Can $\dot x = \sin x$ be equivalently written as $x'(t) = \sin(x(t))$?

Question

Somewhat related to a question I asked earlier. Basically, can we write $\dot x = \sin x$ as, equivalently, $x'(t) = \sin(x(t))$?

Do those two equations have the same meaning, or is there some difference? I'm trying to make sure I understand whether $x$ is a function of time, and, if so, how that affects the way we can write the differential equation.

And also, can $x$ be considered a variable here, or only a function? I find variables a bit more intuitive as, technically speaking, functions cannot "change"; their dependent variable (the "value" of the function?) changes, right?

Answer 1

My first impulse is to say, yes, $\dot x = \sin x$ means $x'(t) = \sin(x(t))$. There is a subtle difference, however, which you brought up when you mentioned that functions cannot "change."

Indeed, that comment is correct in reference to the modern set-theoretic definition of a function, which is a fixed mapping from a domain to a codomain. The application of the function may produce different results depending on which elements of the domain the function is applied to, but the function itself is simply what it is.

The $\dot x$ notation, on the other hand, is a relic of the 17th century, invented by Sir Isaac Newton.

In the 17th century I do not believe the notion of a function was formalized the way it is today. It would have been equally valid to say that $x$ is a variable and a function, provided that it is somehow supposed to vary in a certain way as some other variable (such as $t$) varies. And we still use that language even today in applications where the definition of a function as a mapping is not so prominent.

So if I wrote something like $$ x = x(t), $$ you might parse this as saying the $x$ on the left of the equation is a variable that relates to the variable $t$ on the right side of the equation according to the mapping determined by the function named $x$ on the right side of the equation. And now, confusingly, I have used $x$ to name two different things, a variable and a function, and really the only way you can hope to distinguish them is that the function name is followed by something in parentheses, or if I happen to point out in words when I mean $x$ to be a variable and when a function.

But people do this sort of thing quite often.

If it happens that the function written $x(t)$ is an invertible function, then the inverse function might be written $t(x)$, so \begin{align} x &= x(t), \\ t &= t(x), \end{align} which signifies there is a function that takes $x\mapsto t$ and the inverse of that function takes $t\mapsto x.$

So now we still have just two symbols but there are four objects. Somehow, in practice, people are able to disambiguate the objects that share names.

Now, I happen to share your affinity for variables, partly because of situations where the functions involved are invertible. Here's a real-life example: suppose I want to predict how an airplane will fly as it descends from a high altitude to a low altitude. While it descends, time passes and it travels some distance along the ground. A typical way to calculate the descent would be to make functions that map the elapsed time to altitude and distance traveled along the ground. But there's nothing to say I can't make functions that give elapsed time and altitude as functions of distance along the ground, or time and distance as functions of altitude. And in fact I think time and distance as functions of altitude is useful in some cases. And by thinking of all three things as variables that can be related by functions, not as a variable and two functions, it is easier to change my thinking about how I want to calculate them.

Answer 2

There isn't a whole lot of differences between variables and functions, until you need to start proving complex things. As long as you assume some relationship between $x$ and $t$, it doesn't matter too much if $x$ is $x(t)$ or $t$ is $t(x)$ or if they are both functions of some unknown variable like $q$, so that $x$ is actually $x(q)$ and $t$ is actually $t(q)$. The fact that they are functions instead of variables can come into play when doing partial derivatives or other things where specific dependency relationships matter. For basic differential equations, as long as they are related in a systematic matter is what is important.

Basically, $\dot x$ means $\frac{dx}{dt}$. It is often assumed in calculus that the variable in the differential on the bottom is the independent variable, which is why people will write $x(t)$. If $x$ is dependent and $t$ is independent and there is a strict relationship that doesn't depend on other variables, then that's basically the definition of $x$ being a function of $t$.

However, classical calculus from the Liebniz perspective was more geometric, in that it didn't presuppose which variable was dependent or independent. You treated equations as relations, not functions. Personally, I find this to be a much more powerful perspective, though sometimes things in a direct function relationship are easier to demonstrate.