# Euler-Lagrange Equations for charged particle in Einstein notation

I'm a math guy, just wanting to clarify some notation.

This is the Euler Lagrange equation associated with a charge in an electric field I found in a book. Where $$\phi$$ and $$A$$ are the scalar and vector potential related to Maxwell Equations. I am just curious, this last term, the one with the subscript $$j$$, is this last term suppose to be a sum over $$j=1,2,3$$ of that last quantity, i.e it will be three terms? I just wanted clarification for this, since I'm not sure why this sum of omitted if there should be a summation here. If there is not a summation here. Can someone tell me what $$j$$ is referencing.

I know i is referencing different components of the position vector.

• Yes, it sums over $j = 1, 2, 3$. May 24, 2017 at 2:50
• Thank you for the response. Just curious in physics is this omittance of the sum typical? Does the change in subscript over another variable just imply sum? May 24, 2017 at 2:55
• Yes, the convention is that any index repeated twice is summed. Also, if it's a regular letter (as opposed to Greek) it's over $1, 2, 3$ instead of $0, 1, 2, 3$. May 24, 2017 at 2:56
• The sum is omitted because a lot of equations would become bulky and almost unreadable with all the sums included. Einstein notation is very neat and compact :) May 24, 2017 at 7:46
• Actually, beware, because in Special and General relativity, the implicit sum is only for "up-down" indices in tensors. For example, in $F_{\mu\nu}F^{\mu\nu}$, there is an implicit sum $\sum_{\mu=1}^4\sum_{\nu=1}^4$. But if you write $F_{\mu\nu}F_{\mu\nu}$, there is no implicit sum.
– user154997
May 24, 2017 at 8:05