Einstein's Summation Convention: Questions Answered

[nenyan]

Please see the attached pic.

 

[D H]

The derivative that is being calculated at first is not $\frac{\partial L}{\partial \dot x^1}$. It's $\frac{\partial L}{\partial \dot x^p}$. That p index is what the part in red is addressing.

You miscalculated when you calculated $\frac{\partial L}{\partial \dot x^1}$. You dropped a factor of two in calculating $\frac{\partial}{\partial \dot x^1}g_{11}\dot x^1 \dot x^1$. This should be $2g_{11}\dot x^1$, which means your second batch of stuff in red should be $2g_{11}\dot x^1 + g_{12}\dot x^2 + g_{21}\dot x^2$. This is exactly the same as $g_{l1}\dot x^l + g_{1m}\dot x^m$. Note how this expands upon doing the summation: $g_{l1}\dot x^l + g_{1m}\dot x^m = g_{11}\dot x^1 + g_{21}\dot x^2 + g_{11}\dot x^1 + g_{12}\dot x^2$.

 

[nenyan]

Thank you D H. Your reply is very useful.