The semimajor axis a of the elliptical Kepler orbit and the eccentricity e are functions o...

The semimajor axis a of the elliptical Kepler orbit and the eccentricity e are functions of first integrals of the motion, and therefore of the canonical variables. Similarly, the mean anomaly

is a function of r, θ, and the conjugate momenta. Here T is the time of periapsis passage and is a constant of the motion. Evaluate the Poisson brackets that can be formed of a, e, ϕ, ω, and T. There are in fact only nine nonvanishing distinct Poisson brackets out of these quantities.