I am currently studying general relativity from James Hartle's book and I have trouble understanding how he goes to equation (5.60) from equation (5.58). It's about the variational principle for free particle motion from special relativity.
Here is the problem. First we have $$L = \biggl[\biggl(\frac{dt}{d\sigma}\biggr)^2 - \biggl(\frac{dx}{d\sigma}\biggr)^2 - \biggl(\frac{dy}{d\sigma}\biggr)^2 - \biggl(\frac{dz}{d\sigma}\biggr)^2\biggr]^\frac{1}{2} \tag{5.57}$$ then, $$-\frac{d}{d\sigma}\biggl(\frac{\partial L}{\partial(dx^\alpha/d\sigma)}\biggr) + \frac{\partial L}{\partial x^\alpha} = 0 .\tag{5.58}$$ And considering that $x^1 = x$ we conclude that $$\frac{d}{d\sigma}\biggl[\frac{1}{L}\frac{dx^1}{d\sigma}\biggr] = 0. \tag{5.60}$$ (Till here I am ok, I got help from previous comments. Now I just need to know how to get through the last step.)
Lastly, knowing that $L= \frac{d\tau}{d\sigma}$ and multiplying (5.60) with $\frac{d\sigma}{d\tau}$ we get $$\frac{d^2x^1}{d\tau^2} = 0.$$
I don't see how we come to the last equation with the suggested multiplication.
