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How do we check the geometry of the phase space ? I mean in classical mechanics we use position and conjugate momenta as a space of all possible states of the particle. How do we know that this phase space is flat? In other words, is phase space of classical pendulum flat or curved like a cylinder?

Any reference concerning theory of dynamical systems for physicists and chaos would be useful.

JamalS
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WoofDoggy
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    Do you mean symplectic manifolds? http://en.wikipedia.org/wiki/Symplectic_manifold. I don't think curvature is defined for symplectic geometry . There is a preserved phase space area element, though, but I don't know if that can be used to define a geometry on the manifold. – CuriousOne Dec 30 '14 at 20:46
  • If the classical system you have in mind is Liouville integrable, i.e. there are as many constants of motion in involution as degrees of freedom (say $n$), then the topology is an $n$ dimensional Torus $T^n.$ And since the surface of a torus doesn't have any intrinsic curvature, it is equivalent to a $n$ dimensional flat surface. So the phase space of integrable systems like the simple pendulum is a flat symplectic manifold. You will find many useful posts here on SE by a quick search, on manifolds, topology, phase space, classical chaos. I'll paste a few related ones below. – Ellie Dec 30 '14 at 21:10
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    Related to your questions: http://physics.stackexchange.com/q/142603/, http://physics.stackexchange.com/q/141468/, http://physics.stackexchange.com/q/144615/, http://physics.stackexchange.com/q/4990/ – Ellie Dec 30 '14 at 21:10
  • @Phonon: Does the curvature in the non-integrable example tell me anything about the system? It seems that only the topological characteristics of the manifold makes a difference, but not the local curvature? What am I missing? – CuriousOne Dec 30 '14 at 21:23
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    @CuriousOne well yes, it tells you that it is chaotic, and the curvature, e.g. for a negative Riemannian curvature, neighboring paths diverge exponentially. Or maybe I mis-understood what you meant by "...anything about the system?"? – Ellie Dec 30 '14 at 21:29
  • @Phonon: That is exactly what I meant. So the local curvature is basically a measure for the local stability? – CuriousOne Dec 30 '14 at 21:35
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    @CuriousOne the local stability is determined by the sign of eigenvalues of the Jacobi matrix, a dynamical system generated by smooth ODEs can be unstable without having to be non-integrable. The curvature tells you about the integrability, in other words the predictability of the system and not stability per se. – Ellie Dec 30 '14 at 21:42
  • @Phonon: Integrability is a global feature, though, isn't it? That's why I am confused. Let me read up on it, I don't want to steal too much of your time with my confusion, since I am not even the OP. – CuriousOne Dec 30 '14 at 21:47
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    @CuriousOne Sure, no worries. Well do bear in mind that a dynamic non-linear system can have different regimes, chaotic and regular, e.g. the Lorenz system. These regimes can be visualized in a poincaré map, where regions with periodic orbits (stable or unstable), quasi periodic and isolated ones can be seen. Make sure to read up on the KAM theory, which roughly states that a non-integrable system which is close to an integrable system will have areas where the periodic orbits persist. This is why I'm a bit hesitant about globally defining features of a dynamical system. – Ellie Dec 30 '14 at 21:55
  • @Phonon: Thanks for the pointers! That gives me a couple of hours of homework. :-) – CuriousOne Dec 30 '14 at 22:21
  • from Wikipedia "While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behavior" - do You know any example of this phenomena? – WoofDoggy Dec 30 '14 at 22:26
  • @Nex_Friedrich This is already advanced stuff for a newcomer to chaos theory, try to go through the basic elements first (i.e. non-linearity, integrability, attractors). I don't know much about this theorem but it seems to be referring to mechanical many body problems, for which it is computationally impossible to solve the problem for the entire closed system, i.e. by setting and solving all the differential equations of motion (even for linear ones) with given initial conditions. So we also denote such systems as chaotic because their predictability is only possible via statistical mechanics – Ellie Dec 30 '14 at 22:41
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    As it stands, this post (v2) with several sub-questions ranging from symplectic geometry to reference request for chaos theory seems too broad. – Qmechanic Dec 30 '14 at 22:45

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The phase space of classical mechanics is a cotangent bundle to a manifold $\Gamma$, known as the "configuration space". The latter is locally described by the set of generalised coordinates, so once you know how to patch the whole configuration space with (smooth) charts you get an atlas and therefore a smooth structure on $\Gamma$. At this point you can then use differential geometry to study the properties of the configuration space, the phase space being just $T^*\Gamma$, which carries a natural symplectic structure. Sometimes $\Gamma$, and hence $T^*\Gamma$ has nice topological properties, like not being Hausdorff and stuff (though I can't really recall a specific example, perhaps a double pendulum or something).

Phoenix87
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  • Wait, don't you mean $\Gamma$ IS Hausdorff? A non-Hausdorff phase space would be problematical... – Alex Nelson Dec 30 '14 at 23:00
  • I'm referring to the whole phase space, i.e. $T^*\Gamma$. I believe it is possible to find mechanical systems with such a property – Phoenix87 Dec 30 '14 at 23:02
  • I'm just rather shocked to hear it for mechanical systems. I know in $2+1$-dimensional GR, for example, you can end up with a phase space that's non-Hausdorff...but I assumed it was always just field theories that had such peculiarities. (I may be mistaken, which is why I ask about such things) – Alex Nelson Dec 30 '14 at 23:09