，分别是正常恒定高斯曲率的空间中的$ n+1 $ - 体问题，因为点质量的$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n+n+n+n+1 $ s均可固定。
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.
