Follow along at http://star-www.st-and.ac.uk/~hz4/gr/GRlec4+5+6.pdf and go to PDF page 9 or page 44 of the "slides." I'm trying to see how to go from the first to the third line. If we write the free particle Lagrangian and use q^i-dot and q^j-dot as the velocities and metric g_ij, how is it we cancel out the factor of 1/2 when when take the q^l-dot derivative? It looks like more than just a simple change of variables from j to l. If instead we took dL/dq^j-dot I'd expect to get 0.5m x g_ij x q^i-dot . I don't think I can assume I'm only taking derivatives when i=j, so how does this work?