About the "ambiguity" of explicit and implicit function of a variable Since i started to study calculus, i have been strugled by the same problem many times, and always i got the same conclusion that i didn't understand pretty well the true meaning of it:
Let's say, for example. The Hamiltonian, IF it is not a explicit function of time, it is constant in time. So let's suppose a Hamiltonian here$$H = m \dot x^2/2 + V(x)$$ The hamiltonian evidently is not explicitly a function of time. BUT, here is the main problem, suppose H is an implicit function of time, that is, x is a function. x=x(t), just for the sake of the example, let's say $$x = t^n f(...)$$, now, if we want, couldn't we just substitute it in the hamiltonian? $$H =V(t,f) + m(nt^{n-1}f(...))^2/2$$
As you can see, putting explicitly the function x in the hamiltonian, it become explicitly dependent of t.
So, isn't this ambiguous? I mean, the first hamiltonian is the same as the second, but one is explicitly function of time, and the other not, so one is constant and the other is not. But, shouldn't both describe the same physical situation?
I think that a possible solution to my problem would be something like the terms involving t in the second Hamiltonian just cancelling each other, but i can't prove that. Or maybe the fact that V need to satisfy equations like "mx'' = - \grad V" restric the solutions of x in such way that all the t's cancel.
Could someone help me?
 A: You're confusing the difference between a function, and another function obtained from the first via composition (and as is common in physics, you're overloading the symbol $x$). Just so we're absolutely clear, a function consists of 3 pieces of data $(f,A,B)$, typically written $f:A\to B$, where $A$ is called the domain, $B$ is called the target space/codomain and $f$ is the "rule" which tells us how elements of $A$ are mapped to elements of $B$ (I'm not going to go into the set-theory rabbit hole here). If any of these data changes, we say we have a different function. Even if the "rule" is the same, but the domain is different, we say we have a different function.
Suppose now we have two functions $f:A\to B$ and $g:B\to C$, then we can form their composition $g\circ f:A\to C$, namely by mapping each $a\in A$ to $g(f(a))\in C$. Now, unless, $A=B$ and $f=\text{id}_A$ is the identity function, the two functions $g$ and $g\circ f$ are COMPLETELY DIFFERENT THINGS. Sure, the composite $g\circ f$ may have been obtained from $g$, but that does not mean it is equal to $g$. Otherwise, it's kind of like saying just because the earth and moon are round, they are both the same thing... just utter nonsense.

Now, I'll avoid the term Hamiltonian, because that term is reserved for when we think of a "function of position and momenta", whereas you've written things as "functions of position and velocities". For that reason, I'll use the term "energy" instead. Let me write out explicitly all the functions which are present (assuming we're modelling $n$-dimensional motion):

*

*First we have a function $V:\Bbb{R}^n\to\Bbb{R}$, where we physically regard this as saying for each 'position' $x\in\Bbb{R}^n$, $V(x)$ is the potential energy at $x$.

*Next, we have a function $E:\Bbb{R}^n\times\Bbb{R}^n\to\Bbb{R}$ defined as $E(x,v):=\frac{m\|v\|^2}{2}+V(x)$. Physically, we might regard this as giving us the mechanical energy of a particle at position $x$ moving with velocity $v$, in the presence of the potential function $V$.

*Lastly, suppose we have a curve $\gamma:\Bbb{R}\to\Bbb{R}^n$. Physically, we may like to think of this as saying for each $t\in\Bbb{R}$, $\gamma(t)$ is the position of a particle at time $t$. Typically, physics texts use the letter $x$ in place of $\gamma$; I am against such a notation for beginners, because it completely overloads the symbol $x$ and has different meanings in different equations, which ultimately just causes all sorts of (avoidable) confusion.

From this given data, we can form a composition $\mathcal{E}_{\gamma}:\Bbb{R}\to\Bbb{R}$, given by $\mathcal{E}_{\gamma}=E\circ (\gamma,\gamma')$, or more explicitly, for each $t\in\Bbb{R}$,
\begin{align}
\mathcal{E}_{\gamma}(t)&:= E(\gamma(t),\gamma'(t)):=\frac{m\|\gamma'(t)\|^2}{2}+V(\gamma(t))
\end{align}
What you've done in your post is that you've constantly written everything as $H$ (which like I mentioned above really should be an $E$... but this is just a small issue) without writing out the arguments of the function, and as a result, you're completely mixing up different functions
Now, of course, $\mathcal{E}_{\gamma}$ and $E$ are completely different functions: $\mathcal{E}_{\gamma}$ has domain $\Bbb{R}$ while $E$ has domain $\Bbb{R}^n\times\Bbb{R}^n$. In physics, we say $\mathcal{E}_{\gamma}$ "is a function of time", whereas "$E$ is a function of position and velocity". In this context, saying $E$ depends on "implicitly" on time is just a confusing statement.

So, isn't this ambiguous? I mean, the first hamiltonian is the same as the second, but one is explicitly function of time, and the other not

Now, when one says something like "potential energy depends explicitly on time", what one means is that we have a function $V_1:\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$. The interpretation of this is that for each $x\in\Bbb{R}^n,t\in\Bbb{R}$, $V_1(x,t)$ is the potential energy at position $x$ and time $t$. Correspondingly, one may introduce a function $E_1:\Bbb{R}^n\times\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ defined as
\begin{align}
E_1(x,v,t)&:=\frac{m\|v\|^2}{2}+V_1(x,t).
\end{align}

*

*So, just to compare once again, $V:\Bbb{R}^n\to\Bbb{R}$ and $V_1:\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ are completely different functions. We usually say "$V_1$ depends explicitly on time" because its domain allows for the extra factor of $\Bbb{R}$.


*Likewise, $E:\Bbb{R}^n\times\Bbb{R}^n\to\Bbb{R}$ and $E_1:\Bbb{R}^n\times\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ are completely different functions. We say $E_1$ "depends explicitly on time" because its domain has the extra $\Bbb{R}$ factor.


*Given a curve $\gamma:\Bbb{R}\to\Bbb{R}$ as above, we can introduce a new function $\mathcal{E}_{1,\gamma}:\Bbb{R}\to\Bbb{R}$ defined for each $t\in\Bbb{R}$ as $\mathcal{E}_{1,\gamma}(t):= E_1(\gamma(t),\gamma'(t),t)$. Here, $\mathcal{E}_{\gamma}$ and $\mathcal{E}_{1,\gamma}$ both have the same domain and target space of $\Bbb{R}$, but they are different functions because the "rule" is obviously different.
Just to repeat once again, you should not mix up different functions (especially if they're obtained by composition), such as $E$ vs $\mathcal{E}_{\gamma}$, or $E_1$ vs $\mathcal{E}_{1,\gamma}$. Also, depending explicitly on time simply means that the domain of the function has "an extra factor of $\Bbb{R}$".
