Abstract:
We study relative equilibria ($RE$) for the three-body problem
on $\mathbb{S}^2$,
under the influence of a general potential which only depends on
$\cos\sigma_{ij}$ where $\sigma_{ij}$ are the mutual angles
among the masses.
Explicit conditions for
masses $m_k$ and $\cos\sigma_{ij}$
to form relative equilibrium are shown.
Using the above conditions,
we study the equal masses case
under the cotangent potential.
We show the existence of
scalene, isosceles, and equilateral Euler $RE$, and isosceles
and equilateral Lagrange $RE$.
We also show that
the equilateral Euler $RE$ on a rotating meridian
exists for general potential $\sum_{i<j}m_i m_j U(\cos\sigma_{ij})$
with any mass ratios.
Keywords:relative equilibria, Euler and Lagrange configurations.
References