For the curved n-body problem in \(\mathbb {S}^3\), we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium if and only if n is odd and the masses are equal. The equilibrium is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on \(\mathbb {S}^1\) embedded in \(\mathbb {S}^2\). We then study the stability of the associated relative equilibria on \(\mathbb {S}^1\) and \(\mathbb {S}^2\). We show that they are Lyapunov stable on \(\mathbb {S}^1\), they are Lyapunov stable on \(\mathbb {S}^2\) if the absolute value of angular velocity is larger than a certain value, and that they are linearly unstable on \(\mathbb {S}^2\) if the absolute value of angular velocity is smaller than that certain value.

对于\（\ mathbb {S} ^ 3 \）中的弯曲n体问题，我们证明，当且仅当n为奇数且质量相等时，测地线上n个质量的规则多边形配置为平衡。平衡被相对平衡，这发生在一个一参数家族（取决于角速度）相关联\（\ mathbb {S} ^ 1 \）嵌入在\（\ mathbb {S} ^ 2 \） 。然后，我们研究\（\ mathbb {S} ^ 1 \）和\（\ mathbb {S} ^ 2 \）上的相关相对平衡的稳定性。我们证明它们在\（\ mathbb {S} ^ 1 \）上是Lyapunov稳定的，它们在\（\ mathbb {S} ^ 2 \）上是Lyapunov稳定的如果角速度的绝对值大于某个值，并且它们在\（\ mathbb {S} ^ 2 \）上线性不稳定。