Why is phase space 2D for the timeless double pendulum in Rovelli?

[nrqed]

On page 109 he does the example of the "timeless double pendulum".
He says that the phase space is two dimensional. I think I am getting confused with all the spaces he defined

Why 2?? I would have expected 4 since there are two coordinates, a and b, and their conjugate momenta.

 

[sheaf]

On page 109 he does the example of the "timeless double pendulum".
He says that the phase space is two dimensional. I think I am getting confused with all the spaces he defined

Why 2?? I would have expected 4 since there are two coordinates, a and b, and their conjugate momenta.

It looks like his phase space "relativistic phase space" isn't simply the usual one we're familiar with (cotangent bundle of configuration space). Just a bit below the comment about dimension 2, he gives the evolution equation f(a,b,alpha,beta) and states that the phase space (space of ellipses in (a,b) space) is parametrised by alpha and beta hence 2 dimensional.