# Determining the geometry of the phase space of a system [closed]

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.

## closed as too broad by Danu, Kyle Kanos, ACuriousMind♦, David Hammen, JamalSJan 1 '15 at 19:44

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Do you mean symplectic manifolds? 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
• – Phonon Dec 30 '14 at 21:10
• @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?"? – Phonon Dec 30 '14 at 21:29
• @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. – Phonon Dec 30 '14 at 21:42
• 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

## 1 Answer

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).

• 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