In the Heisenberg picture, I can define the velocity Operator $\hat{V}$ as the operator which satisfies $\hat{V}(t) = \frac{\partial \hat{x}}{\partial t}(t)$ for all $t$. The Heisenberg equation then tells me that $ \hat{V}(t) = \frac{i}{\hbar}[\hat{H}(t), \hat{x}(t)]$. If I want to calculate the time evolution of the velocity operator, a straight forward approach yields:
$$\dot{\hat{V}} = \frac{i}{ \hbar}(\dot{\hat{H}}\hat{x}-\hat{x}\dot{\hat{H}} + \hat{H}\dot{\hat{x}} -\dot{\hat{x}}\hat{H} ) = \frac{i}{ \hbar}(\dot{\hat{H}}\hat{x}-\hat{x}\dot{\hat{H}} +[\hat{H}(t), \hat{V}(t)]) $$
What I find strange about that is that the time evolution of $V$ is no longer described by the Heisenberg-equation, but instead by some more complicated form, at least if $\dot{\hat{H}}$ doesn't commute with $\hat{x}$. I see problems arising, since I now can't in general make statements on the time evolution of functions $\hat{f}(\hat{x}, \hat{V})$, like for example the lagrangian $\hat{L}(\hat{x}, \hat{V}, t)$. Since the time evolution of $\hat{V}$ isn't given by the Heisenberg equation, the time evolution of for example $\hat{p}(t) = \frac{\partial L}{\partial t}(\hat{x}(t), \hat{v}(t), t)$ also isn't necessarily given anymore by the Heisenberg equation.
I know that this is a unusual approach to the topic, since I assume observables to be functions of $\hat{x}$ and $\hat{v}$, and not functions of $\hat{x}$ and $\hat{p}$, but maybe somebody can still tell me where the mistakes are that I do in this derivations, or if it is an additional assumption that $\dot{\hat{H}}$ and $\hat{x}$ do always commute. If I did a mistake, what should I do different to still approach the topic from the lagrangian point of view?
