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I've edited my question to clear any possible confusing parts:

This an exercise from the book "The Theoretical minimum", I'm paraphrasing.

In classical mechanics, dynamical laws must be reversible and deterministic.

Consider the function N(n) where:

N is a point on a infinite line, and n symbolizes time.

Here are some possible dynamical laws of this system:

  1. N(n+1) = N(n) + 1

  2. N(n+1) = N(n) - 1

  3. N(n+1) = N(n) + 2

  4. N(n+1) = N(n)²

Determine which equations are allowable. (deterministic and reversible)


The solutions are on:

What I don't understand is why equation 4, N(n+1) = N(n)² is not allowable.

If N is 2 we get 4 and then 16. If we "reverse the arrows" we get 16, 4 and 2. If N is 1, it will never change.

How is this not deterministic and reversible?

Thanks in advance!

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perhaps you can rephrase what your question exactly is? Are you saying $N$ is a function of $n$, which itself is time (usually $t$)? What exactly does 'allowable' mean in this case? As it stands now it's a bit hard to understand what you're asking... – ALB Nov 28 '13 at 21:36
Hey sorry I thought I might be a bit confusing. I'm very new to physics. Yes N is a function of n, which is time. It isn't t because t is for continuous evolution of time, whereas here time evolves in a stroboscopic manner. (I'm paraphrasing the book). Allowable means that the system is both deterministic and reversible. My question is, how is the equation N(n+1) = N(n)² not allowable. Hope this helps. – user34914 Nov 28 '13 at 21:41
up vote 1 down vote accepted

Is it because there are two arrows into each state (which happen to not be shown in the solution)? Both $N(n)$ and $-N(n)$ go to $N(n+1)=N^2(n)$ under time advancement. Thus, under time reversal, you're not sure one to go back to.

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Actually, if you know that your current state is $N(n)=16$, there's no way for you to discern under that law whether $N(n-1)=4$ or $N(n-1)=-4$. There isn't a constraint stating that $N(k)\ge0$ in the problem statement.

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