Why the Lagrangian doesn't have an explicit time dependence?

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?

• No, it’s implicitly time-dependent. Explicit means $t$ itself actually appearing in $L$. Dec 1, 2019 at 7:55
• Can't we write $\dot x(t)$, and then $L$ becomes explicitly time-dependent?
– Naps
Dec 1, 2019 at 7:58
• Dec 1, 2019 at 8:00
• You can write that, but it doesn’t make $t$ “explicit”. It’s just a function argument. Dec 1, 2019 at 8:00
• Thank you so much @G.Smith
– Naps
Dec 1, 2019 at 8:10

Explicit dependence would mean $$\partial_tL\ne0$$. Note that $$\partial_t\dot{x}=0$$.
In the Lagrangian formalism, the Lagrangian is a function $$L(x, \dot{x}, t)$$. The notation $$\frac{\partial L}{\partial t}$$ means nothing but "the partial derivative of L with respect to its third argument". The partial derivative notation always means differentiating a function with respect to a certain "argument slot", regardless of what you put in the slot afterwards. (Personally, I have been confused about this for several years before realising how to think about it.)