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I'm trying to understand the classical derivation of Friedmann equation but I'm missing one step.
So, I start with accelerations, where $a$ is a scale factor
$\ddot{a}=-\frac{GM}{a^{2}}$
$\ddot{a}=-\frac{4\pi G}{3}\rho a$
Now multiply both sides by $\dot{a}$, and here is the question: Why LHS is
$\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$
Not sure I see the problem.
$\frac{1}{2}\frac{{\rm d}}{{\rm d}t}\big((\dot{a})^{2}\big) = \frac{1}{2}*2*\dot{a}*\frac{{\rm d}}{{\rm d}t}\big(\dot{a}\big) = \dot{a}\ddot{a}$. Maybe you don't recognize the notation - a dot is often used to denote a time derivative. Thus two dots indicates a double time derivative.
