Solving Summations: Tips & Tricks for Homework

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The discussion revolves around understanding the transition from the summation notation \sum_{b\neq a} to \sum_{ab} in the context of particle interactions. Participants clarify that this transformation involves taking the product of two summations, where the notation \sum_{ab} is used to simplify expressions while neglecting self-interaction terms. An example with three particles illustrates how the sums can be expanded, highlighting that self-interaction terms like F_{11}, F_{22}, and F_{33} are zero and thus omitted. There is a consensus that while the notation \sum_{ab} is convenient, it may not be strictly accurate without acknowledging the neglect of self-interactions. The conversation emphasizes the importance of clarity in mathematical notation when discussing summations in physics.
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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.
 

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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.
 
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
 
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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.
 
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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!
 
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