I Einstein summation convention confusion

AI Thread Summary
The discussion centers on the confusion surrounding the Einstein summation convention, particularly whether xixi is equivalent to xi². While xixi represents the sum of the squares of vector components, xi² refers to the square of a single component, leading to potential misunderstanding. Additionally, the conversation addresses the limitations of the summation convention, noting that certain quantities, like the kinetic energy of multiple particles, may not be expressible in this format due to their complexity. The participants emphasize the importance of context in applying the convention, as clarity can vary based on notation. Overall, the Einstein summation convention is not universally applicable in all scenarios, particularly in classical mechanics.
dyn
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Hi
If i have a vector r = ( x1 , x2 , x3) then i can write r2 as xixi where the i is summed over because it occurs twice. Now is xixi the same as xi2 ? I have come across an example where they are used as equivalent but i am confused because xi2 seems to be the square of just one component of r but xi2 also seems to be logically the same as xixi

My other question is ; are there some quantities that cannot be written in summation convention ? Such the kinetic energy of many particles . I have seen it written using sigma notation as the sum over k from 1 to N as mkvkvk but obviously k appears 3 times here. This applies to small oscillations where the rk is differentiated with respect to different variables . Are some quantities impossible to write in summation convention ?

Thanks
 
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My understanding of the Einstein convention is that it would be xixi.
 
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Thanks. My questions are just in reference to classical mechanics so in both questions i have asked all indices are lower indices
 
Use whatever can be read unambiguously without confusing the reader too much. I wouldn't expect the Einstein sum convention in classical mechanics at all, so a footnote or other comment would be useful anyway. Specify how you want to use it there.
 
I think a lot of this depends on context too. If you wrote ##y_i=x_i^2## it's pretty clear you're not summing, and if you write ##y=x_i^2## then you are. Assuming the book doesn't have a typo 😬
 
dyn said:
My other question is ; are there some quantities that cannot be written in summation convention ? Such the kinetic energy of many particles . I have seen it written using sigma notation as the sum over k from 1 to N as mkvkvk but obviously k appears 3 times here. This applies to small oscillations where the rk is differentiated with respect to different variables . Are some quantities impossible to write in summation convention ?
Because the convention assumes the universal quantifier, it can't express the existential quantifier. You can't say: $$\exists i: x_i = y_i$$
 
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