I'm studying for my math physics final tomorrow and I'm going through a derivation done in our book, but I'm stuck on this one step. The derivation is of the geodesic equation using variational calculus (this is done in the Arfken and Weber book, on page 156 if you have it). Anyways, I follow the derivation up to this point:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{1}{2}\int{(\frac{dq^i}{ds}\frac{dq^j}{ds}\partial{g_{ij}}{q^k}-\frac{d}{ds}(g_{ik}\frac{dq^i}{ds}+g_kj\frac{dq^j}{ds}))\delta q^kds}=0[/tex]

since there is an independant variation with the [tex]\delta q^k[/tex], the rest of the integral is zero:

[tex]\frac{1}{2}( \frac{dq^i}{ds} \frac{dq^j}{ds} \frac{\partial{g_{ij}}}{\partial{q^k}} - \frac{d}{ds} (g_{ik} \frac{dq^i}{ds} + g_kj \frac{dq^j}{ds}))=0[/tex]

He then goes on to expand the derivatives of g in the last term:

[tex]\frac{dg_{ik}}{ds}=\frac{\partial{g_{ik}}}{\partial{q^j}}\frac{dq^j}{ds}[/tex]

[tex]\frac{dg_{kj}}{ds}=\frac{\partial{g_{kj}}}{\partial{q^i}}\frac{dq^i}{ds}[/tex]

The next step is the one I don't understand. He gets:

[tex]\frac{1}{2} \frac{dq^i}{ds} \frac{dq^j}{ds} (\frac{\partial {g_{ij}}} {\partial {q^k}} - \frac{\partial {g_{ik}}}{\partial {q^j}} - \frac{\partial {g_{jk}}}{\partial {q^i}})-g_{ik} \frac{d^2q^i}{ds^2}=0[/tex]

I don't know where this last term is coming from. This seems a lot like something my professor would ask on the test, so I would like to understand what is going on but I don't have time to talk to him before the test tomorrow. Can anyone explain where that second derivative is coming from?

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# Geodesic Derivation Question

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