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I'm watching Susskind's video lectures and he says in the first lecture on classical mechanics that for a physical law to be allowable in classical mechanics it must be reversible, in the sense that for any given state $S\in \mathcal{M}$ where $\mathcal{M}$ is the configuration space there should be only one state $S_0\in \mathcal{M}$ such that $S_0\mapsto S$ in the evolution of the system.

Now, why is this? Why do we really need this reversibility? I can't understand what are the reasons for us to wish it from a physical law. What are the consequences of not having it?

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The sense you describe is not reversibility as usually meant in mechanics. Reversibility in mechanics is meant in this way: keep all the positions $\mathbf r_i$, reverse all velocities $\mathbf v_i$ and calculate the evolution of positions $\mathbf r_i(t)$ according to the equations of motion. The equations of motion (physical laws of motion) are reversible if the coordinates retrace their past values. – Ján Lalinský Feb 5 '14 at 21:04
Damped motion described by the equation of motion $m\ddot x + m\gamma \dot x = 0$ evolves position $S_0$ to single position $S$, but is not reversible in mechanical sense, due to the term $m\gamma\dot x$. – Ján Lalinský Feb 5 '14 at 21:06

Pretty much the entire field of Classical Mechanics comes down to one thing: prediction.

Given the initial conditions of a system, and a set of mathematical laws that model reality, we want to be able to tell what state the system will be in after a given time.

The principal of reversibility that Susskind mentions is essential to Classical Mechanics because it ensures that every time have the same initial conditions, and same laws, we will predict the same outcome. In other words, our model needs to work "in reverse". If we start at the end, and reverse our laws, we need to end up at our system with the initial conditions. Otherwise, we could have a system with different initial conditions, using the same laws, that gives us the same exact final state. In the same way, if our laws didn't work in reverse, we could have a system with the same initial conditions that gives us different final states.

We couldn't predict anything using a model like that.

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It may also be worth adding that your answer is also sound for everything in quantum mechanics aside from measurement (the measurement problem of whether or not state "collapse" happens). Likewise for prediction of all statistics of measured quantities. And if some of the hunches about things like einselection prove right, then even measurement in QM becomes a particular kind of unitary state evolution, and we're back to something not too unlike what Laplace thought, though a great deal more complicated! I think this is important, particularly .... – WetSavannaAnimal aka Rod Vance Feb 6 '14 at 1:32
.... since Suskind has in mind quantum states just as well as classical ones. – WetSavannaAnimal aka Rod Vance Feb 6 '14 at 1:33

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