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On the Physics.SE I haven't found a simple definition of the invariance for time-reversal. Is it possible to have a simple explanation (also using differential equations) of the invariance for time-reversal?

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    $\begingroup$ have you seen en.wikipedia.org/wiki/T-symmetry#Invariance $\endgroup$ – Adrian Howard Aug 8 '19 at 15:16
  • $\begingroup$ @AdrianHoward In all honesty, no. But I have seen your link but for me it is all complicated. I asked for something very simple. $\endgroup$ – Sebastiano Aug 8 '19 at 15:20
  • $\begingroup$ It simply means that an equations of motion are the same if you replace t by -t $\endgroup$ – Wolphram jonny Aug 8 '19 at 15:56
  • $\begingroup$ @Wolphramjonny If $t\geq 0$, what is $-t$? If the law is $md^2\mathbf {r}/dt^2=\mathbf F$ what happen replacing $t$ by $-t$ if it has a significance? $\endgroup$ – Sebastiano Aug 8 '19 at 16:03
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    $\begingroup$ it is better to call it $-t_p$ instead of just -t. $t_p$ grows into the past. the equation above will stay the same, with $t_p$ instead of t. which means that the system evolves in the same way into the past than into the future. It means that if I show you a movie of two planets orbiting you cannot tell if the movie is running forwards or backwards, there is nothing in the motion that lets you tell. $\endgroup$ – Wolphram jonny Aug 8 '19 at 16:22
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I don't know if this is simple enough, but is how I view it:

Imagine any physical system (try a pendulum, without friction), on a given time (let's call this instant in time '$t$') the system is in a state you can know things about and make a note of the state, if you let advance time for a while ant take a note of the state of the system again, you'll notice it has changed, and label the new entry $t+1$.

A law of physics is said to be symmetric with respect of something if it has a parameter you can change in the functions that model the behavior and not change the result, for example if you label $t$ as $12$:$03\,$ and $t+1$ as $12$:$05\,$ but you discover your watch is two minutes fast and relabel them as $t' = 12$:$01$ and $t'+1 = 12$:$03$ nothing else in your experiment will change that's because the laws of physics are symmetric with respect of time translation.

Now if laws of physics are time reversible that would mean that they can be 'played backwards' so that you can relabel $t' = 12$:$05\,$ and $t'+1 = 12$:$03$ and the only change is that you get the notes on the states backwards, that would imply that you can always know what the past of the system by knowing its current state; we still don't know if this is true (but physicists would love that it was).

Time reversibility would imply that entropy is invariant, and the second law of thermodynamics says that entropy increases for any closed system, that's how they are related.

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