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.
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).
