Understanding Centre of Mass Motion: E + $\sum_{i < j}(F_j_i + F_j_j)$

[ElDavidas]

I was looking over my notes for centre of mass for a system and it says:

$$c = \frac {1} {M} \sum_{i} m_i\ddot{r}_i = \sum_{i}(E_i + \sum_{j \neq i}F_i_j)$$

where M is the total mass of the system.

Then it considers the centre of Mass in motion:

$$M \ddot{c} = \sum_i m_i \ddot{r}_i = \sum_{i}(E_i + \sum_{j \neq i}F_i_j)$$

$$= \sum_{i}E_i + \sum_{j \neq i}F_i_j = E + \sum_{i < j}(F_j_i + F_j_j)$$

$$= E$$

The thing is, I don't understand the line:

$$E + \sum_{i < j}(F_j_i + F_j_j)$$

and how it comes about (especially the i < j) part.

Any help would be grateful!

 

[Doc Al]

Could there be a typo? I'd think that:
$$\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)$$

(And this term disappears due to Newton's 3rd law.)

 

[ElDavidas]

$$\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)$$

Is this something you just need to know? It suddenly appears in my Uni notes without any explanation.

 

[Doc Al]

I'm not sure what you're asking. Are you asking "Why is that true?" or "How am I supposed to know that?"

 

[ElDavidas]

I'm not sure what you're asking. Are you asking "Why is that true?" or "How am I supposed to know that?"

What I mean is why is this true? Can this be shown from previous statements regarding the interaction of forces. I don't understand where

$$\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)$$

is coming from.

 

[Doc Al]

The truth of this has nothing to do with physics; it's just a mathematical truism. To see that it's true, make up a small example (where i,j < 5, say) and confirm that both sides are equal.