Do physicists use a different way of dealing with Function notation? I've started studying physics concepts, in Mathematics we tend to define $f$ to be some function that we apply to a variable $x$, where as in Physics and Mechanics work I sometimes find that people often define $F$ to be the force and use notation like $F(t)$, and will perform an re-arrangement operation using a function for example the position $S(t)$ to determine what they title $F(s)$, a function which is equal to $F(t)$ as if $F$ is just a free variable and not simply denoting a function - where it perhaps should be $F(G(s))$ where $t=G(s)$ is this a mistake in my reasoning or a different way of doing things?
 A: I would not say that this is a particular quirk of physicists, rather it is something what happens when you have to talk about several similar functions that are "morally" the same but in the strict sense are different functions.
For example, when you encounter "the force" once expressed as $F(s)$ and once as $F(t)$, indeed these are two different functions in the strict sense, one from the space of spatial displacements to the space of force vectors and one from the space of temporal displacements to the space of force vectors. They are connected in that you have a (locally) invertible function $s(t)$ from the space of temporal displacements to the space of spatial displacements that describes a motion and so you can describe the force either by defining $F(s)$ and then "$F(t)$" is really $F(s(t))$ or by defining $F(t)$ and then "$F(s)$" is really $F(t(s))$, where you'll also note that we call the inverse of $s(t)$ just $t(s)$ and not $s^{-1}$ or something.
What we're physically trying to say here is that there's "a force" along "the motion". We don't really care whether we describe a point in the motion spatially or temporally, the two descriptions are equivalent via the invertible $s(t)$. The particulars of how the specific functions are defined in the strict sense are not always relevant to what we're doing, and so they are conveniently elided by this notation - but it requires being careful about how things "really" work when you have to do math like the chain rule for derivatives!
More generally, you will see this sort of elision also in other mathematical contexts where invertible functions (or more abstractly isomorphisms) are involved and the difference between a function and a function concatenated with that isomorphism becomes murky in the text: When you have $f$ and an invertible $g$, particularly when $g$ is "obvious" or "natural" in some sense, then the "difference" between talking about $f$ and $f\circ g$ or $f' := f\circ g$ and $g^{-1}\circ f'$ or whatever becomes pretty inconsequential as along as everyone is aware what's happening.
One example of this is when we talk about functions on manifolds and charts - we talk about a scalar function $f : M\to \mathbb{R}$ "on the manifold" and then someone will write down some expression e.g. $f(x_1,x_2) = x_2$ for it in a particular coordinate system but strictly speaking of course the function on the coordinate system is a function $\mathbb{R}^n \to \mathbb{R}$ and cannot be "the same function" as $f$ but after the first week of learning how charts and coordinates work no one will ever use a different symbol for these two "different" functions unless it is somehow specifically relevant to their point. Think about the function $f(x,y) = x^2 + y^2$ on $\mathbb{R}^2$ and how people might also write $f(r,\theta) = r^2$ for it in spherical coordinates - if you want, you can complain we used the same symbol for formally "different" functions, but that's just not how we actually think about functions.
A: You can decide for yourself how idiosyncratic the practices I summarize below are:

*

*$F=F(t)$ for all $t$ means there is a function $F_t$ from $t$ values to $F$ values such that, for all times $t$, $F=F_t(t)$.

*$F=F(s)$ for all $s$ means the same thing but for $s$, viz. a function $F_s$. Writing $s$ as a function $s_t$ of $t$, $F_t=F_s\circ s_t$. This equation identifies the functions $F_t,\,F_s$ in a sense, which motivates just writing $F$ all the time.

*Just for completeness: $F=F(t)$ with a specific $t$ in mind means that, for the function $F_t$ mentioned in the first bullet point, $F$ has value $F_t(t)$ at that time. Similarly with $F=F(s)$ or $s=s(t)$.

