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For equilibrium systems, we expect the Liouville theorem to hold. This theorem states that the density function of the states of the system is a constant of motion, which in turn can be translated into the phase volume consistent with the system's energy being constant in time. My question is rather related to the choice of measure, namely the phase space volume $d\mathbf{\Gamma}=d\mathbf{q}d\mathbf{p}.$ Denoting the measure by $\mu$ and the time transformation by $T,$ Liouville's theorem can be expressed as: $$ \mu(T V_0) = \mu(V_0) $$ where $V_0$ is the initial ($t=0$) volume of our system in phase space.

  • Is the choice of "volume" as the Lebesgue measure of choice arises mainly from physical considerations?
  • I understand that it is a valid measure as it (at least intuitively) fulfills the usual axioms of a measure function (monotonicity, countable additivity etc).
  • Finally, a more specific question: in the context of dynamical systems and discussions of accessibility of different parts of phase space, it is often said that almost all points in the space are accessible, except for a set of measure zero initial conditions (points). In such contexts, both mathematically and physically, what does it mean that a certain set of phase space points have measure zero?
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  • $\begingroup$ I'm not really sure what your questions are: Liouville's theorem says that the phase space volume is constant. If you want to express this with a measure, you have to choose the Lebesgue measure, because the Lebesgue measure measures volume rather by its definition. And a set of measure zero...is a set whose Lebesgue measure is zero, i.e. which has no volume. $\endgroup$ – ACuriousMind Dec 3 '15 at 14:07
  • $\begingroup$ @ACuriousMind Thanks for your comment. About the measure zero bit, what does it mean physically though? i.e. when initial conditions have 0 measure in phase space. Are they still valid initial conditions? $\endgroup$ – user929304 Dec 3 '15 at 14:10
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[Disclaimer: I give my answer, however I am not an expert so feel free to comment/edit if I have made some mistake ;-) ]

Mathematically, you can think of evolution in classical mechanics as a (symplecto)morphism on the cotangent bundle $T^*M$ of some smooth $n$-dimensional manifold $M$.

The cotangent bundle $T^* M$ is a symplectic manifold, and thus carries a natural volume form $\omega$, that is the $2n$-th exterior power of the symplectic form. This volume form $\omega$ induces naturally a measure $\mu_\omega$ defining the measure of a Borel set $B\in \mathscr{B}$ on $T^*M$ as $$\mu_\omega(B)=\int_B \omega\; .$$ Apart from the mathematical technicalities, the idea is the following: what in physics is called phase space is a particular geometrical object (the cotangent bundle) endowed with a "natural" measure. In the simplest case, where the coordinate space $M=\mathbb{R}^n$ and the phase space $T^*M = \mathbb{R}^{2n}$, then the natural (symplectic) measure $\mu_\omega$ is exactly the Lebesgue measure.

The dynamics is described by a symplectomorphism $\phi$ (i.e. preserves the symplectic form) in the phase space: $\phi:\mathbb{R}\to T^* M$ (also called the Hamiltonian flow). This map preserves also the symplectic measure: $$(\forall B\in\mathscr{K})\; \phi_\#\mu_\omega(B)=\mu_\omega(B)\; ,$$ where $\phi_\#\mu_\omega$ is the push forward of the measure by the flow $\phi$ and $\mathscr{K}$ is the set of compact Borel subsets of $T^*M$ (this is essentially a restatement of the relation you wrote).

This is, however, a consequence of the special feature "the dynamics preserves the symplectic form", and is therefore not true for general measures. In my opinion, this is physically relevant, for the evolution being a symplectomorphism is closely related to the Hamilton-Jacobi form of the equations of motion (and in turn to the least action principle).

Concerning the last point, the idea is that in "chaotic" dynamical systems closed (periodic) phase-space-trajectories are possible but "unstable", in the sense that are isolated points on the phase space. That means, physically, that a slight perturbation of the initial conditions (either position or momentum) of a periodic motion results in a non periodic motion that never returns to the initial condition (and in suitable situations, densely covers all the phase space or a region of the phase space). Of course each point of the phase space is still an admissible initial condition, but only a set of measure zero of them gives rise to periodic solutions, while the others to non periodic ones.

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    $\begingroup$ Thanks for your answer. I have to admit that from a mathematical point of view this answer goes way beyond me, although I am sure it is correct and don't doubt its rigor. It would be very helpful if you could add a tad more friendly explanations, e.g. regarding the volume form $\omega$ "the 2n-th exterior power of the symplectic form...induces a measure $\mu_\omega$" I really did not get this part, and I feel it is very important. Finally, do you have some comments for the last question? Many thanks $\endgroup$ – user929304 Dec 3 '15 at 14:54
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    $\begingroup$ @user929304 I have edited my answer a little bit ;-) $\endgroup$ – yuggib Dec 3 '15 at 16:10
  • $\begingroup$ Thanks a lot, the edit was of great help, specially the last paragraph. One question though: so in the more usual sense I could write $\int_B \omega$ as $\int_B dx_1\cdots dp_{2n}$ (assuming n degrees of freedom), is this correct? finally, now I understand that: periodic orbits are made of isolated points in phase space, which in turn implies that all sets of initial conditions that lead to periodic orbits must be of measure zero, since isolated points have volume 0. Just to be clear, what exactly qualifies a point in phase space as isolated? $\endgroup$ – user929304 Dec 4 '15 at 11:26
  • $\begingroup$ Yes, the volume form for a "flat" phase space is the usual $dx_1\dotsm dx_ndp_1\dotsm dp_n$. In this context, a point with a periodic orbit is "isolated" if it has a neighbourhood of points (an open ball that contains it) that do not yield periodic orbits. $\endgroup$ – yuggib Dec 4 '15 at 11:59

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