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I'm reading Harbovsky and Susskind's The Theoretical Minimum Vol I.

In Lecture 3: Dynamics, under Aristotle's Law of Motion is mentioned Aristotle's (fallacious) law $F(t) = m {dx\over dt}$. Later, time is reversed by "changing $t$ to $-t$" and the result is mentioned as $F(-t) = -m{dx\over dt}$. This made absolutely no sense to me. So I tried to formulate this in precise mathematical terms. The following is my result.

Let $\tau: \mathbb{R\to R}$ be a function defined by $\tau (t):=-t$ and let $\dot{f}$ denote the derivative of a differentiable function $f$. Let $F$ and $x$ be differentiable functions satisfying $F(t) = m\dot x(t)$ for some constant $m$. Then $F(-t) =-m\dot{(x\circ\tau)}(t)$.

Question: Am I correct?

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  • $\begingroup$ I found the original statement by Susskind much easier to understand. What is wrong with it? He simply replaces $t$ by $-t$ in his equation. $\endgroup$
    – mike stone
    Commented Jun 22, 2020 at 13:35
  • $\begingroup$ @mikestone I don’t like abusing notation. (And that’s a curse!) $\endgroup$
    – Atom
    Commented Jun 22, 2020 at 13:37
  • $\begingroup$ How is replacing $t$ by $-t$ an abuse of notation? That the deriviatve of $f(-t)$ wrt $t$ is minus the derivative of $f(t)$ wrt $t$ s just the chain rule of elementary calculus. $\endgroup$
    – mike stone
    Commented Jun 22, 2020 at 13:40
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    $\begingroup$ First, one doesn't talk of derivative of $f(t)$. It is derivative of $f$. Second, (in sloppy language) the derivative of $f(-t)$ wrt $t$ evaluated at $t$ is minus derivative of $f(t)$ wrt $t$ evaluated at $-t$. You can see how bad language can easily confuse. $\endgroup$
    – Atom
    Commented Jun 22, 2020 at 13:47

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