lagrangian formalism - Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be written as \begin{equation} \delta L = L\left( q(t),\frac{dq(t)}{dt},t\right) - L\left( q(t+ \epsilon),\frac{dq(t+ \epsilon)}{dt},t+\epsilon \right) = 0. \end{equation} Using Taylor series, keeping only first order terms this gives \begin{equation}\rightarrow \delta L =- \frac{\partial L }{\partial q} \frac{\partial q}{\partial t} \epsilon- \frac{\partial L }{\partial \dot{q}} \frac{\partial \dot{q}}{\partial t} \epsilon - \frac{\partial L }{\partial t} \epsilon = 0. \end{equation} Using the Euler-Lagrange equation and assuming that the Lagrangian does not depend explicitly on time we get \begin{equation}\rightarrow \delta L =- \frac{d}{d t} \left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} \right) \frac{\partial q}{\partial t} \epsilon- \frac{\partial L }{\partial \dot{q}} \frac{\partial \dot{q}}{\partial t} \epsilon =0. \end{equation} Which we can write as \begin{equation}\rightarrow \delta L = - \frac{d}{d t} \left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} \frac{\partial q}{\partial t} \right) \epsilon = - \frac{d}{d t} \left(p \frac{\partial q}{\partial t} \right) \epsilon = 0. \end{equation} But unfortunatly this is not the Hamiltonian. This computation should yield \begin{equation} \rightarrow \frac{d}{dt} \left( p \dot{q} - L \right) = 0. \end{equation} But I can't find no reason why and how the the extra $-L$ should emerge. I can see that this term can be written at the place where it is written because we have $\delta L = - \frac{d L}{dt } \epsilon$ and therefore \begin{equation} \rightarrow \delta L = - \frac{d}{d t} \left(\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} \frac{\partial q}{\partial t} \right) \epsilon = - \frac{d L}{dt } \epsilon. \end{equation} And then the desired equation would only say $0-0=0$. Any idea where i did a mistake would be much appreciated.