Practical use for negative $dt.$

Question

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would be negative is in theoretical time travel, or in the extrapolation of what might have happened prior to a known event. SO here's my question:

When is it practical, or even possible, to utilize a negative value for $dt?$

Answer 1

I think this is related to classical physics, yes? It seems that you have 3 degrees of freedom or coordinates and basis. The displacement vectors or velocity or what ever contravariant vector you have exists independent of the coordinates and can change it's inherent position. This change is most often seen as continuous and t is nothing more than a parameterization the objects path inside 3 dimensions. So $-t$ makes sense classically. Very generally the object may exist in $\Bbb R^3$ so considering time independent relations (like gradient which is only concerned with the spatial quantities) you can describe some situation sometimes with only respect to the 3 spatial cooridnates.

You can consider interaction $f:\Bbb R^3 \to \Bbb R^3$ the intuitive motion of the some body or transformation of it's position would typically be continous and is clearly only a function of it's coordinates current location.

But you can consider $g:\Bbb R \to \Bbb R^3$ and then consider $f \circ g :\Bbb R \to \Bbb R^3$. Also it does make sense in classical mechanics to consider $-\frac{d}{dt}x$ and you do it all the time in classical mechanics. I think the notions of negative time or time differential only start to break down in systems were thermodynamics effects are taken into account due to unrecoverable losses of energy which make it extremely difficult to predict the reverse reactions. But mathematically in typical calculus I think it is certainly well defined and meaningful, and in classical mechanics as well.

Answer 2

I think it depends on how you organize your problem.
Suppose your problem takes place in a time interval $a \le t \le b$. Some things you know are $f'(t)$ and $f(b)$. Then to find $f(t)$, you may think of integrating backward from $b$. And maybe some ways to write that would involve thinking of negative $dt$.