# Einstein summation convention confusion

• I
• dyn
In summary, the conversation discusses the use of the Einstein sum convention in classical mechanics and whether certain quantities, such as kinetic energy, can be expressed in summation notation. It is noted that the convention assumes the universal quantifier and cannot express the existential quantifier. There is also a mention of using footnotes or comments to clarify usage of the convention.

#### dyn

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

• Delta2
My understanding of the Einstein convention is that it would be xixi.

• dextercioby
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: $$\exists i: x_i = y_i$$