The sum of all reciprocal forces in a system of particles is always zero

up:: Newton’s Third Law

Given a system of particles $\{m_i\}$, with reciprocal forces $\{{\vec{F}}_{i\to j}{\}}_{i,j\ne i}$, we have that these forces can be seen as an antisymmetrical matrix

$$A=\left(\begin{array}{cccc}0& F_{1\to 2}& \dots & F_{1\to n}\\ -F_{1\to 2}& 0& \dots & F_{2\to n}\\ ⋮& ⋮& \ddots & ⋮\\ -F_{1\to n}& -F_{2\to n}& \dots & 0\end{array}\right)$$

The sum of these reciprocal forces is the sum of all elements of $A$,

$$\sum _{\begin{array}{c}i,j\\ i\ne j\end{array}}^n{F}_{i\to j}=\sum _{\begin{array}{c}i,j\\ i\ne j\end{array}}^n{A}_{ij}=0$$

which has to be $0$.

Corollaries

• Linear Momentum Conservation Theorem, which states that the Center of Mass behaves like a particle under the influence of (the sum of) all external forces