Solving Summations: Tips & Tricks for Homework

[CMBR]

Homework Statement

See the attachment, I am stuck as to how the summation sign \sum_{b\neq a}^{} in (2.1.1) ends up as \sum_{ab}^{} in the term with the red dot above (2.1.5).

Homework Equations

The Attempt at a Solution

As I understand you end up taking the product of two summations such that \sum_{a}^{}(\sum_{b\neq a}^{})=\sum_{ab}^{}, but I don't really understand the logic here.

just trying to understand, thanks in advance.

 

[Orodruin]

There is a subindex $a$ missing from what should be $\dot{\bf r}_a$ from equation 2.1.4 and subsequently in $\ddot{\bf r}_a$ in 2.1.5. The rest is just inserting 2.1.1.

 

[CMBR]

yeh I got that but once you insert 2.1.1 i don't get how the summation in front of the red dot term is \sum_{ab}^{} once you sub 2.1.1 you get \sum_{a}^{}(\sum_{b\neq a}^{}), I don't really understand how that works

 

[Orodruin]

You are making a sum of sums. It may help to write the sums out for a small number of particles, let us say 3:
$$
\sum_{a} \sum_{b\neq a} F_{ab} = (F_{12} + F_ {13}) + [F_{21} + F_{23}] + \{F_{31} + F_{21}\}
$$
where the term in () is the term originating from the sum over $b \neq 1$ for $a = 1$, [] for $a = 2$, and {} for $a = 3$. Now $\sum_{ab}$ is a bit of a bastard notation. If assuming that we by this mean $\sum_{a=1}^3 \sum_{b=1}^3$, then we get some additional terms $F_{11} + F_{22} + F_{33}$, but the particles do not exert forces onto themselves so these can be taken to be zero.

 

[CMBR]

Yeh I sort of came to a similar conclusion myself, the issue was that I didn't get why you can use the notation \sum_{ab}^{} if you neglect particle self interactions, the notation in that case is not strictly true then? Wouldn't it be better to keep it in the form \sum_{a}^{}(\sum_{b\neq a}^{}), anyway thanks for the clarifications!