I was trying to work out the differential equation for a simple pendulum, and got a sign mistake along the way (among other things I guess). Here's my reasoning:
assuming that the friction forces are negligible, one may use the conservation of energy principle. When $\theta(t)=\theta_0$ (position 1), we have
$E_{mech_1}=U_1$ $\quad$ (since $K_1=0$)
and that at the equilibrium point (position 2)
$E_{mech_2}=K_2$ $\quad$ (since $U_2=0$)
and therefore
$K_2=U_1 \iff \frac{1}{2}mL^2\dot{\theta}^2\color{red}{\boldsymbol{-}}mgL(1-\cos\theta)=0.$
It does however make sense to me that (ref.)
$E_{mech}=U+K$
but I don't really see my mistake. Is it my interpretation of the potential $U$?
Write the equation as $$ K_2=U_1 \iff \frac{1}{2}mL^2\dot{\theta_2}^2\color{red}{\boldsymbol{-}}mgL(1-\cos\theta_1)=0, $$ then it is correct. You are computing the relation of maximum angle and maximal velocity, these events happen at different positions, as you also wrote above.