Editing Center of mass (section)

===Barycentric coordinates===
{{Further|Barycentric coordinate system}}
The coordinates '''R''' of the center of mass of a two-particle system, ''P''<sub>1</sub> and ''P''<sub>2</sub>, with masses ''m''<sub>1</sub> and ''m''<sub>2</sub> is given by
<math display="block"> \mathbf{R} = {{m_1 \mathbf{r}_1 + m_2\mathbf{r}_2} \over m_1 + m_2}.</math>

Let the [[percentage]] of the total mass divided between these two [[particle]]s vary from 100% ''P''<sub>1</sub> and 0% ''P''<sub>2</sub> through 50% ''P''<sub>1</sub> and 50% ''P''<sub>2</sub> to 0% ''P''<sub>1</sub> and 100% ''P''<sub>2</sub>, then the center of mass '''R''' moves along the line from ''P''<sub>1</sub> to ''P''<sub>2</sub>.  The percentages of mass at each point can be viewed as projective coordinates of the point '''R''' on this line, and are termed [[Barycentric coordinate system|barycentric coordinates]]. Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively.

==={{anchor|Cluster straddling}}Systems with periodic boundary conditions===
For particles in a system with [[periodic boundary conditions]] two particles can be neighbours even though they are on opposite sides of the system. This occurs often in [[molecular dynamics]] simulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate, ''x'' and ''y'' and/or ''z'', as if it were on a circle instead of a line.{{sfn|Bai|Breen|2008}} The calculation takes every particle's ''x'' coordinate and maps it to an angle,
<math display="block">\theta_i = \frac{x_i}{x_\max} 2 \pi </math>
where ''x''<sub>max</sub> is the system size in the ''x'' direction and <math>x_i \in [0, x_\max)</math>. From this angle, two new points <math>(\xi_i, \zeta_i)</math> can be generated, which can be weighted by the mass of the particle <math>x_i</math> for the center of mass or given a value of 1 for the geometric center:
<math display="block">\begin{align}
    \xi_i &= \cos(\theta_i) \\
  \zeta_i &= \sin(\theta_i)
\end{align}</math>

In the <math>(\xi, \zeta)</math> plane, these coordinates lie on a circle of radius 1. From the collection of <math>\xi_i</math> and <math>\zeta_i</math> values from all the particles, the averages <math>\overline{\xi}</math> and <math>\overline{\zeta}</math> are calculated.

<math display="block">\begin{align}
    \overline{\xi} &= \frac 1 M \sum_{i=1}^n m_i \xi_i, \\
  \overline{\zeta} &= \frac 1 M \sum_{i=1}^n m_i \zeta_i,
\end{align}</math>
where {{mvar|M}} is the sum of the masses of all of the particles.

These values are mapped back into a new angle, <math>\overline{\theta}</math>, from which the ''x'' coordinate of the center of mass can be obtained:

<math display="block">\begin{align}
  \overline{\theta} &= \operatorname{atan2}\left(-\overline{\zeta}, -\overline{\xi}\right) + \pi \\
       x_\text{com} &= x_\max \frac{\overline{\theta}}{2 \pi}
\end{align}</math>

The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using [[cluster analysis]] to "unfold" a cluster straddling the periodic boundaries. If both average values are zero, <math>\left(\overline{\xi}, \overline{\zeta}\right) = (0, 0)</math>, then <math>\overline{\theta}</math> is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, their ''x'' coordinates are mathematically identical in a [[periodic boundary conditions#Practical implementation: continuity and the minimum image convention|periodic system]].