A system of particles distributed on the surface of a ball, what is the "center of mass" of them on the surface? Suppose a system of $n$ particles distributed on the surface of a ball, what is the "center of mass" of them on the surface?
Does the following optimization problem have an analytical solution?
\begin{equation}
\boldsymbol {P}     =\operatorname*{argmin}_{c\in  {\Omega} } \;\sum _{i=1}^{n}m_{i}  d^2 {\left(c, p_i \right)}  
\end{equation}
Where $\Omega$ is the surface of the ball, $m_i$ is particle $p_i$'s mass.
$d^2 \left(c, p_i \right)$ is the squared distanced from  $c$ to particle $p_i$ on the surface. That is to say, it is the squared great-circle distance.
Suppose we know each particle's coordinate $(x_i,y_i,z_i)$.
 A: The great circle distance between two points $\ c\ $ and $\ p_i\ $ on the surface of a ball is proportional to the angle subtended by the points at centre of the ball—that is, to
$$
\arccos\left(\frac{\langle c,p_i\rangle}{\|c\|\,\|p_i\|}\right)\ .
$$
To simplify matters, choose your unit of distance to be the radius of the ball, so that $\ \|p_i\|=1\ $ for all $\ i\ $, and the condition $\ c\in\Omega\ $ is equivalent to $\ \|c\|^2=1\ $.  Your optimisation problem then reduces to
\begin{align}
&\min_{c\in\mathbb{R}^3}\sum_{i=1}^n m_i\arccos\big(\langle c,p_i\rangle\big)^2\\
&\text{subject to: }\ \|c\|^2=1\ .
\end{align}
The Lagrange condition for the minimum is
$$
{\Large\sum_{i=1}^n}\frac{m_i\arccos\big(\langle c,p_i\rangle\big) p_i}{\sqrt{1-\langle c,p_i\rangle^2}}-\lambda c=0\ .
$$
From this and the condition $\ \|c\|^2=1\ $, we get
$$
\lambda^2={\Large\sum_{i=1}^n}{\Large\sum_{j=1}^n}\frac{m_im_j\arccos\big(\langle c,p_i\rangle\big)\arccos\big(\langle c,p_j\rangle\big)\langle p_i,p_j\rangle}{\sqrt{1-\langle c,p_i\rangle^2}\sqrt{1-\langle c,p_j\rangle^2}}\ .
$$
Since we're looking for the minimum of the objective, the optimal $\ c\ $ must be in the opposite direction to its gradient, so \begin{align}
\lambda&=-\sqrt{{\Large\sum_{i=1}^n}{\Large\sum_{j=1}^n}\frac{m_im_j\arccos\big(\langle c,p_i\rangle\big)\arccos\big(\langle c,p_j\rangle\big)\langle p_i,p_j\rangle}{\sqrt{1-\langle c,p_i\rangle^2}\sqrt{1-\langle c,p_j\rangle^2}}}\ .
\end{align}
If we take $\ w_i=\langle c,p_i\rangle\ $ as a set of $\ n\ $ unknowns in the above equations, they must satisfy the following $\ n\ $ simultaneous non-linear equations
$$
w_j=\lambda(w)^{-1}{\Large\sum_{i=1}^n}\frac{m_i\arccos\big(w_i\big) \langle p_i,p_j\rangle}{\sqrt{1-w_i^2}}\ .
$$
I doubt if there's any simple expression for the  solution.  I expect the best you'd be able to do is solve them numerically for any given values of the $\ m_i\ $ and $\ p_i\ $.  Alternatively, it's probably going to be simpler to solve the optimisation problem directly with a gradient projection method.
