I have a simple question regarding an example presented by Leonard Susskind and George Hrabovsky in their book on Classical Mechanics The Theoretical Minimum. In page 151, they state:

> 
> "*If there is no explicit time dependence in the Lagrangian, then the energy $H$ is conserved. If, however, the Lagrangian is explicitly time-dependent, then the Hamiltonian is not conserved.*"
> 
> 
> 

An example is given by the authors: suppose that a charged particle is moving between the plates of a capacitor with potential difference $\epsilon x$. If the field $\epsilon$ is constant, the Lagrangian is written as
$$L= \frac{1}2m\dot x^2 + \epsilon x.$$
In this case, the energy is conserved. If the field $\epsilon$ is not constant (i.e. the capacitor is charging), the Lagrangian has an explicit time dependence and it is written as
$$L= \frac{1}2m\dot x^2 + \epsilon(t) x.$$

My question: why the first Lagrangian doesn't have an explicit time dependence? Don't we have an explicit time dependence through $\dot x$? Even if $\dot x$ is constant in this case, isn't $\dot x$ generally explicitly time-dependent?