Luminosity L(t) for a homologously contracting star I'm dealing with a homologously contracting star with Mass M, Radius R and a gravitational binding energy of 

$E = -a GM^2 / R $ 

(a is a constant). 

I was looking for a way to find an expression for L(t), the luminosity as a function of time. Unfortunately, my attempts using time derivatives have not taken me far. 

Thank you for your help!
Fairly sure this is standard bookwork if you are talking about a PMS star on the Hayashi track.

Differentiate your GPE wrt time, assuming that the mass is constant. Take half of this as the luminosity (via the virial theorem). This gives $L$ in terms of $dR/dt$.

Then if you are on the Hayashi track you can assume surface temperature $T$ is constant and Stefan's law gives you $dR/dt$ in terms of $L$, $T$ and $dL/dt$.

Substitute that in and integrate the resulting differential equation to get $L(t)$.

Using this approach, I managed to get
$$ L = \left(\frac{\alpha GM^2}{6}\right)^{2/3} \left(4\pi \sigma T^4\right)^{1/3} t^{-2/3},$$
where $T$ is the temperature of the Hayashi track in question.

If you don't want to make the Hayashi track assumption, you can say that $L = AM^B T^C$ defines tracks in the HR diagram and this can be used in addition to Stefan's law to eliminate $dT/dt$ and give your result in terms of constants $A$, $B$, $C$. These can come from polytropic theory or by fitting the $L,M,T$ function to numerical calculations. (E.g. see Jackson &amp; Jeffries 2014 https://arxiv.org/abs/1404.0683 ).