# Phase space of double pendulum

Let $$M$$ be the phase space of a double pendulum system. What is known about the manifold $$M$$? Do we know it's homotopy typed? Do we know its (co)homology? Are there any references tackling these questions?

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• Generally it depends on your pendulum design, but for a typical textbook example, it would be the product of two circles. The cohomology of this is computed in almost any introductory book that describes what cohomology is. – Ryan Budney Apr 30 at 2:16
• @RyanBudney That would be the configuration space. The phase space I am assuming is a 4-manifold. The Hamiltonian equations would describe the motion of a particle starting from initial conditions, traveling around on this manifold. – LostTopologist Apr 30 at 2:23
• Phase space is just the tangent bundle of the underlying configuration space. That would be a product of two circles, and two copies of the real line. This has the same homotopy-type as the configuration space, so the same cohomology. – Ryan Budney Apr 30 at 2:24
• I think some of my confusion is stemming from this discussion. In which case I would be interested in the cotangent bundle on the torus. – LostTopologist Apr 30 at 2:30
• If all you care about is cohomology, you can throw the vector spaces away. i.e. effectively you want the configuration space. – Ryan Budney Apr 30 at 2:35