On mathematical level, what exactly is time in Newtonian mechanics?

Question

One easy answer would be that in practical purposes, it is a very special sort of parameter. This point comes up quite clearly when we do the derivation for the work-energy theorem:

$$ W = \int\mathbf{ F} \cdot \text d\mathbf s = \int m \left(\frac{\text dv}{\text dx}\cdot \frac{\text dx}{\text dt} \right)\, \text dx = \int m v\, \text dv =\frac12 mv^2 +C$$

When we go from result of first equality to the one of second equality, we are implicitly choosing a very specific parameterization which is of the actual time experienced ticking as the object moves through the path, but, on a mathematical level, no matter how fast we would have run through the force field (assuming force is indep of time), it would be that the work integral gives same answer.

So, this made me wonder, what properties does time have beyond just being a parameter when doing integrals in Newtonian mechanics?

Answer 1

Here is one way to address your question.

"Time is defined so that motion looks simple." - Misner, Thorne, and Wheeler in Gravitation, p.23.

Continue through to p. 26 where they say
"Good clocks make spacetime trajectories of free particles look straight".

Misner, Thorne, and Wheeler (Gravitation, p.26)

Look at a bad clock for a good view of how time is defined. Let $t$ be time on a "good" clock (time coordinate of a local inertial frame); it makes the tracks of free particles through the local region of spacetime look straight. Let $T(t)$ be the reading of the "bad" clock; it makes the world lines of free particles through the local region of spacetime look curved (Figure 1.9). The old value of the acceleration, translated into the new ("bad") time, becomes
$$0=\frac{d^2x}{dt^2}=\frac{d}{dt}\left(\frac{dT}{dt}\frac{dx}{dT}\right)=\frac{d^2T}{dt^2}\frac{dx}{dT}+\left(\frac{dT}{dt}\right)^2\frac{d^2x}{dT^2}$$ To explain the apparent accelerations of the particles, the user of the new time introduces a force that one knows to be fictitious: $$ F_x=m\frac{d^2x}{dT^2}=-m\frac{\left(\frac{dx}{dT}\right)\left(\frac{d^2T}{dt^2}\right)}{\left(\frac{dT}{dt}\right)^2} $$ It is clear from this example of a "bad" time that Newton thought of a "good" time when he set up the principle that "Time flows uniformly" ($d^2T/dt^2 = 0$). Time is defined to make motion look simple!

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UPDATE:

For a similar argument,
refer to p.415 in Trautman's "Comparison of Newtonian and Relativistic Theories of Space-Time"
in Perspectives in Geometry and Relativity, Essays in honor of V. Hlavaty, ed. by B. Hoffmann, 1966

http://trautman.fuw.edu.pl/publications/Papers-in-pdf/22.pdf

(item 95 from http://trautman.fuw.edu.pl/publications/scientific-articles.html )

Answer 2

In my opinion the idea of time is related to a fixed relation between some physical events. The numbers of full moons, or day/night cycles between the repetition of a sky configuration, which we call year for example.

The idea is regularity. The oscillation of the same pendulum can be compared with the period of a day with a good precision. But our pulse rate fails the test. Even the frequency of atomic clocks are based on the same principle.

It is this parameter that works in the Newtonian equations.