I am struggling with seemingly simple algebraic manipulations with expressions containing finite sums. It is a physics based case but the interpretation of terms does not play any role now. Just note that the dot denotes the time derivative.
It starts with
$$ \left[ \sum_{n=1}^N n q_n \cos (n \sigma) \right] \left[ \sum_{n=1}^N n \dot{q}_n \cos (n \sigma) \right] + \left[ \sum_{n=1}^N \dot{q}_n \sin (n \sigma) \right] \left[ \sum_{n=1}^N n^2 q_n \sin (n \sigma) \right] = \ ... $$
There are multiple ways to deal with these expressions. I have chosen a dummy way on purpose since it has the lowest risk of an error:
$$ ... = \sum_{r=1}^N \sum_{s=1}^N r s q_r \dot{q}_s \cos (r \sigma) \cos (s \sigma) + \sum_{r=1}^N \sum_{s=1}^N r^2 q_r \dot{q}_s \sin(r\sigma) \sin(s\sigma) $$
Up to this step I am pretty confident. But in the next one I have made a mistake surely:
$$ ... = \sum_{r=1}^N \sum_{s=1}^N q_r \dot{q}_s \left[ r s \cos (r \sigma) \cos (s \sigma) + r^2 \sin(r\sigma) \sin(s\sigma) \right] $$
I cannot merge the sums since the index choice must remain arbitrary and I have made it bound.
- Am I correct?
- What is a proper remedy? Leaving four independent indices?
I don't see any mistake. You add two products of sums. Each product, when you use the distributive law (first step that you are confident in) will get a double sum that has terms $q_r\dot{q}_s$ with a factor where both $r$ and $s$ vary from 1 to $N$. And why do you think you cannot combine those sums into one, as you did?
It may not make any sense for the physics involved, but mathematically this is sound.
Addition to answer Victor's remark: @Victor: You are right, you could choose $r,s$ the opposite way for the second sum, or could choose $r,s$ for the first sum and $i,j$ for the second sum. Then you could not apply the mathematical formalism that you used.
But the choice of index name is arbitrary, it doesn't matter what you call them, the term $1\times2q_1\dot{q}_2\cos(\sigma)\cos(2\sigma)$ will always be a part of the first sum, and $1^2q_1\dot{q}_2\sin(\sigma)\sin(2\sigma)$ will always be a part of the second sum, and you can always combine them to $q_1\dot{q}_2(2\cos(\sigma)\cos(2\sigma)+\sin(\sigma)\sin(2\sigma))$.
The 'right' choise of index names makes the formalism applicable and leads to the combined formula. It's like choosing the 'right' coordinate system for a mechanical probem (like a tilted plane): Doing so allows the math (like the differential equation from Neton's laws) to be simple and easily solvable. That doesn't mean that using the 'wrong' one leads to a different solution. It just means that the math may be more complicated (like coupled variables) and that you may not be able to solve it with what you know.