Order of indices using index notation

Question

I'm trying to solve a problem in Lagrangian mechanics involving index notation. I'm wondering if the expression $$ A_{ij}\dot{q^i}q^j = A_{ji}\dot{q^j}q^i $$ is true. Our professor skipped over this notation convention and I think the problem is made much easier if it is true.

Thanks in advance for any help.

Answer 1

If you have trouble thinking about this sort of thing, take it in stages:

$$A_{ij}\dot{q}^iq^j=A_{kl}\dot{q}^kq^l=A_{ji}\dot{q}^jq^i,$$where at the first equals I impose $i,\,j\mapsto k,\,l$ while at the second I impose $k,\,l\mapsto j,\,i$.

Answer 2

Yes, it is the same. Indices don't carry any meaning, they are just labels. As an example

$$ \sum_i a_i b^i = \sum_p a_p b^p $$

or in Einstein's notation

$$ a_i b^i = a_p b^p $$

So, coming back to your example

$$ A_{ij}\dot{q}^i q^j = A_{m n}\dot{q}^m q^n = A_{j i}\dot{q}^j q^i $$