Einstein summation convention confusion

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Discussion Overview

The discussion revolves around the Einstein summation convention, particularly its application in classical mechanics. Participants explore the equivalence of expressions involving summed indices and the limitations of the convention in representing certain quantities.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions whether the expression xixi is equivalent to xi2, noting that xi2 appears to represent the square of a single component while xixi suggests a summation over components.
  • Another participant asserts that the Einstein convention would indeed be represented as xixi.
  • A participant emphasizes that their questions pertain specifically to classical mechanics, clarifying that all indices in their examples are lower indices.
  • There is a suggestion that clarity in notation is important, particularly in classical mechanics, where the Einstein summation convention may not be expected.
  • One participant points out that context matters in notation, indicating that the expression y_i=x_i^2 is clear in its meaning, while y=x_i^2 could imply summation.
  • Participants discuss the limitations of the summation convention, particularly in expressing certain quantities like kinetic energy for multiple particles, which may require sigma notation due to repeated indices.
  • A later reply mentions that the convention assumes a universal quantifier, making it incapable of expressing existential quantifiers, such as stating the existence of an index satisfying a condition.

Areas of Agreement / Disagreement

Participants express varying views on the equivalence of xixi and xi2, and there is no consensus on whether certain quantities can be represented in summation convention. The discussion remains unresolved regarding the limitations of the convention.

Contextual Notes

Participants note that the application of the Einstein summation convention may depend on the context, particularly in classical mechanics, where clarity in notation is emphasized. There are unresolved questions about the ability to express certain mathematical relationships using the convention.

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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